3 /*****************************************************************************/
5 /* 888888888 ,o, / 888 */
6 /* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */
7 /* 888 888 888 88b 888 888 888 888 888 d888 88b */
8 /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */
9 /* 888 888 888 C888 888 888 888 / 888 q888 */
10 /* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */
13 /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */
20 /* Jonathan Richard Shewchuk */
21 /* School of Computer Science */
22 /* Carnegie Mellon University */
23 /* 5000 Forbes Avenue */
24 /* Pittsburgh, Pennsylvania 15213-3891 */
27 /* This program may be freely redistributed under the condition that the */
28 /* copyright notices (including this entire header and the copyright */
29 /* notice printed when the `-h' switch is selected) are not removed, and */
30 /* no compensation is received. Private, research, and institutional */
31 /* use is free. You may distribute modified versions of this code UNDER */
32 /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */
33 /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */
34 /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */
35 /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */
36 /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */
37 /* WITH THE AUTHOR. (If you are not directly supplying this code to a */
38 /* customer, and you are instead telling them how they can obtain it for */
39 /* free, then you are not required to make any arrangement with me.) */
41 /* Hypertext instructions for Triangle are available on the Web at */
43 /* http://www.cs.cmu.edu/~quake/triangle.html */
45 /* Some of the references listed below are marked [*]. These are available */
46 /* for downloading from the Web page */
48 /* http://www.cs.cmu.edu/~quake/triangle.research.html */
50 /* A paper discussing some aspects of Triangle is available. See Jonathan */
51 /* Richard Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator */
52 /* and Delaunay Triangulator," First Workshop on Applied Computational */
53 /* Geometry, ACM, May 1996. [*] */
55 /* Triangle was created as part of the Archimedes project in the School of */
56 /* Computer Science at Carnegie Mellon University. Archimedes is a */
57 /* system for compiling parallel finite element solvers. For further */
58 /* information, see Anja Feldmann, Omar Ghattas, John R. Gilbert, Gary L. */
59 /* Miller, David R. O'Hallaron, Eric J. Schwabe, Jonathan R. Shewchuk, */
60 /* and Shang-Hua Teng, "Automated Parallel Solution of Unstructured PDE */
61 /* Problems." To appear in Communications of the ACM, we hope. */
63 /* The quality mesh generation algorithm is due to Jim Ruppert, "A */
64 /* Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh */
65 /* Generation," Journal of Algorithms 18(3):548-585, May 1995. [*] */
67 /* My implementation of the divide-and-conquer and incremental Delaunay */
68 /* triangulation algorithms follows closely the presentation of Guibas */
69 /* and Stolfi, even though I use a triangle-based data structure instead */
70 /* of their quad-edge data structure. (In fact, I originally implemented */
71 /* Triangle using the quad-edge data structure, but switching to a */
72 /* triangle-based data structure sped Triangle by a factor of two.) The */
73 /* mesh manipulation primitives and the two aforementioned Delaunay */
74 /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */
75 /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */
76 /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */
77 /* 4(2):74-123, April 1985. */
79 /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */
80 /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */
81 /* Delaunay Triangulation," International Journal of Computer and */
82 /* Information Science 9(3):219-242, 1980. The idea to improve the */
83 /* divide-and-conquer algorithm by alternating between vertical and */
84 /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */
85 /* Conquer Algorithm for Constructing Delaunay Triangulations," */
86 /* Algorithmica 2(2):137-151, 1987. */
88 /* The incremental insertion algorithm was first proposed by C. L. Lawson, */
89 /* "Software for C1 Surface Interpolation," in Mathematical Software III, */
90 /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */
91 /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */
92 /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */
93 /* Preprocessing in Two- and Three-dimensional Delaunay Triangulations," */
94 /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */
95 /* ACM, May 1996. [*] If I were to randomize the order of point */
96 /* insertion (I currently don't bother), their result combined with the */
97 /* result of Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, */
98 /* "Randomized Incremental Construction of Delaunay and Voronoi */
99 /* Diagrams," Algorithmica 7(4):381-413, 1992, would yield an expected */
100 /* O(n^{4/3}) bound on running time. */
102 /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */
103 /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */
104 /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */
105 /* boundary of the triangulation are maintained in a splay tree for the */
106 /* purpose of point location. Splay trees are described by Daniel */
107 /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */
108 /* Trees," Journal of the ACM 32(3):652-686, July 1985. */
110 /* The algorithms for exact computation of the signs of determinants are */
111 /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */
112 /* Point Arithmetic and Fast Robust Geometric Predicates," Technical */
113 /* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */
114 /* University, Pittsburgh, Pennsylvania, May 1996. [*] (Submitted to */
115 /* Discrete & Computational Geometry.) An abbreviated version appears as */
116 /* Jonathan Richard Shewchuk, "Robust Adaptive Floating-Point Geometric */
117 /* Predicates," Proceedings of the Twelfth Annual Symposium on Computa- */
118 /* tional Geometry, ACM, May 1996. [*] Many of the ideas for my exact */
119 /* arithmetic routines originate with Douglas M. Priest, "Algorithms for */
120 /* Arbitrary Precision Floating Point Arithmetic," Tenth Symposium on */
121 /* Computer Arithmetic, 132-143, IEEE Computer Society Press, 1991. [*] */
122 /* Many of the ideas for the correct evaluation of the signs of */
123 /* determinants are taken from Steven Fortune and Christopher J. Van Wyk, */
124 /* "Efficient Exact Arithmetic for Computational Geometry," Proceedings */
125 /* of the Ninth Annual Symposium on Computational Geometry, ACM, */
126 /* pp. 163-172, May 1993, and from Steven Fortune, "Numerical Stability */
127 /* of Algorithms for 2D Delaunay Triangulations," International Journal */
128 /* of Computational Geometry & Applications 5(1-2):193-213, March-June */
131 /* For definitions of and results involving Delaunay triangulations, */
132 /* constrained and conforming versions thereof, and other aspects of */
133 /* triangular mesh generation, see the excellent survey by Marshall Bern */
134 /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */
135 /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */
136 /* editors, World Scientific, Singapore, pp. 23-90, 1992. */
138 /* The time for incrementally adding PSLG (planar straight line graph) */
139 /* segments to create a constrained Delaunay triangulation is probably */
140 /* O(n^2) per segment in the worst case and O(n) per edge in the common */
141 /* case, where n is the number of triangles that intersect the segment */
142 /* before it is inserted. This doesn't count point location, which can */
143 /* be much more expensive. (This note does not apply to conforming */
144 /* Delaunay triangulations, for which a different method is used to */
145 /* insert segments.) */
147 /* The time for adding segments to a conforming Delaunay triangulation is */
148 /* not clear, but does not depend upon n alone. In some cases, very */
149 /* small features (like a point lying next to a segment) can cause a */
150 /* single segment to be split an arbitrary number of times. Of course, */
151 /* floating-point precision is a practical barrier to how much this can */
154 /* The time for deleting a point from a Delaunay triangulation is O(n^2) in */
155 /* the worst case and O(n) in the common case, where n is the degree of */
156 /* the point being deleted. I could improve this to expected O(n) time */
157 /* by "inserting" the neighboring vertices in random order, but n is */
158 /* usually quite small, so it's not worth the bother. (The O(n) time */
159 /* for random insertion follows from L. Paul Chew, "Building Voronoi */
160 /* Diagrams for Convex Polygons in Linear Expected Time," Technical */
161 /* Report PCS-TR90-147, Department of Mathematics and Computer Science, */
162 /* Dartmouth College, 1990. */
164 /* Ruppert's Delaunay refinement algorithm typically generates triangles */
165 /* at a linear rate (constant time per triangle) after the initial */
166 /* triangulation is formed. There may be pathological cases where more */
167 /* time is required, but these never arise in practice. */
169 /* The segment intersection formulae are straightforward. If you want to */
170 /* see them derived, see Franklin Antonio. "Faster Line Segment */
171 /* Intersection." In Graphics Gems III (David Kirk, editor), pp. 199- */
172 /* 202. Academic Press, Boston, 1992. */
174 /* If you make any improvements to this code, please please please let me */
175 /* know, so that I may obtain the improvements. Even if you don't change */
176 /* the code, I'd still love to hear what it's being used for. */
178 /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */
179 /* whatsoever. This code is provided "as-is". Use at your own risk. */
181 /*****************************************************************************/
183 /* For single precision (which will save some memory and reduce paging), */
184 /* define the symbol SINGLE by using the -DSINGLE compiler switch or by */
185 /* writing "#define SINGLE" below. */
187 /* For double precision (which will allow you to refine meshes to a smaller */
188 /* edge length), leave SINGLE undefined. */
190 /* Double precision uses more memory, but improves the resolution of the */
191 /* meshes you can generate with Triangle. It also reduces the likelihood */
192 /* of a floating exception due to overflow. Finally, it is much faster */
193 /* than single precision on 64-bit architectures like the DEC Alpha. I */
194 /* recommend double precision unless you want to generate a mesh for which */
195 /* you do not have enough memory. */
204 #else /* not SINGLE */
207 #endif /* not SINGLE */
209 /* If yours is not a Unix system, define the NO_TIMER compiler switch to */
210 /* remove the Unix-specific timing code. */
215 /* To insert lots of self-checks for internal errors, define the SELF_CHECK */
216 /* symbol. This will slow down the program significantly. It is best to */
217 /* define the symbol using the -DSELF_CHECK compiler switch, but you could */
218 /* write "#define SELF_CHECK" below. If you are modifying this code, I */
219 /* recommend you turn self-checks on. */
221 /* #define SELF_CHECK */
223 /* To compile Triangle as a callable object library (triangle.o), define the */
224 /* TRILIBRARY symbol. Read the file triangle.h for details on how to call */
225 /* the procedure triangulate() that results. */
230 /* It is possible to generate a smaller version of Triangle using one or */
231 /* both of the following symbols. Define the REDUCED symbol to eliminate */
232 /* all features that are primarily of research interest; specifically, the */
233 /* -i, -F, -s, and -C switches. Define the CDT_ONLY symbol to eliminate */
234 /* all meshing algorithms above and beyond constrained Delaunay */
235 /* triangulation; specifically, the -r, -q, -a, -S, and -s switches. */
236 /* These reductions are most likely to be useful when generating an object */
237 /* library (triangle.o) by defining the TRILIBRARY symbol. */
244 /* On some machines, the exact arithmetic routines might be defeated by the */
245 /* use of internal extended precision floating-point registers. Sometimes */
246 /* this problem can be fixed by defining certain values to be volatile, */
247 /* thus forcing them to be stored to memory and rounded off. This isn't */
248 /* a great solution, though, as it slows Triangle down. */
250 /* To try this out, write "#define INEXACT volatile" below. Normally, */
251 /* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */
254 INEXACT /* Nothing */
255 /* #define INEXACT volatile */
257 /* Maximum number of characters in a file name (including the null). */
262 /* Maximum number of characters in a line read from a file (including the */
268 /* For efficiency, a variety of data structures are allocated in bulk. The */
269 /* following constants determine how many of each structure is allocated */
273 TRIPERBLOCK 4092 /* Number of triangles allocated at once. */
275 SHELLEPERBLOCK 508 /* Number of shell edges allocated at once. */
277 POINTPERBLOCK 4092 /* Number of points allocated at once. */
279 VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */
280 /* Number of encroached segments allocated at once. */
282 BADSEGMENTPERBLOCK 252
283 /* Number of skinny triangles allocated at once. */
286 /* Number of splay tree nodes allocated at once. */
288 SPLAYNODEPERBLOCK 508
290 /* The point marker DEADPOINT is an arbitrary number chosen large enough to */
291 /* (hopefully) not conflict with user boundary markers. Make sure that it */
292 /* is small enough to fit into your machine's integer size. */
295 DEADPOINT -1073741824
297 /* The next line is used to outsmart some very stupid compilers. If your */
298 /* compiler is smarter, feel free to replace the "int" with "void". */
299 /* Not that it matters. */
304 /* Two constants for algorithms based on random sampling. Both constants */
305 /* have been chosen empirically to optimize their respective algorithms. */
307 /* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */
308 /* how large a random sample of triangles to inspect. */
311 /* Used in Fortune's sweepline Delaunay algorithm to determine what fraction */
312 /* of boundary edges should be maintained in the splay tree for point */
313 /* location on the front. */
317 /* A number that speaks for itself, every kissable digit. */
320 PI 3.141592653589793238462643383279502884197169399375105820974944592308
325 SQUAREROOTTWO 1.4142135623730950488016887242096980785696718753769480732
327 /* And here's one for those of you who are intimidated by math. */
330 ONETHIRD 0.333333333333333333333333333333333333333333333333333333333333
342 #endif /* NO_TIMER */
347 #endif /* TRILIBRARY */
349 /* The following obscenity seems to be necessary to ensure that this program */
350 /* will port to Dec Alphas running OSF/1, because their stdio.h file commits */
351 /* the unpardonable sin of including stdlib.h. Hence, malloc(), free(), and */
352 /* exit() may or may not already be defined at this point. I declare these */
353 /* functions explicitly because some non-ANSI C compilers lack stdlib.h. */
357 extern void *malloc();
360 extern double strtod();
361 extern long strtol();
362 #endif /* _STDLIB_H_ */
364 /* A few forward declarations. */
371 #endif /* not TRILIBRARY */
373 /* Labels that signify whether a record consists primarily of pointers or of */
374 /* floating-point words. Used to make decisions about data alignment. */
377 POINTER, FLOATINGPOINT
380 /* Labels that signify the result of point location. The result of a */
381 /* search indicates that the point falls in the interior of a triangle, on */
382 /* an edge, on a vertex, or outside the mesh. */
384 enum locateresult { INTRIANGLE, ONEDGE, ONVERTEX, OUTSIDE };
386 /* Labels that signify the result of site insertion. The result indicates */
387 /* that the point was inserted with complete success, was inserted but */
388 /* encroaches on a segment, was not inserted because it lies on a segment, */
389 /* or was not inserted because another point occupies the same location. */
391 enum insertsiteresult {
392 SUCCESSFULPOINT, ENCROACHINGPOINT, VIOLATINGPOINT,
396 /* Labels that signify the result of direction finding. The result */
397 /* indicates that a segment connecting the two query points falls within */
398 /* the direction triangle, along the left edge of the direction triangle, */
399 /* or along the right edge of the direction triangle. */
401 enum finddirectionresult { WITHIN, LEFTCOLLINEAR, RIGHTCOLLINEAR };
403 /* Labels that signify the result of the circumcenter computation routine. */
404 /* The return value indicates which edge of the triangle is shortest. */
406 enum circumcenterresult { OPPOSITEORG, OPPOSITEDEST, OPPOSITEAPEX };
408 /*****************************************************************************/
410 /* The basic mesh data structures */
412 /* There are three: points, triangles, and shell edges (abbreviated */
413 /* `shelle'). These three data structures, linked by pointers, comprise */
414 /* the mesh. A point simply represents a point in space and its properties.*/
415 /* A triangle is a triangle. A shell edge is a special data structure used */
416 /* to represent impenetrable segments in the mesh (including the outer */
417 /* boundary, boundaries of holes, and internal boundaries separating two */
418 /* triangulated regions). Shell edges represent boundaries defined by the */
419 /* user that triangles may not lie across. */
421 /* A triangle consists of a list of three vertices, a list of three */
422 /* adjoining triangles, a list of three adjoining shell edges (when shell */
423 /* edges are used), an arbitrary number of optional user-defined floating- */
424 /* point attributes, and an optional area constraint. The latter is an */
425 /* upper bound on the permissible area of each triangle in a region, used */
426 /* for mesh refinement. */
428 /* For a triangle on a boundary of the mesh, some or all of the neighboring */
429 /* triangles may not be present. For a triangle in the interior of the */
430 /* mesh, often no neighboring shell edges are present. Such absent */
431 /* triangles and shell edges are never represented by NULL pointers; they */
432 /* are represented by two special records: `dummytri', the triangle that */
433 /* fills "outer space", and `dummysh', the omnipresent shell edge. */
434 /* `dummytri' and `dummysh' are used for several reasons; for instance, */
435 /* they can be dereferenced and their contents examined without causing the */
436 /* memory protection exception that would occur if NULL were dereferenced. */
438 /* However, it is important to understand that a triangle includes other */
439 /* information as well. The pointers to adjoining vertices, triangles, and */
440 /* shell edges are ordered in a way that indicates their geometric relation */
441 /* to each other. Furthermore, each of these pointers contains orientation */
442 /* information. Each pointer to an adjoining triangle indicates which face */
443 /* of that triangle is contacted. Similarly, each pointer to an adjoining */
444 /* shell edge indicates which side of that shell edge is contacted, and how */
445 /* the shell edge is oriented relative to the triangle. */
447 /* Shell edges are found abutting edges of triangles; either sandwiched */
448 /* between two triangles, or resting against one triangle on an exterior */
449 /* boundary or hole boundary. */
451 /* A shell edge consists of a list of two vertices, a list of two */
452 /* adjoining shell edges, and a list of two adjoining triangles. One of */
453 /* the two adjoining triangles may not be present (though there should */
454 /* always be one), and neighboring shell edges might not be present. */
455 /* Shell edges also store a user-defined integer "boundary marker". */
456 /* Typically, this integer is used to indicate what sort of boundary */
457 /* conditions are to be applied at that location in a finite element */
460 /* Like triangles, shell edges maintain information about the relative */
461 /* orientation of neighboring objects. */
463 /* Points are relatively simple. A point is a list of floating point */
464 /* numbers, starting with the x, and y coordinates, followed by an */
465 /* arbitrary number of optional user-defined floating-point attributes, */
466 /* followed by an integer boundary marker. During the segment insertion */
467 /* phase, there is also a pointer from each point to a triangle that may */
468 /* contain it. Each pointer is not always correct, but when one is, it */
469 /* speeds up segment insertion. These pointers are assigned values once */
470 /* at the beginning of the segment insertion phase, and are not used or */
471 /* updated at any other time. Edge swapping during segment insertion will */
472 /* render some of them incorrect. Hence, don't rely upon them for */
473 /* anything. For the most part, points do not have any information about */
474 /* what triangles or shell edges they are linked to. */
476 /*****************************************************************************/
478 /*****************************************************************************/
482 /* The oriented triangle (`triedge') and oriented shell edge (`edge') data */
483 /* structures defined below do not themselves store any part of the mesh. */
484 /* The mesh itself is made of `triangle's, `shelle's, and `point's. */
486 /* Oriented triangles and oriented shell edges will usually be referred to */
487 /* as "handles". A handle is essentially a pointer into the mesh; it */
488 /* allows you to "hold" one particular part of the mesh. Handles are used */
489 /* to specify the regions in which one is traversing and modifying the mesh.*/
490 /* A single `triangle' may be held by many handles, or none at all. (The */
491 /* latter case is not a memory leak, because the triangle is still */
492 /* connected to other triangles in the mesh.) */
494 /* A `triedge' is a handle that holds a triangle. It holds a specific side */
495 /* of the triangle. An `edge' is a handle that holds a shell edge. It */
496 /* holds either the left or right side of the edge. */
498 /* Navigation about the mesh is accomplished through a set of mesh */
499 /* manipulation primitives, further below. Many of these primitives take */
500 /* a handle and produce a new handle that holds the mesh near the first */
501 /* handle. Other primitives take two handles and glue the corresponding */
502 /* parts of the mesh together. The exact position of the handles is */
503 /* important. For instance, when two triangles are glued together by the */
504 /* bond() primitive, they are glued by the sides on which the handles lie. */
506 /* Because points have no information about which triangles they are */
507 /* attached to, I commonly represent a point by use of a handle whose */
508 /* origin is the point. A single handle can simultaneously represent a */
509 /* triangle, an edge, and a point. */
511 /*****************************************************************************/
513 /* The triangle data structure. Each triangle contains three pointers to */
514 /* adjoining triangles, plus three pointers to vertex points, plus three */
515 /* pointers to shell edges (defined below; these pointers are usually */
516 /* `dummysh'). It may or may not also contain user-defined attributes */
517 /* and/or a floating-point "area constraint". It may also contain extra */
518 /* pointers for nodes, when the user asks for high-order elements. */
519 /* Because the size and structure of a `triangle' is not decided until */
520 /* runtime, I haven't simply defined the type `triangle' to be a struct. */
522 typedef REAL **triangle; /* Really: typedef triangle *triangle */
524 /* An oriented triangle: includes a pointer to a triangle and orientation. */
525 /* The orientation denotes an edge of the triangle. Hence, there are */
526 /* three possible orientations. By convention, each edge is always */
527 /* directed to point counterclockwise about the corresponding triangle. */
531 int orient; /* Ranges from 0 to 2. */
534 /* The shell data structure. Each shell edge contains two pointers to */
535 /* adjoining shell edges, plus two pointers to vertex points, plus two */
536 /* pointers to adjoining triangles, plus one shell marker. */
538 typedef REAL **shelle; /* Really: typedef shelle *shelle */
540 /* An oriented shell edge: includes a pointer to a shell edge and an */
541 /* orientation. The orientation denotes a side of the edge. Hence, there */
542 /* are two possible orientations. By convention, the edge is always */
543 /* directed so that the "side" denoted is the right side of the edge. */
547 int shorient; /* Ranges from 0 to 1. */
550 /* The point data structure. Each point is actually an array of REALs. */
551 /* The number of REALs is unknown until runtime. An integer boundary */
552 /* marker, and sometimes a pointer to a triangle, is appended after the */
557 /* A queue used to store encroached segments. Each segment's vertices are */
558 /* stored so that one can check whether a segment is still the same. */
561 struct edge encsegment; /* An encroached segment. */
562 point segorg, segdest; /* The two vertices. */
563 struct badsegment *nextsegment; /* Pointer to next encroached segment. */
566 /* A queue used to store bad triangles. The key is the square of the cosine */
567 /* of the smallest angle of the triangle. Each triangle's vertices are */
568 /* stored so that one can check whether a triangle is still the same. */
571 struct triedge badfacetri; /* A bad triangle. */
572 REAL key; /* cos^2 of smallest (apical) angle. */
573 point faceorg, facedest, faceapex; /* The three vertices. */
574 struct badface *nextface; /* Pointer to next bad triangle. */
577 /* A node in a heap used to store events for the sweepline Delaunay */
578 /* algorithm. Nodes do not point directly to their parents or children in */
579 /* the heap. Instead, each node knows its position in the heap, and can */
580 /* look up its parent and children in a separate array. The `eventptr' */
581 /* points either to a `point' or to a triangle (in encoded format, so that */
582 /* an orientation is included). In the latter case, the origin of the */
583 /* oriented triangle is the apex of a "circle event" of the sweepline */
584 /* algorithm. To distinguish site events from circle events, all circle */
585 /* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */
588 REAL xkey, ykey; /* Coordinates of the event. */
589 VOID *eventptr; /* Can be a point or the location of a circle event. */
590 int heapposition; /* Marks this event's position in the heap. */
593 /* A node in the splay tree. Each node holds an oriented ghost triangle */
594 /* that represents a boundary edge of the growing triangulation. When a */
595 /* circle event covers two boundary edges with a triangle, so that they */
596 /* are no longer boundary edges, those edges are not immediately deleted */
597 /* from the tree; rather, they are lazily deleted when they are next */
598 /* encountered. (Since only a random sample of boundary edges are kept */
599 /* in the tree, lazy deletion is faster.) `keydest' is used to verify */
600 /* that a triangle is still the same as when it entered the splay tree; if */
601 /* it has been rotated (due to a circle event), it no longer represents a */
602 /* boundary edge and should be deleted. */
605 struct triedge keyedge; /* Lprev of an edge on the front. */
606 point keydest; /* Used to verify that splay node is still live. */
607 struct splaynode *lchild, *rchild; /* Children in splay tree. */
610 /* A type used to allocate memory. firstblock is the first block of items. */
611 /* nowblock is the block from which items are currently being allocated. */
612 /* nextitem points to the next slab of free memory for an item. */
613 /* deaditemstack is the head of a linked list (stack) of deallocated items */
614 /* that can be recycled. unallocateditems is the number of items that */
615 /* remain to be allocated from nowblock. */
617 /* Traversal is the process of walking through the entire list of items, and */
618 /* is separate from allocation. Note that a traversal will visit items on */
619 /* the "deaditemstack" stack as well as live items. pathblock points to */
620 /* the block currently being traversed. pathitem points to the next item */
621 /* to be traversed. pathitemsleft is the number of items that remain to */
622 /* be traversed in pathblock. */
624 /* itemwordtype is set to POINTER or FLOATINGPOINT, and is used to suggest */
625 /* what sort of word the record is primarily made up of. alignbytes */
626 /* determines how new records should be aligned in memory. itembytes and */
627 /* itemwords are the length of a record in bytes (after rounding up) and */
628 /* words. itemsperblock is the number of items allocated at once in a */
629 /* single block. items is the number of currently allocated items. */
630 /* maxitems is the maximum number of items that have been allocated at */
631 /* once; it is the current number of items plus the number of records kept */
632 /* on deaditemstack. */
635 VOID **firstblock, **nowblock;
640 enum wordtype itemwordtype;
642 int itembytes, itemwords;
644 long items, maxitems;
645 int unallocateditems;
649 /* Variables used to allocate memory for triangles, shell edges, points, */
650 /* viri (triangles being eaten), bad (encroached) segments, bad (skinny */
651 /* or too large) triangles, and splay tree nodes. */
653 static struct memorypool triangles;
654 static struct memorypool shelles;
655 static struct memorypool points;
656 static struct memorypool viri;
657 static struct memorypool badsegments;
658 static struct memorypool badtriangles;
659 static struct memorypool splaynodes;
661 /* Variables that maintain the bad triangle queues. The tails are pointers */
662 /* to the pointers that have to be filled in to enqueue an item. */
664 static struct badface *queuefront[64];
665 static struct badface **queuetail[64];
667 static REAL xmin, xmax, ymin, ymax; /* x and y bounds. */
668 static REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */
669 static int inpoints; /* Number of input points. */
670 static int inelements; /* Number of input triangles. */
671 static int insegments; /* Number of input segments. */
672 static int holes; /* Number of input holes. */
673 static int regions; /* Number of input regions. */
674 static long edges; /* Number of output edges. */
675 static int mesh_dim; /* Dimension (ought to be 2). */
676 static int nextras; /* Number of attributes per point. */
677 static int eextras; /* Number of attributes per triangle. */
678 static long hullsize; /* Number of edges of convex hull. */
679 static int triwords; /* Total words per triangle. */
680 static int shwords; /* Total words per shell edge. */
681 static int pointmarkindex; /* Index to find boundary marker of a point. */
682 static int point2triindex; /* Index to find a triangle adjacent to a point. */
683 static int highorderindex; /* Index to find extra nodes for high-order elements. */
684 static int elemattribindex; /* Index to find attributes of a triangle. */
685 static int areaboundindex; /* Index to find area bound of a triangle. */
686 static int checksegments; /* Are there segments in the triangulation yet? */
687 static int readnodefile; /* Has a .node file been read? */
688 static long samples; /* Number of random samples for point location. */
689 static unsigned long randomseed; /* Current random number seed. */
691 static REAL splitter; /* Used to split REAL factors for exact multiplication. */
692 static REAL epsilon; /* Floating-point machine epsilon. */
693 static REAL resulterrbound;
694 static REAL ccwerrboundA, ccwerrboundB, ccwerrboundC;
695 static REAL iccerrboundA, iccerrboundB, iccerrboundC;
697 static long incirclecount; /* Number of incircle tests performed. */
698 static long counterclockcount; /* Number of counterclockwise tests performed. */
699 static long hyperbolacount; /* Number of right-of-hyperbola tests performed. */
700 static long circumcentercount; /* Number of circumcenter calculations performed. */
701 static long circletopcount; /* Number of circle top calculations performed. */
703 /* Switches for the triangulator. */
704 /* poly: -p switch. refine: -r switch. */
705 /* quality: -q switch. */
706 /* minangle: minimum angle bound, specified after -q switch. */
707 /* goodangle: cosine squared of minangle. */
708 /* vararea: -a switch without number. */
709 /* fixedarea: -a switch with number. */
710 /* maxarea: maximum area bound, specified after -a switch. */
711 /* regionattrib: -A switch. convex: -c switch. */
712 /* firstnumber: inverse of -z switch. All items are numbered starting */
713 /* from firstnumber. */
714 /* edgesout: -e switch. voronoi: -v switch. */
715 /* neighbors: -n switch. geomview: -g switch. */
716 /* nobound: -B switch. nopolywritten: -P switch. */
717 /* nonodewritten: -N switch. noelewritten: -E switch. */
718 /* noiterationnum: -I switch. noholes: -O switch. */
719 /* noexact: -X switch. */
720 /* order: element order, specified after -o switch. */
721 /* nobisect: count of how often -Y switch is selected. */
722 /* steiner: maximum number of Steiner points, specified after -S switch. */
723 /* steinerleft: number of Steiner points not yet used. */
724 /* incremental: -i switch. sweepline: -F switch. */
725 /* dwyer: inverse of -l switch. */
726 /* splitseg: -s switch. */
727 /* docheck: -C switch. */
728 /* quiet: -Q switch. verbose: count of how often -V switch is selected. */
729 /* useshelles: -p, -r, -q, or -c switch; determines whether shell edges */
730 /* are used at all. */
732 /* Read the instructions to find out the meaning of these switches. */
734 static int poly, refine, quality, vararea, fixedarea, regionattrib, convex;
735 static int firstnumber;
736 static int edgesout, voronoi, neighbors, geomview;
737 static int nobound, nopolywritten, nonodewritten, noelewritten, noiterationnum;
738 static int noholes, noexact;
739 static int incremental, sweepline, dwyer;
742 static int quiet, verbose;
743 static int useshelles;
746 static int steiner, steinerleft;
747 static REAL minangle, goodangle;
750 /* Variables for file names. */
754 char innodefilename[FILENAMESIZE];
755 char inelefilename[FILENAMESIZE];
756 char inpolyfilename[FILENAMESIZE];
757 char areafilename[FILENAMESIZE];
758 char outnodefilename[FILENAMESIZE];
759 char outelefilename[FILENAMESIZE];
760 char outpolyfilename[FILENAMESIZE];
761 char edgefilename[FILENAMESIZE];
762 char vnodefilename[FILENAMESIZE];
763 char vedgefilename[FILENAMESIZE];
764 char neighborfilename[FILENAMESIZE];
765 char offfilename[FILENAMESIZE];
766 #endif /* not TRILIBRARY */
768 /* Triangular bounding box points. */
770 static point infpoint1, infpoint2, infpoint3;
772 /* Pointer to the `triangle' that occupies all of "outer space". */
774 static triangle *dummytri;
775 static triangle *dummytribase; /* Keep base address so we can free() it later. */
777 /* Pointer to the omnipresent shell edge. Referenced by any triangle or */
778 /* shell edge that isn't really connected to a shell edge at that */
781 static shelle *dummysh;
782 static shelle *dummyshbase; /* Keep base address so we can free() it later. */
784 /* Pointer to a recently visited triangle. Improves point location if */
785 /* proximate points are inserted sequentially. */
787 static struct triedge recenttri;
789 /*****************************************************************************/
791 /* Mesh manipulation primitives. Each triangle contains three pointers to */
792 /* other triangles, with orientations. Each pointer points not to the */
793 /* first byte of a triangle, but to one of the first three bytes of a */
794 /* triangle. It is necessary to extract both the triangle itself and the */
795 /* orientation. To save memory, I keep both pieces of information in one */
796 /* pointer. To make this possible, I assume that all triangles are aligned */
797 /* to four-byte boundaries. The `decode' routine below decodes a pointer, */
798 /* extracting an orientation (in the range 0 to 2) and a pointer to the */
799 /* beginning of a triangle. The `encode' routine compresses a pointer to a */
800 /* triangle and an orientation into a single pointer. My assumptions that */
801 /* triangles are four-byte-aligned and that the `unsigned long' type is */
802 /* long enough to hold a pointer are two of the few kludges in this program.*/
804 /* Shell edges are manipulated similarly. A pointer to a shell edge */
805 /* carries both an address and an orientation in the range 0 to 1. */
807 /* The other primitives take an oriented triangle or oriented shell edge, */
808 /* and return an oriented triangle or oriented shell edge or point; or they */
809 /* change the connections in the data structure. */
811 /*****************************************************************************/
813 /********* Mesh manipulation primitives begin here *********/
817 /* Fast lookup arrays to speed some of the mesh manipulation primitives. */
819 int plus1mod3[3] = { 1, 2, 0 };
820 int minus1mod3[3] = { 2, 0, 1 };
822 /********* Primitives for triangles *********/
826 /* decode() converts a pointer to an oriented triangle. The orientation is */
827 /* extracted from the two least significant bits of the pointer. */
830 decode( ptr, triedge ) \
831 ( triedge ).orient = (int) ((unsigned long) ( ptr ) & (unsigned long) 3l ); \
832 ( triedge ).tri = (triangle *) \
833 ((unsigned long) ( ptr ) ^ (unsigned long) ( triedge ).orient )
835 /* encode() compresses an oriented triangle into a single pointer. It */
836 /* relies on the assumption that all triangles are aligned to four-byte */
837 /* boundaries, so the two least significant bits of (triedge).tri are zero.*/
841 (triangle) ((unsigned long) ( triedge ).tri | (unsigned long) ( triedge ).orient )
843 /* The following edge manipulation primitives are all described by Guibas */
844 /* and Stolfi. However, they use an edge-based data structure, whereas I */
845 /* am using a triangle-based data structure. */
847 /* sym() finds the abutting triangle, on the same edge. Note that the */
848 /* edge direction is necessarily reversed, because triangle/edge handles */
849 /* are always directed counterclockwise around the triangle. */
852 sym( triedge1, triedge2 ) \
853 ptr = ( triedge1 ).tri[( triedge1 ).orient]; \
854 decode( ptr, triedge2 );
858 ptr = ( triedge ).tri[( triedge ).orient]; \
859 decode( ptr, triedge );
861 /* lnext() finds the next edge (counterclockwise) of a triangle. */
864 lnext( triedge1, triedge2 ) \
865 ( triedge2 ).tri = ( triedge1 ).tri; \
866 ( triedge2 ).orient = plus1mod3[( triedge1 ).orient]
869 lnextself( triedge ) \
870 ( triedge ).orient = plus1mod3[( triedge ).orient]
872 /* lprev() finds the previous edge (clockwise) of a triangle. */
875 lprev( triedge1, triedge2 ) \
876 ( triedge2 ).tri = ( triedge1 ).tri; \
877 ( triedge2 ).orient = minus1mod3[( triedge1 ).orient]
880 lprevself( triedge ) \
881 ( triedge ).orient = minus1mod3[( triedge ).orient]
883 /* onext() spins counterclockwise around a point; that is, it finds the next */
884 /* edge with the same origin in the counterclockwise direction. This edge */
885 /* will be part of a different triangle. */
888 onext( triedge1, triedge2 ) \
889 lprev( triedge1, triedge2 ); \
893 onextself( triedge ) \
894 lprevself( triedge ); \
897 /* oprev() spins clockwise around a point; that is, it finds the next edge */
898 /* with the same origin in the clockwise direction. This edge will be */
899 /* part of a different triangle. */
902 oprev( triedge1, triedge2 ) \
903 sym( triedge1, triedge2 ); \
904 lnextself( triedge2 );
907 oprevself( triedge ) \
908 symself( triedge ); \
909 lnextself( triedge );
911 /* dnext() spins counterclockwise around a point; that is, it finds the next */
912 /* edge with the same destination in the counterclockwise direction. This */
913 /* edge will be part of a different triangle. */
916 dnext( triedge1, triedge2 ) \
917 sym( triedge1, triedge2 ); \
918 lprevself( triedge2 );
921 dnextself( triedge ) \
922 symself( triedge ); \
923 lprevself( triedge );
925 /* dprev() spins clockwise around a point; that is, it finds the next edge */
926 /* with the same destination in the clockwise direction. This edge will */
927 /* be part of a different triangle. */
930 dprev( triedge1, triedge2 ) \
931 lnext( triedge1, triedge2 ); \
935 dprevself( triedge ) \
936 lnextself( triedge ); \
939 /* rnext() moves one edge counterclockwise about the adjacent triangle. */
940 /* (It's best understood by reading Guibas and Stolfi. It involves */
941 /* changing triangles twice.) */
944 rnext( triedge1, triedge2 ) \
945 sym( triedge1, triedge2 ); \
946 lnextself( triedge2 ); \
950 rnextself( triedge ) \
951 symself( triedge ); \
952 lnextself( triedge ); \
955 /* rnext() moves one edge clockwise about the adjacent triangle. */
956 /* (It's best understood by reading Guibas and Stolfi. It involves */
957 /* changing triangles twice.) */
960 rprev( triedge1, triedge2 ) \
961 sym( triedge1, triedge2 ); \
962 lprevself( triedge2 ); \
966 rprevself( triedge ) \
967 symself( triedge ); \
968 lprevself( triedge ); \
971 /* These primitives determine or set the origin, destination, or apex of a */
975 org( triedge, pointptr ) \
976 pointptr = (point) ( triedge ).tri[plus1mod3[( triedge ).orient] + 3]
979 dest( triedge, pointptr ) \
980 pointptr = (point) ( triedge ).tri[minus1mod3[( triedge ).orient] + 3]
983 apex( triedge, pointptr ) \
984 pointptr = (point) ( triedge ).tri[( triedge ).orient + 3]
987 setorg( triedge, pointptr ) \
988 ( triedge ).tri[plus1mod3[( triedge ).orient] + 3] = (triangle) pointptr
991 setdest( triedge, pointptr ) \
992 ( triedge ).tri[minus1mod3[( triedge ).orient] + 3] = (triangle) pointptr
995 setapex( triedge, pointptr ) \
996 ( triedge ).tri[( triedge ).orient + 3] = (triangle) pointptr
999 setvertices2null( triedge ) \
1000 ( triedge ).tri[3] = (triangle) NULL; \
1001 ( triedge ).tri[4] = (triangle) NULL; \
1002 ( triedge ).tri[5] = (triangle) NULL;
1004 /* Bond two triangles together. */
1007 bond( triedge1, triedge2 ) \
1008 ( triedge1 ).tri[( triedge1 ).orient] = encode( triedge2 ); \
1009 ( triedge2 ).tri[( triedge2 ).orient] = encode( triedge1 )
1011 /* Dissolve a bond (from one side). Note that the other triangle will still */
1012 /* think it's connected to this triangle. Usually, however, the other */
1013 /* triangle is being deleted entirely, or bonded to another triangle, so */
1014 /* it doesn't matter. */
1017 dissolve( triedge ) \
1018 ( triedge ).tri[( triedge ).orient] = (triangle) dummytri
1020 /* Copy a triangle/edge handle. */
1023 triedgecopy( triedge1, triedge2 ) \
1024 ( triedge2 ).tri = ( triedge1 ).tri; \
1025 ( triedge2 ).orient = ( triedge1 ).orient
1027 /* Test for equality of triangle/edge handles. */
1030 triedgeequal( triedge1, triedge2 ) \
1031 ((( triedge1 ).tri == ( triedge2 ).tri ) && \
1032 (( triedge1 ).orient == ( triedge2 ).orient ))
1034 /* Primitives to infect or cure a triangle with the virus. These rely on */
1035 /* the assumption that all shell edges are aligned to four-byte boundaries.*/
1039 ( triedge ).tri[6] = (triangle) \
1040 ((unsigned long) ( triedge ).tri[6] | (unsigned long) 2l )
1043 uninfect( triedge ) \
1044 ( triedge ).tri[6] = (triangle) \
1045 ((unsigned long) ( triedge ).tri[6] & ~(unsigned long) 2l )
1047 /* Test a triangle for viral infection. */
1050 infected( triedge ) \
1051 (((unsigned long) ( triedge ).tri[6] & (unsigned long) 2l ) != 0 )
1053 /* Check or set a triangle's attributes. */
1056 elemattribute( triedge, attnum ) \
1057 ((REAL *) ( triedge ).tri )[elemattribindex + ( attnum )]
1060 setelemattribute( triedge, attnum, value ) \
1061 ((REAL *) ( triedge ).tri )[elemattribindex + ( attnum )] = (REAL)value
1063 /* Check or set a triangle's maximum area bound. */
1066 areabound( triedge ) ((REAL *) ( triedge ).tri )[areaboundindex]
1069 setareabound( triedge, value ) \
1070 ((REAL *) ( triedge ).tri )[areaboundindex] = (REAL)value
1072 /********* Primitives for shell edges *********/
1076 /* sdecode() converts a pointer to an oriented shell edge. The orientation */
1077 /* is extracted from the least significant bit of the pointer. The two */
1078 /* least significant bits (one for orientation, one for viral infection) */
1079 /* are masked out to produce the real pointer. */
1082 sdecode( sptr, edge ) \
1083 ( edge ).shorient = (int) ((unsigned long) ( sptr ) & (unsigned long) 1l ); \
1084 ( edge ).sh = (shelle *) \
1085 ((unsigned long) ( sptr ) & ~(unsigned long) 3l )
1087 /* sencode() compresses an oriented shell edge into a single pointer. It */
1088 /* relies on the assumption that all shell edges are aligned to two-byte */
1089 /* boundaries, so the least significant bit of (edge).sh is zero. */
1093 (shelle) ((unsigned long) ( edge ).sh | (unsigned long) ( edge ).shorient )
1095 /* ssym() toggles the orientation of a shell edge. */
1098 ssym( edge1, edge2 ) \
1099 ( edge2 ).sh = ( edge1 ).sh; \
1100 ( edge2 ).shorient = 1 - ( edge1 ).shorient
1104 ( edge ).shorient = 1 - ( edge ).shorient
1106 /* spivot() finds the other shell edge (from the same segment) that shares */
1107 /* the same origin. */
1110 spivot( edge1, edge2 ) \
1111 sptr = ( edge1 ).sh[( edge1 ).shorient]; \
1112 sdecode( sptr, edge2 )
1115 spivotself( edge ) \
1116 sptr = ( edge ).sh[( edge ).shorient]; \
1117 sdecode( sptr, edge )
1119 /* snext() finds the next shell edge (from the same segment) in sequence; */
1120 /* one whose origin is the input shell edge's destination. */
1123 snext( edge1, edge2 ) \
1124 sptr = ( edge1 ).sh[1 - ( edge1 ).shorient]; \
1125 sdecode( sptr, edge2 )
1129 sptr = ( edge ).sh[1 - ( edge ).shorient]; \
1130 sdecode( sptr, edge )
1132 /* These primitives determine or set the origin or destination of a shell */
1136 sorg( edge, pointptr ) \
1137 pointptr = (point) ( edge ).sh[2 + ( edge ).shorient]
1140 sdest( edge, pointptr ) \
1141 pointptr = (point) ( edge ).sh[3 - ( edge ).shorient]
1144 setsorg( edge, pointptr ) \
1145 ( edge ).sh[2 + ( edge ).shorient] = (shelle) pointptr
1148 setsdest( edge, pointptr ) \
1149 ( edge ).sh[3 - ( edge ).shorient] = (shelle) pointptr
1151 /* These primitives read or set a shell marker. Shell markers are used to */
1152 /* hold user boundary information. */
1155 mark( edge ) ( *(int *) (( edge ).sh + 6 ))
1158 setmark( edge, value ) \
1159 *(int *) (( edge ).sh + 6 ) = value
1161 /* Bond two shell edges together. */
1164 sbond( edge1, edge2 ) \
1165 ( edge1 ).sh[( edge1 ).shorient] = sencode( edge2 ); \
1166 ( edge2 ).sh[( edge2 ).shorient] = sencode( edge1 )
1168 /* Dissolve a shell edge bond (from one side). Note that the other shell */
1169 /* edge will still think it's connected to this shell edge. */
1173 ( edge ).sh[( edge ).shorient] = (shelle) dummysh
1175 /* Copy a shell edge. */
1178 shellecopy( edge1, edge2 ) \
1179 ( edge2 ).sh = ( edge1 ).sh; \
1180 ( edge2 ).shorient = ( edge1 ).shorient
1182 /* Test for equality of shell edges. */
1185 shelleequal( edge1, edge2 ) \
1186 ((( edge1 ).sh == ( edge2 ).sh ) && \
1187 (( edge1 ).shorient == ( edge2 ).shorient ))
1189 /********* Primitives for interacting triangles and shell edges *********/
1193 /* tspivot() finds a shell edge abutting a triangle. */
1196 tspivot( triedge, edge ) \
1197 sptr = (shelle) ( triedge ).tri[6 + ( triedge ).orient]; \
1198 sdecode( sptr, edge )
1200 /* stpivot() finds a triangle abutting a shell edge. It requires that the */
1201 /* variable `ptr' of type `triangle' be defined. */
1204 stpivot( edge, triedge ) \
1205 ptr = (triangle) ( edge ).sh[4 + ( edge ).shorient]; \
1206 decode( ptr, triedge )
1208 /* Bond a triangle to a shell edge. */
1211 tsbond( triedge, edge ) \
1212 ( triedge ).tri[6 + ( triedge ).orient] = (triangle) sencode( edge ); \
1213 ( edge ).sh[4 + ( edge ).shorient] = (shelle) encode( triedge )
1215 /* Dissolve a bond (from the triangle side). */
1218 tsdissolve( triedge ) \
1219 ( triedge ).tri[6 + ( triedge ).orient] = (triangle) dummysh
1221 /* Dissolve a bond (from the shell edge side). */
1224 stdissolve( edge ) \
1225 ( edge ).sh[4 + ( edge ).shorient] = (shelle) dummytri
1227 /********* Primitives for points *********/
1232 pointmark( pt ) ((int *) ( pt ))[pointmarkindex]
1235 setpointmark( pt, value ) \
1236 ((int *) ( pt ))[pointmarkindex] = value
1239 point2tri( pt ) ((triangle *) ( pt ))[point2triindex]
1242 setpoint2tri( pt, value ) \
1243 ((triangle *) ( pt ))[point2triindex] = value
1247 /********* Mesh manipulation primitives end here *********/
1249 /********* User interaction routines begin here *********/
1253 /*****************************************************************************/
1255 /* syntax() Print list of command line switches. */
1257 /*****************************************************************************/
1267 printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n" );
1268 #else /* not REDUCED */
1269 printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n" );
1270 #endif /* not REDUCED */
1271 #else /* not CDT_ONLY */
1274 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n" );
1275 #else /* not REDUCED */
1276 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n" );
1277 #endif /* not REDUCED */
1278 #endif /* not CDT_ONLY */
1280 printf( " -p Triangulates a Planar Straight Line Graph (.poly file).\n" );
1283 printf( " -r Refines a previously generated mesh.\n" );
1285 " -q Quality mesh generation. A minimum angle may be specified.\n" );
1286 printf( " -a Applies a maximum triangle area constraint.\n" );
1287 #endif /* not CDT_ONLY */
1289 " -A Applies attributes to identify elements in certain regions.\n" );
1290 printf( " -c Encloses the convex hull with segments.\n" );
1291 printf( " -e Generates an edge list.\n" );
1292 printf( " -v Generates a Voronoi diagram.\n" );
1293 printf( " -n Generates a list of triangle neighbors.\n" );
1294 printf( " -g Generates an .off file for Geomview.\n" );
1295 printf( " -B Suppresses output of boundary information.\n" );
1296 printf( " -P Suppresses output of .poly file.\n" );
1297 printf( " -N Suppresses output of .node file.\n" );
1298 printf( " -E Suppresses output of .ele file.\n" );
1299 printf( " -I Suppresses mesh iteration numbers.\n" );
1300 printf( " -O Ignores holes in .poly file.\n" );
1301 printf( " -X Suppresses use of exact arithmetic.\n" );
1302 printf( " -z Numbers all items starting from zero (rather than one).\n" );
1303 printf( " -o2 Generates second-order subparametric elements.\n" );
1306 printf( " -Y Suppresses boundary segment splitting.\n" );
1307 printf( " -S Specifies maximum number of added Steiner points.\n" );
1308 #endif /* not CDT_ONLY */
1311 printf( " -i Uses incremental method, rather than divide-and-conquer.\n" );
1312 printf( " -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n" );
1313 #endif /* not REDUCED */
1314 printf( " -l Uses vertical cuts only, rather than alternating cuts.\n" );
1320 " -s Force segments into mesh by splitting (instead of using CDT).\n" );
1321 #endif /* not CDT_ONLY */
1322 printf( " -C Check consistency of final mesh.\n" );
1323 #endif /* not REDUCED */
1324 printf( " -Q Quiet: No terminal output except errors.\n" );
1325 printf( " -V Verbose: Detailed information on what I'm doing.\n" );
1326 printf( " -h Help: Detailed instructions for Triangle.\n" );
1330 #endif /* not TRILIBRARY */
1332 /*****************************************************************************/
1334 /* info() Print out complete instructions. */
1336 /*****************************************************************************/
1342 printf( "Triangle\n" );
1344 "A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n" );
1345 printf( "Version 1.3\n\n" );
1347 "Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)\n"
1349 printf( "School of Computer Science / Carnegie Mellon University\n" );
1350 printf( "5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891\n" );
1352 "Created as part of the Archimedes project (tools for parallel FEM).\n" );
1354 "Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n" );
1355 printf( "There is no warranty whatsoever. Use at your own risk.\n" );
1358 printf( "This executable is compiled for single precision arithmetic.\n\n\n" );
1359 #else /* not SINGLE */
1360 printf( "This executable is compiled for double precision arithmetic.\n\n\n" );
1361 #endif /* not SINGLE */
1363 "Triangle generates exact Delaunay triangulations, constrained Delaunay\n" );
1365 "triangulations, and quality conforming Delaunay triangulations. The latter\n"
1368 "can be generated with no small angles, and are thus suitable for finite\n" );
1370 "element analysis. If no command line switches are specified, your .node\n" );
1372 "input file will be read, and the Delaunay triangulation will be returned in\n"
1374 printf( ".node and .ele output files. The command syntax is:\n\n" );
1379 printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n\n" );
1380 #else /* not REDUCED */
1381 printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n\n" );
1382 #endif /* not REDUCED */
1383 #else /* not CDT_ONLY */
1386 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n\n" );
1387 #else /* not REDUCED */
1388 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n" );
1389 #endif /* not REDUCED */
1390 #endif /* not CDT_ONLY */
1392 "Underscores indicate that numbers may optionally follow certain switches;\n" );
1394 "do not leave any space between a switch and its numeric parameter.\n" );
1396 "input_file must be a file with extension .node, or extension .poly if the\n" );
1398 "-p switch is used. If -r is used, you must supply .node and .ele files,\n" );
1400 "and possibly a .poly file and .area file as well. The formats of these\n" );
1401 printf( "files are described below.\n\n" );
1402 printf( "Command Line Switches:\n\n" );
1404 " -p Reads a Planar Straight Line Graph (.poly file), which can specify\n"
1407 " points, segments, holes, and regional attributes and area\n" );
1409 " constraints. Will generate a constrained Delaunay triangulation\n" );
1411 " fitting the input; or, if -s, -q, or -a is used, a conforming\n" );
1413 " Delaunay triangulation. If -p is not used, Triangle reads a .node\n"
1415 printf( " file by default.\n" );
1417 " -r Refines a previously generated mesh. The mesh is read from a .node\n"
1420 " file and an .ele file. If -p is also used, a .poly file is read\n" );
1422 " and used to constrain edges in the mesh. Further details on\n" );
1423 printf( " refinement are given below.\n" );
1425 " -q Quality mesh generation by Jim Ruppert's Delaunay refinement\n" );
1427 " algorithm. Adds points to the mesh to ensure that no angles\n" );
1429 " smaller than 20 degrees occur. An alternative minimum angle may be\n"
1432 " specified after the `q'. If the minimum angle is 20.7 degrees or\n" );
1434 " smaller, the triangulation algorithm is theoretically guaranteed to\n"
1437 " terminate (assuming infinite precision arithmetic - Triangle may\n" );
1439 " fail to terminate if you run out of precision). In practice, the\n" );
1441 " algorithm often succeeds for minimum angles up to 33.8 degrees.\n" );
1443 " For highly refined meshes, however, it may be necessary to reduce\n" );
1445 " the minimum angle to well below 20 to avoid problems associated\n" );
1447 " with insufficient floating-point precision. The specified angle\n" );
1448 printf( " may include a decimal point.\n" );
1450 " -a Imposes a maximum triangle area. If a number follows the `a', no\n" );
1452 " triangle will be generated whose area is larger than that number.\n" );
1454 " If no number is specified, an .area file (if -r is used) or .poly\n" );
1456 " file (if -r is not used) specifies a number of maximum area\n" );
1458 " constraints. An .area file contains a separate area constraint for\n"
1461 " each triangle, and is useful for refining a finite element mesh\n" );
1463 " based on a posteriori error estimates. A .poly file can optionally\n"
1466 " contain an area constraint for each segment-bounded region, thereby\n"
1469 " enforcing triangle densities in a first triangulation. You can\n" );
1471 " impose both a fixed area constraint and a varying area constraint\n" );
1473 " by invoking the -a switch twice, once with and once without a\n" );
1475 " number following. Each area specified may include a decimal point.\n"
1478 " -A Assigns an additional attribute to each triangle that identifies\n" );
1480 " what segment-bounded region each triangle belongs to. Attributes\n" );
1482 " are assigned to regions by the .poly file. If a region is not\n" );
1484 " explicitly marked by the .poly file, triangles in that region are\n" );
1486 " assigned an attribute of zero. The -A switch has an effect only\n" );
1487 printf( " when the -p switch is used and the -r switch is not.\n" );
1489 " -c Creates segments on the convex hull of the triangulation. If you\n" );
1491 " are triangulating a point set, this switch causes a .poly file to\n" );
1493 " be written, containing all edges in the convex hull. (By default,\n"
1496 " a .poly file is written only if a .poly file is read.) If you are\n"
1499 " triangulating a PSLG, this switch specifies that the interior of\n" );
1501 " the convex hull of the PSLG should be triangulated. If you do not\n"
1504 " use this switch when triangulating a PSLG, it is assumed that you\n" );
1506 " have identified the region to be triangulated by surrounding it\n" );
1508 " with segments of the input PSLG. Beware: if you are not careful,\n"
1511 " this switch can cause the introduction of an extremely thin angle\n" );
1513 " between a PSLG segment and a convex hull segment, which can cause\n" );
1515 " overrefinement or failure if Triangle runs out of precision. If\n" );
1517 " you are refining a mesh, the -c switch works differently; it\n" );
1519 " generates the set of boundary edges of the mesh, rather than the\n" );
1520 printf( " convex hull.\n" );
1522 " -e Outputs (to an .edge file) a list of edges of the triangulation.\n" );
1524 " -v Outputs the Voronoi diagram associated with the triangulation.\n" );
1525 printf( " Does not attempt to detect degeneracies.\n" );
1527 " -n Outputs (to a .neigh file) a list of triangles neighboring each\n" );
1528 printf( " triangle.\n" );
1530 " -g Outputs the mesh to an Object File Format (.off) file, suitable for\n"
1532 printf( " viewing with the Geometry Center's Geomview package.\n" );
1534 " -B No boundary markers in the output .node, .poly, and .edge output\n" );
1536 " files. See the detailed discussion of boundary markers below.\n" );
1538 " -P No output .poly file. Saves disk space, but you lose the ability\n" );
1540 " to impose segment constraints on later refinements of the mesh.\n" );
1541 printf( " -N No output .node file.\n" );
1542 printf( " -E No output .ele file.\n" );
1544 " -I No iteration numbers. Suppresses the output of .node and .poly\n" );
1546 " files, so your input files won't be overwritten. (If your input is\n"
1549 " a .poly file only, a .node file will be written.) Cannot be used\n" );
1551 " with the -r switch, because that would overwrite your input .ele\n" );
1553 " file. Shouldn't be used with the -s, -q, or -a switch if you are\n" );
1555 " using a .node file for input, because no .node file will be\n" );
1556 printf( " written, so there will be no record of any added points.\n" );
1557 printf( " -O No holes. Ignores the holes in the .poly file.\n" );
1559 " -X No exact arithmetic. Normally, Triangle uses exact floating-point\n"
1562 " arithmetic for certain tests if it thinks the inexact tests are not\n"
1565 " accurate enough. Exact arithmetic ensures the robustness of the\n" );
1567 " triangulation algorithms, despite floating-point roundoff error.\n" );
1569 " Disabling exact arithmetic with the -X switch will cause a small\n" );
1571 " improvement in speed and create the possibility (albeit small) that\n"
1574 " Triangle will fail to produce a valid mesh. Not recommended.\n" );
1576 " -z Numbers all items starting from zero (rather than one). Note that\n"
1579 " this switch is normally overrided by the value used to number the\n" );
1581 " first point of the input .node or .poly file. However, this switch\n"
1583 printf( " is useful when calling Triangle from another program.\n" );
1585 " -o2 Generates second-order subparametric elements with six nodes each.\n"
1588 " -Y No new points on the boundary. This switch is useful when the mesh\n"
1591 " boundary must be preserved so that it conforms to some adjacent\n" );
1593 " mesh. Be forewarned that you will probably sacrifice some of the\n" );
1595 " quality of the mesh; Triangle will try, but the resulting mesh may\n"
1598 " contain triangles of poor aspect ratio. Works well if all the\n" );
1600 " boundary points are closely spaced. Specify this switch twice\n" );
1602 " (`-YY') to prevent all segment splitting, including internal\n" );
1603 printf( " boundaries.\n" );
1605 " -S Specifies the maximum number of Steiner points (points that are not\n"
1608 " in the input, but are added to meet the constraints of minimum\n" );
1610 " angle and maximum area). The default is to allow an unlimited\n" );
1612 " number. If you specify this switch with no number after it,\n" );
1614 " the limit is set to zero. Triangle always adds points at segment\n" );
1616 " intersections, even if it needs to use more points than the limit\n" );
1618 " you set. When Triangle inserts segments by splitting (-s), it\n" );
1620 " always adds enough points to ensure that all the segments appear in\n"
1623 " the triangulation, again ignoring the limit. Be forewarned that\n" );
1625 " the -S switch may result in a conforming triangulation that is not\n"
1628 " truly Delaunay, because Triangle may be forced to stop adding\n" );
1630 " points when the mesh is in a state where a segment is non-Delaunay\n"
1633 " and needs to be split. If so, Triangle will print a warning.\n" );
1635 " -i Uses an incremental rather than divide-and-conquer algorithm to\n" );
1637 " form a Delaunay triangulation. Try it if the divide-and-conquer\n" );
1638 printf( " algorithm fails.\n" );
1640 " -F Uses Steven Fortune's sweepline algorithm to form a Delaunay\n" );
1642 " triangulation. Warning: does not use exact arithmetic for all\n" );
1643 printf( " calculations. An exact result is not guaranteed.\n" );
1645 " -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n" );
1647 " default, Triangle uses alternating vertical and horizontal cuts,\n" );
1649 " which usually improve the speed except with point sets that are\n" );
1651 " small or short and wide. This switch is primarily of theoretical\n" );
1652 printf( " interest.\n" );
1654 " -s Specifies that segments should be forced into the triangulation by\n"
1657 " recursively splitting them at their midpoints, rather than by\n" );
1659 " generating a constrained Delaunay triangulation. Segment splitting\n"
1662 " is true to Ruppert's original algorithm, but can create needlessly\n"
1664 printf( " small triangles near external small features.\n" );
1666 " -C Check the consistency of the final mesh. Uses exact arithmetic for\n"
1669 " checking, even if the -X switch is used. Useful if you suspect\n" );
1670 printf( " Triangle is buggy.\n" );
1672 " -Q Quiet: Suppresses all explanation of what Triangle is doing, unless\n"
1674 printf( " an error occurs.\n" );
1676 " -V Verbose: Gives detailed information about what Triangle is doing.\n" );
1678 " Add more `V's for increasing amount of detail. `-V' gives\n" );
1680 " information on algorithmic progress and more detailed statistics.\n" );
1682 " `-VV' gives point-by-point details, and will print so much that\n" );
1684 " Triangle will run much more slowly. `-VVV' gives information only\n"
1686 printf( " a debugger could love.\n" );
1687 printf( " -h Help: Displays these instructions.\n" );
1689 printf( "Definitions:\n" );
1692 " A Delaunay triangulation of a point set is a triangulation whose vertices\n"
1695 " are the point set, having the property that no point in the point set\n" );
1697 " falls in the interior of the circumcircle (circle that passes through all\n"
1699 printf( " three vertices) of any triangle in the triangulation.\n\n" );
1701 " A Voronoi diagram of a point set is a subdivision of the plane into\n" );
1703 " polygonal regions (some of which may be infinite), where each region is\n" );
1705 " the set of points in the plane that are closer to some input point than\n" );
1707 " to any other input point. (The Voronoi diagram is the geometric dual of\n"
1709 printf( " the Delaunay triangulation.)\n\n" );
1711 " A Planar Straight Line Graph (PSLG) is a collection of points and\n" );
1713 " segments. Segments are simply edges, whose endpoints are points in the\n" );
1715 " PSLG. The file format for PSLGs (.poly files) is described below.\n" );
1718 " A constrained Delaunay triangulation of a PSLG is similar to a Delaunay\n" );
1720 " triangulation, but each PSLG segment is present as a single edge in the\n" );
1722 " triangulation. (A constrained Delaunay triangulation is not truly a\n" );
1723 printf( " Delaunay triangulation.)\n\n" );
1725 " A conforming Delaunay triangulation of a PSLG is a true Delaunay\n" );
1727 " triangulation in which each PSLG segment may have been subdivided into\n" );
1729 " several edges by the insertion of additional points. These inserted\n" );
1731 " points are necessary to allow the segments to exist in the mesh while\n" );
1732 printf( " maintaining the Delaunay property.\n\n" );
1733 printf( "File Formats:\n\n" );
1735 " All files may contain comments prefixed by the character '#'. Points,\n" );
1737 " triangles, edges, holes, and maximum area constraints must be numbered\n" );
1739 " consecutively, starting from either 1 or 0. Whichever you choose, all\n" );
1741 " input files must be consistent; if the nodes are numbered from 1, so must\n"
1744 " be all other objects. Triangle automatically detects your choice while\n" );
1746 " reading the .node (or .poly) file. (When calling Triangle from another\n" );
1748 " program, use the -z switch if you wish to number objects from zero.)\n" );
1749 printf( " Examples of these file formats are given below.\n\n" );
1750 printf( " .node files:\n" );
1752 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n" );
1754 " <# of boundary markers (0 or 1)>\n"
1757 " Remaining lines: <point #> <x> <y> [attributes] [boundary marker]\n" );
1760 " The attributes, which are typically floating-point values of physical\n" );
1762 " quantities (such as mass or conductivity) associated with the nodes of\n"
1765 " a finite element mesh, are copied unchanged to the output mesh. If -s,\n"
1768 " -q, or -a is selected, each new Steiner point added to the mesh will\n" );
1769 printf( " have attributes assigned to it by linear interpolation.\n\n" );
1771 " If the fourth entry of the first line is `1', the last column of the\n" );
1773 " remainder of the file is assumed to contain boundary markers. Boundary\n"
1776 " markers are used to identify boundary points and points resting on PSLG\n"
1779 " segments; a complete description appears in a section below. The .node\n"
1782 " file produced by Triangle will contain boundary markers in the last\n" );
1783 printf( " column unless they are suppressed by the -B switch.\n\n" );
1784 printf( " .ele files:\n" );
1786 " First line: <# of triangles> <points per triangle> <# of attributes>\n" );
1788 " Remaining lines: <triangle #> <point> <point> <point> ... [attributes]\n"
1792 " Points are indices into the corresponding .node file. The first three\n"
1795 " points are the corners, and are listed in counterclockwise order around\n"
1798 " each triangle. (The remaining points, if any, depend on the type of\n" );
1800 " finite element used.) The attributes are just like those of .node\n" );
1802 " files. Because there is no simple mapping from input to output\n" );
1804 " triangles, an attempt is made to interpolate attributes, which may\n" );
1806 " result in a good deal of diffusion of attributes among nearby triangles\n"
1809 " as the triangulation is refined. Diffusion does not occur across\n" );
1811 " segments, so attributes used to identify segment-bounded regions remain\n"
1814 " intact. In output .ele files, all triangles have three points each\n" );
1816 " unless the -o2 switch is used, in which case they have six, and the\n" );
1818 " fourth, fifth, and sixth points lie on the midpoints of the edges\n" );
1819 printf( " opposite the first, second, and third corners.\n\n" );
1820 printf( " .poly files:\n" );
1822 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n" );
1824 " <# of boundary markers (0 or 1)>\n"
1827 " Following lines: <point #> <x> <y> [attributes] [boundary marker]\n" );
1828 printf( " One line: <# of segments> <# of boundary markers (0 or 1)>\n" );
1830 " Following lines: <segment #> <endpoint> <endpoint> [boundary marker]\n" );
1831 printf( " One line: <# of holes>\n" );
1832 printf( " Following lines: <hole #> <x> <y>\n" );
1834 " Optional line: <# of regional attributes and/or area constraints>\n" );
1836 " Optional following lines: <constraint #> <x> <y> <attrib> <max area>\n" );
1839 " A .poly file represents a PSLG, as well as some additional information.\n"
1842 " The first section lists all the points, and is identical to the format\n"
1845 " of .node files. <# of points> may be set to zero to indicate that the\n"
1848 " points are listed in a separate .node file; .poly files produced by\n" );
1850 " Triangle always have this format. This has the advantage that a point\n"
1853 " set may easily be triangulated with or without segments. (The same\n" );
1855 " effect can be achieved, albeit using more disk space, by making a copy\n"
1858 " of the .poly file with the extension .node; all sections of the file\n" );
1859 printf( " but the first are ignored.)\n\n" );
1861 " The second section lists the segments. Segments are edges whose\n" );
1863 " presence in the triangulation is enforced. Each segment is specified\n" );
1865 " by listing the indices of its two endpoints. This means that you must\n"
1868 " include its endpoints in the point list. If -s, -q, and -a are not\n" );
1870 " selected, Triangle will produce a constrained Delaunay triangulation,\n" );
1872 " in which each segment appears as a single edge in the triangulation.\n" );
1874 " If -q or -a is selected, Triangle will produce a conforming Delaunay\n" );
1876 " triangulation, in which segments may be subdivided into smaller edges.\n"
1878 printf( " Each segment, like each point, may have a boundary marker.\n\n" );
1880 " The third section lists holes (and concavities, if -c is selected) in\n" );
1882 " the triangulation. Holes are specified by identifying a point inside\n" );
1884 " each hole. After the triangulation is formed, Triangle creates holes\n" );
1886 " by eating triangles, spreading out from each hole point until its\n" );
1888 " progress is blocked by PSLG segments; you must be careful to enclose\n" );
1890 " each hole in segments, or your whole triangulation may be eaten away.\n" );
1892 " If the two triangles abutting a segment are eaten, the segment itself\n" );
1894 " is also eaten. Do not place a hole directly on a segment; if you do,\n" );
1895 printf( " Triangle will choose one side of the segment arbitrarily.\n\n" );
1897 " The optional fourth section lists regional attributes (to be assigned\n" );
1899 " to all triangles in a region) and regional constraints on the maximum\n" );
1901 " triangle area. Triangle will read this section only if the -A switch\n" );
1903 " is used or the -a switch is used without a number following it, and the\n"
1906 " -r switch is not used. Regional attributes and area constraints are\n" );
1908 " propagated in the same manner as holes; you specify a point for each\n" );
1910 " attribute and/or constraint, and the attribute and/or constraint will\n" );
1912 " affect the whole region (bounded by segments) containing the point. If\n"
1915 " two values are written on a line after the x and y coordinate, the\n" );
1917 " former is assumed to be a regional attribute (but will only be applied\n"
1920 " if the -A switch is selected), and the latter is assumed to be a\n" );
1922 " regional area constraint (but will only be applied if the -a switch is\n"
1925 " selected). You may also specify just one value after the coordinates,\n"
1928 " which can serve as both an attribute and an area constraint, depending\n"
1931 " on the choice of switches. If you are using the -A and -a switches\n" );
1933 " simultaneously and wish to assign an attribute to some region without\n" );
1934 printf( " imposing an area constraint, use a negative maximum area.\n\n" );
1936 " When a triangulation is created from a .poly file, you must either\n" );
1938 " enclose the entire region to be triangulated in PSLG segments, or\n" );
1940 " use the -c switch, which encloses the convex hull of the input point\n" );
1942 " set. If you do not use the -c switch, Triangle will eat all triangles\n"
1945 " on the outer boundary that are not protected by segments; if you are\n" );
1947 " not careful, your whole triangulation may be eaten away. If you do\n" );
1949 " use the -c switch, you can still produce concavities by appropriate\n" );
1950 printf( " placement of holes just inside the convex hull.\n\n" );
1952 " An ideal PSLG has no intersecting segments, nor any points that lie\n" );
1954 " upon segments (except, of course, the endpoints of each segment.) You\n"
1957 " aren't required to make your .poly files ideal, but you should be aware\n"
1960 " of what can go wrong. Segment intersections are relatively safe -\n" );
1962 " Triangle will calculate the intersection points for you and add them to\n"
1965 " the triangulation - as long as your machine's floating-point precision\n"
1968 " doesn't become a problem. You are tempting the fates if you have three\n"
1971 " segments that cross at the same location, and expect Triangle to figure\n"
1974 " out where the intersection point is. Thanks to floating-point roundoff\n"
1977 " error, Triangle will probably decide that the three segments intersect\n"
1980 " at three different points, and you will find a minuscule triangle in\n" );
1982 " your output - unless Triangle tries to refine the tiny triangle, uses\n" );
1984 " up the last bit of machine precision, and fails to terminate at all.\n" );
1986 " You're better off putting the intersection point in the input files,\n" );
1988 " and manually breaking up each segment into two. Similarly, if you\n" );
1990 " place a point at the middle of a segment, and hope that Triangle will\n" );
1992 " break up the segment at that point, you might get lucky. On the other\n"
1995 " hand, Triangle might decide that the point doesn't lie precisely on the\n"
1998 " line, and you'll have a needle-sharp triangle in your output - or a lot\n"
2000 printf( " of tiny triangles if you're generating a quality mesh.\n\n" );
2002 " When Triangle reads a .poly file, it also writes a .poly file, which\n" );
2004 " includes all edges that are part of input segments. If the -c switch\n" );
2006 " is used, the output .poly file will also include all of the edges on\n" );
2008 " the convex hull. Hence, the output .poly file is useful for finding\n" );
2010 " edges associated with input segments and setting boundary conditions in\n"
2013 " finite element simulations. More importantly, you will need it if you\n"
2016 " plan to refine the output mesh, and don't want segments to be missing\n" );
2017 printf( " in later triangulations.\n\n" );
2018 printf( " .area files:\n" );
2019 printf( " First line: <# of triangles>\n" );
2020 printf( " Following lines: <triangle #> <maximum area>\n\n" );
2022 " An .area file associates with each triangle a maximum area that is used\n"
2025 " for mesh refinement. As with other file formats, every triangle must\n" );
2027 " be represented, and they must be numbered consecutively. A triangle\n" );
2029 " may be left unconstrained by assigning it a negative maximum area.\n" );
2031 printf( " .edge files:\n" );
2032 printf( " First line: <# of edges> <# of boundary markers (0 or 1)>\n" );
2034 " Following lines: <edge #> <endpoint> <endpoint> [boundary marker]\n" );
2037 " Endpoints are indices into the corresponding .node file. Triangle can\n"
2040 " produce .edge files (use the -e switch), but cannot read them. The\n" );
2042 " optional column of boundary markers is suppressed by the -B switch.\n" );
2045 " In Voronoi diagrams, one also finds a special kind of edge that is an\n" );
2047 " infinite ray with only one endpoint. For these edges, a different\n" );
2048 printf( " format is used:\n\n" );
2049 printf( " <edge #> <endpoint> -1 <direction x> <direction y>\n\n" );
2051 " The `direction' is a floating-point vector that indicates the direction\n"
2053 printf( " of the infinite ray.\n\n" );
2054 printf( " .neigh files:\n" );
2056 " First line: <# of triangles> <# of neighbors per triangle (always 3)>\n"
2059 " Following lines: <triangle #> <neighbor> <neighbor> <neighbor>\n" );
2062 " Neighbors are indices into the corresponding .ele file. An index of -1\n"
2065 " indicates a mesh boundary, and therefore no neighbor. Triangle can\n" );
2067 " produce .neigh files (use the -n switch), but cannot read them.\n" );
2070 " The first neighbor of triangle i is opposite the first corner of\n" );
2071 printf( " triangle i, and so on.\n\n" );
2072 printf( "Boundary Markers:\n\n" );
2074 " Boundary markers are tags used mainly to identify which output points and\n"
2077 " edges are associated with which PSLG segment, and to identify which\n" );
2079 " points and edges occur on a boundary of the triangulation. A common use\n"
2082 " is to determine where boundary conditions should be applied to a finite\n" );
2084 " element mesh. You can prevent boundary markers from being written into\n" );
2085 printf( " files produced by Triangle by using the -B switch.\n\n" );
2087 " The boundary marker associated with each segment in an output .poly file\n"
2089 printf( " or edge in an output .edge file is chosen as follows:\n" );
2091 " - If an output edge is part or all of a PSLG segment with a nonzero\n" );
2093 " boundary marker, then the edge is assigned the same marker.\n" );
2095 " - Otherwise, if the edge occurs on a boundary of the triangulation\n" );
2097 " (including boundaries of holes), then the edge is assigned the marker\n"
2099 printf( " one (1).\n" );
2100 printf( " - Otherwise, the edge is assigned the marker zero (0).\n" );
2102 " The boundary marker associated with each point in an output .node file is\n"
2104 printf( " chosen as follows:\n" );
2106 " - If a point is assigned a nonzero boundary marker in the input file,\n" );
2108 " then it is assigned the same marker in the output .node file.\n" );
2110 " - Otherwise, if the point lies on a PSLG segment (including the\n" );
2112 " segment's endpoints) with a nonzero boundary marker, then the point\n" );
2114 " is assigned the same marker. If the point lies on several such\n" );
2115 printf( " segments, one of the markers is chosen arbitrarily.\n" );
2117 " - Otherwise, if the point occurs on a boundary of the triangulation,\n" );
2118 printf( " then the point is assigned the marker one (1).\n" );
2119 printf( " - Otherwise, the point is assigned the marker zero (0).\n" );
2122 " If you want Triangle to determine for you which points and edges are on\n" );
2124 " the boundary, assign them the boundary marker zero (or use no markers at\n"
2127 " all) in your input files. Alternatively, you can mark some of them and\n" );
2128 printf( " leave others marked zero, allowing Triangle to label them.\n\n" );
2129 printf( "Triangulation Iteration Numbers:\n\n" );
2131 " Because Triangle can read and refine its own triangulations, input\n" );
2133 " and output files have iteration numbers. For instance, Triangle might\n" );
2135 " read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n" );
2137 " triangulation, and output the files mesh.4.node, mesh.4.ele, and\n" );
2138 printf( " mesh.4.poly. Files with no iteration number are treated as if\n" );
2140 " their iteration number is zero; hence, Triangle might read the file\n" );
2142 " points.node, triangulate it, and produce the files points.1.node and\n" );
2143 printf( " points.1.ele.\n\n" );
2145 " Iteration numbers allow you to create a sequence of successively finer\n" );
2147 " meshes suitable for multigrid methods. They also allow you to produce a\n"
2150 " sequence of meshes using error estimate-driven mesh refinement.\n" );
2153 " If you're not using refinement or quality meshing, and you don't like\n" );
2155 " iteration numbers, use the -I switch to disable them. This switch will\n" );
2157 " also disable output of .node and .poly files to prevent your input files\n"
2160 " from being overwritten. (If the input is a .poly file that contains its\n"
2162 printf( " own points, a .node file will be written.)\n\n" );
2163 printf( "Examples of How to Use Triangle:\n\n" );
2165 " `triangle dots' will read points from dots.node, and write their Delaunay\n"
2168 " triangulation to dots.1.node and dots.1.ele. (dots.1.node will be\n" );
2170 " identical to dots.node.) `triangle -I dots' writes the triangulation to\n"
2173 " dots.ele instead. (No additional .node file is needed, so none is\n" );
2174 printf( " written.)\n\n" );
2176 " `triangle -pe object.1' will read a PSLG from object.1.poly (and possibly\n"
2179 " object.1.node, if the points are omitted from object.1.poly) and write\n" );
2180 printf( " their constrained Delaunay triangulation to object.2.node and\n" );
2182 " object.2.ele. The segments will be copied to object.2.poly, and all\n" );
2183 printf( " edges will be written to object.2.edge.\n\n" );
2185 " `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and\n" );
2187 " possibly object.node), generate a mesh whose angles are all greater than\n"
2190 " 31.5 degrees and whose triangles all have area smaller than 0.1, and\n" );
2192 " write the mesh to object.1.node and object.1.ele. Each segment may have\n"
2195 " been broken up into multiple edges; the resulting constrained edges are\n" );
2196 printf( " written to object.1.poly.\n\n" );
2198 " Here is a sample file `box.poly' describing a square with a square hole:\n"
2202 " # A box with eight points in 2D, no attributes, one boundary marker.\n" );
2203 printf( " 8 2 0 1\n" );
2204 printf( " # Outer box has these vertices:\n" );
2205 printf( " 1 0 0 0\n" );
2206 printf( " 2 0 3 0\n" );
2207 printf( " 3 3 0 0\n" );
2208 printf( " 4 3 3 33 # A special marker for this point.\n" );
2209 printf( " # Inner square has these vertices:\n" );
2210 printf( " 5 1 1 0\n" );
2211 printf( " 6 1 2 0\n" );
2212 printf( " 7 2 1 0\n" );
2213 printf( " 8 2 2 0\n" );
2214 printf( " # Five segments with boundary markers.\n" );
2216 printf( " 1 1 2 5 # Left side of outer box.\n" );
2217 printf( " 2 5 7 0 # Segments 2 through 5 enclose the hole.\n" );
2218 printf( " 3 7 8 0\n" );
2219 printf( " 4 8 6 10\n" );
2220 printf( " 5 6 5 0\n" );
2221 printf( " # One hole in the middle of the inner square.\n" );
2223 printf( " 1 1.5 1.5\n\n" );
2225 " Note that some segments are missing from the outer square, so one must\n" );
2227 " use the `-c' switch. After `triangle -pqc box.poly', here is the output\n"
2230 " file `box.1.node', with twelve points. The last four points were added\n" );
2232 " to meet the angle constraint. Points 1, 2, and 9 have markers from\n" );
2234 " segment 1. Points 6 and 8 have markers from segment 4. All the other\n" );
2236 " points but 4 have been marked to indicate that they lie on a boundary.\n" );
2238 printf( " 12 2 0 1\n" );
2239 printf( " 1 0 0 5\n" );
2240 printf( " 2 0 3 5\n" );
2241 printf( " 3 3 0 1\n" );
2242 printf( " 4 3 3 33\n" );
2243 printf( " 5 1 1 1\n" );
2244 printf( " 6 1 2 10\n" );
2245 printf( " 7 2 1 1\n" );
2246 printf( " 8 2 2 10\n" );
2247 printf( " 9 0 1.5 5\n" );
2248 printf( " 10 1.5 0 1\n" );
2249 printf( " 11 3 1.5 1\n" );
2250 printf( " 12 1.5 3 1\n" );
2251 printf( " # Generated by triangle -pqc box.poly\n\n" );
2252 printf( " Here is the output file `box.1.ele', with twelve triangles.\n\n" );
2253 printf( " 12 3 0\n" );
2254 printf( " 1 5 6 9\n" );
2255 printf( " 2 10 3 7\n" );
2256 printf( " 3 6 8 12\n" );
2257 printf( " 4 9 1 5\n" );
2258 printf( " 5 6 2 9\n" );
2259 printf( " 6 7 3 11\n" );
2260 printf( " 7 11 4 8\n" );
2261 printf( " 8 7 5 10\n" );
2262 printf( " 9 12 2 6\n" );
2263 printf( " 10 8 7 11\n" );
2264 printf( " 11 5 1 10\n" );
2265 printf( " 12 8 4 12\n" );
2266 printf( " # Generated by triangle -pqc box.poly\n\n" );
2268 " Here is the output file `box.1.poly'. Note that segments have been added\n"
2271 " to represent the convex hull, and some segments have been split by newly\n"
2274 " added points. Note also that <# of points> is set to zero to indicate\n" );
2275 printf( " that the points should be read from the .node file.\n\n" );
2276 printf( " 0 2 0 1\n" );
2277 printf( " 12 1\n" );
2278 printf( " 1 1 9 5\n" );
2279 printf( " 2 5 7 1\n" );
2280 printf( " 3 8 7 1\n" );
2281 printf( " 4 6 8 10\n" );
2282 printf( " 5 5 6 1\n" );
2283 printf( " 6 3 10 1\n" );
2284 printf( " 7 4 11 1\n" );
2285 printf( " 8 2 12 1\n" );
2286 printf( " 9 9 2 5\n" );
2287 printf( " 10 10 1 1\n" );
2288 printf( " 11 11 3 1\n" );
2289 printf( " 12 12 4 1\n" );
2291 printf( " 1 1.5 1.5\n" );
2292 printf( " # Generated by triangle -pqc box.poly\n\n" );
2293 printf( "Refinement and Area Constraints:\n\n" );
2295 " The -r switch causes a mesh (.node and .ele files) to be read and\n" );
2297 " refined. If the -p switch is also used, a .poly file is read and used to\n"
2300 " specify edges that are constrained and cannot be eliminated (although\n" );
2302 " they can be divided into smaller edges) by the refinement process.\n" );
2305 " When you refine a mesh, you generally want to impose tighter quality\n" );
2307 " constraints. One way to accomplish this is to use -q with a larger\n" );
2309 " angle, or -a followed by a smaller area than you used to generate the\n" );
2311 " mesh you are refining. Another way to do this is to create an .area\n" );
2313 " file, which specifies a maximum area for each triangle, and use the -a\n" );
2315 " switch (without a number following). Each triangle's area constraint is\n"
2318 " applied to that triangle. Area constraints tend to diffuse as the mesh\n" );
2320 " is refined, so if there are large variations in area constraint between\n" );
2321 printf( " adjacent triangles, you may not get the results you want.\n\n" );
2323 " If you are refining a mesh composed of linear (three-node) elements, the\n"
2326 " output mesh will contain all the nodes present in the input mesh, in the\n"
2329 " same order, with new nodes added at the end of the .node file. However,\n"
2332 " there is no guarantee that each output element is contained in a single\n" );
2334 " input element. Often, output elements will overlap two input elements,\n" );
2336 " and input edges are not present in the output mesh. Hence, a sequence of\n"
2339 " refined meshes will form a hierarchy of nodes, but not a hierarchy of\n" );
2341 " elements. If you a refining a mesh of higher-order elements, the\n" );
2343 " hierarchical property applies only to the nodes at the corners of an\n" );
2344 printf( " element; other nodes may not be present in the refined mesh.\n\n" );
2346 " It is important to understand that maximum area constraints in .poly\n" );
2348 " files are handled differently from those in .area files. A maximum area\n"
2351 " in a .poly file applies to the whole (segment-bounded) region in which a\n"
2354 " point falls, whereas a maximum area in an .area file applies to only one\n"
2357 " triangle. Area constraints in .poly files are used only when a mesh is\n" );
2359 " first generated, whereas area constraints in .area files are used only to\n"
2362 " refine an existing mesh, and are typically based on a posteriori error\n" );
2364 " estimates resulting from a finite element simulation on that mesh.\n" );
2367 " `triangle -rq25 object.1' will read object.1.node and object.1.ele, then\n"
2370 " refine the triangulation to enforce a 25 degree minimum angle, and then\n" );
2372 " write the refined triangulation to object.2.node and object.2.ele.\n" );
2375 " `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and\n" );
2377 " z.3.area. After reconstructing the mesh and its segments, Triangle will\n"
2380 " refine the mesh so that no triangle has area greater than 6.2, and\n" );
2382 " furthermore the triangles satisfy the maximum area constraints in\n" );
2384 " z.3.area. The output is written to z.4.node, z.4.ele, and z.4.poly.\n" );
2387 " The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n" );
2389 " x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n" );
2390 printf( " suitable for multigrid.\n\n" );
2391 printf( "Convex Hulls and Mesh Boundaries:\n\n" );
2393 " If the input is a point set (rather than a PSLG), Triangle produces its\n" );
2395 " convex hull as a by-product in the output .poly file if you use the -c\n" );
2397 " switch. There are faster algorithms for finding a two-dimensional convex\n"
2400 " hull than triangulation, of course, but this one comes for free. If the\n"
2403 " input is an unconstrained mesh (you are using the -r switch but not the\n" );
2405 " -p switch), Triangle produces a list of its boundary edges (including\n" );
2406 printf( " hole boundaries) as a by-product if you use the -c switch.\n\n" );
2407 printf( "Voronoi Diagrams:\n\n" );
2409 " The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n" );
2411 " .v.edge. For example, `triangle -v points' will read points.node,\n" );
2413 " produce its Delaunay triangulation in points.1.node and points.1.ele,\n" );
2415 " and produce its Voronoi diagram in points.1.v.node and points.1.v.edge.\n" );
2417 " The .v.node file contains a list of all Voronoi vertices, and the .v.edge\n"
2420 " file contains a list of all Voronoi edges, some of which may be infinite\n"
2423 " rays. (The choice of filenames makes it easy to run the set of Voronoi\n" );
2424 printf( " vertices through Triangle, if so desired.)\n\n" );
2426 " This implementation does not use exact arithmetic to compute the Voronoi\n"
2429 " vertices, and does not check whether neighboring vertices are identical.\n"
2432 " Be forewarned that if the Delaunay triangulation is degenerate or\n" );
2434 " near-degenerate, the Voronoi diagram may have duplicate points, crossing\n"
2437 " edges, or infinite rays whose direction vector is zero. Also, if you\n" );
2439 " generate a constrained (as opposed to conforming) Delaunay triangulation,\n"
2442 " or if the triangulation has holes, the corresponding Voronoi diagram is\n" );
2443 printf( " likely to have crossing edges and unlikely to make sense.\n\n" );
2444 printf( "Mesh Topology:\n\n" );
2446 " You may wish to know which triangles are adjacent to a certain Delaunay\n" );
2448 " edge in an .edge file, which Voronoi regions are adjacent to a certain\n" );
2450 " Voronoi edge in a .v.edge file, or which Voronoi regions are adjacent to\n"
2453 " each other. All of this information can be found by cross-referencing\n" );
2455 " output files with the recollection that the Delaunay triangulation and\n" );
2456 printf( " the Voronoi diagrams are planar duals.\n\n" );
2458 " Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n" );
2460 " the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n" );
2462 " wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n" );
2464 " vertex j of the corresponding .v.node file; and Voronoi region k is the\n" );
2465 printf( " dual of point k of the corresponding .node file.\n\n" );
2467 " Hence, to find the triangles adjacent to a Delaunay edge, look at the\n" );
2469 " vertices of the corresponding Voronoi edge; their dual triangles are on\n" );
2471 " the left and right of the Delaunay edge, respectively. To find the\n" );
2473 " Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the\n"
2476 " corresponding Delaunay edge; their dual regions are on the right and left\n"
2479 " of the Voronoi edge, respectively. To find which Voronoi regions are\n" );
2480 printf( " adjacent to each other, just read the list of Delaunay edges.\n" );
2482 printf( "Statistics:\n" );
2485 " After generating a mesh, Triangle prints a count of the number of points,\n"
2488 " triangles, edges, boundary edges, and segments in the output mesh. If\n" );
2490 " you've forgotten the statistics for an existing mesh, the -rNEP switches\n"
2493 " (or -rpNEP if you've got a .poly file for the existing mesh) will\n" );
2494 printf( " regenerate these statistics without writing any output.\n\n" );
2496 " The -V switch produces extended statistics, including a rough estimate\n" );
2498 " of memory use and a histogram of triangle aspect ratios and angles in the\n"
2500 printf( " mesh.\n\n" );
2501 printf( "Exact Arithmetic:\n\n" );
2503 " Triangle uses adaptive exact arithmetic to perform what computational\n" );
2505 " geometers call the `orientation' and `incircle' tests. If the floating-\n"
2508 " point arithmetic of your machine conforms to the IEEE 754 standard (as\n" );
2510 " most workstations do), and does not use extended precision internal\n" );
2512 " registers, then your output is guaranteed to be an absolutely true\n" );
2513 printf( " Delaunay or conforming Delaunay triangulation, roundoff error\n" );
2515 " notwithstanding. The word `adaptive' implies that these arithmetic\n" );
2517 " routines compute the result only to the precision necessary to guarantee\n"
2520 " correctness, so they are usually nearly as fast as their approximate\n" );
2522 " counterparts. The exact tests can be disabled with the -X switch. On\n" );
2524 " most inputs, this switch will reduce the computation time by about eight\n"
2527 " percent - it's not worth the risk. There are rare difficult inputs\n" );
2529 " (having many collinear and cocircular points), however, for which the\n" );
2531 " difference could be a factor of two. These are precisely the inputs most\n"
2533 printf( " likely to cause errors if you use the -X switch.\n\n" );
2535 " Unfortunately, these routines don't solve every numerical problem. Exact\n"
2538 " arithmetic is not used to compute the positions of points, because the\n" );
2540 " bit complexity of point coordinates would grow without bound. Hence,\n" );
2542 " segment intersections aren't computed exactly; in very unusual cases,\n" );
2544 " roundoff error in computing an intersection point might actually lead to\n"
2547 " an inverted triangle and an invalid triangulation. (This is one reason\n" );
2549 " to compute your own intersection points in your .poly files.) Similarly,\n"
2552 " exact arithmetic is not used to compute the vertices of the Voronoi\n" );
2553 printf( " diagram.\n\n" );
2555 " Underflow and overflow can also cause difficulties; the exact arithmetic\n"
2558 " routines do not ameliorate out-of-bounds exponents, which can arise\n" );
2560 " during the orientation and incircle tests. As a rule of thumb, you\n" );
2562 " should ensure that your input values are within a range such that their\n" );
2564 " third powers can be taken without underflow or overflow. Underflow can\n" );
2566 " silently prevent the tests from being performed exactly, while overflow\n" );
2567 printf( " will typically cause a floating exception.\n\n" );
2568 printf( "Calling Triangle from Another Program:\n\n" );
2569 printf( " Read the file triangle.h for details.\n\n" );
2570 printf( "Troubleshooting:\n\n" );
2571 printf( " Please read this section before mailing me bugs.\n\n" );
2572 printf( " `My output mesh has no triangles!'\n\n" );
2574 " If you're using a PSLG, you've probably failed to specify a proper set\n"
2577 " of bounding segments, or forgotten to use the -c switch. Or you may\n" );
2579 " have placed a hole badly. To test these possibilities, try again with\n"
2582 " the -c and -O switches. Alternatively, all your input points may be\n" );
2584 " collinear, in which case you can hardly expect to triangulate them.\n" );
2586 printf( " `Triangle doesn't terminate, or just crashes.'\n" );
2589 " Bad things can happen when triangles get so small that the distance\n" );
2591 " between their vertices isn't much larger than the precision of your\n" );
2593 " machine's arithmetic. If you've compiled Triangle for single-precision\n"
2596 " arithmetic, you might do better by recompiling it for double-precision.\n"
2599 " Then again, you might just have to settle for more lenient constraints\n"
2602 " on the minimum angle and the maximum area than you had planned.\n" );
2605 " You can minimize precision problems by ensuring that the origin lies\n" );
2607 " inside your point set, or even inside the densest part of your\n" );
2609 " mesh. On the other hand, if you're triangulating an object whose x\n" );
2611 " coordinates all fall between 6247133 and 6247134, you're not leaving\n" );
2612 printf( " much floating-point precision for Triangle to work with.\n\n" );
2614 " Precision problems can occur covertly if the input PSLG contains two\n" );
2616 " segments that meet (or intersect) at a very small angle, or if such an\n"
2619 " angle is introduced by the -c switch, which may occur if a point lies\n" );
2621 " ever-so-slightly inside the convex hull, and is connected by a PSLG\n" );
2623 " segment to a point on the convex hull. If you don't realize that a\n" );
2625 " small angle is being formed, you might never discover why Triangle is\n" );
2627 " crashing. To check for this possibility, use the -S switch (with an\n" );
2629 " appropriate limit on the number of Steiner points, found by trial-and-\n"
2632 " error) to stop Triangle early, and view the output .poly file with\n" );
2634 " Show Me (described below). Look carefully for small angles between\n" );
2636 " segments; zoom in closely, as such segments might look like a single\n" );
2637 printf( " segment from a distance.\n\n" );
2639 " If some of the input values are too large, Triangle may suffer a\n" );
2641 " floating exception due to overflow when attempting to perform an\n" );
2643 " orientation or incircle test. (Read the section on exact arithmetic\n" );
2645 " above.) Again, I recommend compiling Triangle for double (rather\n" );
2646 printf( " than single) precision arithmetic.\n\n" );
2648 " `The numbering of the output points doesn't match the input points.'\n" );
2651 " You may have eaten some of your input points with a hole, or by placing\n"
2653 printf( " them outside the area enclosed by segments.\n\n" );
2655 " `Triangle executes without incident, but when I look at the resulting\n" );
2657 " mesh, it has overlapping triangles or other geometric inconsistencies.'\n" );
2660 " If you select the -X switch, Triangle's divide-and-conquer Delaunay\n" );
2662 " triangulation algorithm occasionally makes mistakes due to floating-\n" );
2664 " point roundoff error. Although these errors are rare, don't use the -X\n"
2666 printf( " switch. If you still have problems, please report the bug.\n" );
2669 " Strange things can happen if you've taken liberties with your PSLG. Do\n" );
2671 " you have a point lying in the middle of a segment? Triangle sometimes\n" );
2673 " copes poorly with that sort of thing. Do you want to lay out a collinear\n"
2676 " row of evenly spaced, segment-connected points? Have you simply defined\n"
2679 " one long segment connecting the leftmost point to the rightmost point,\n" );
2681 " and a bunch of points lying along it? This method occasionally works,\n" );
2683 " especially with horizontal and vertical lines, but often it doesn't, and\n"
2686 " you'll have to connect each adjacent pair of points with a separate\n" );
2687 printf( " segment. If you don't like it, tough.\n\n" );
2689 " Furthermore, if you have segments that intersect other than at their\n" );
2691 " endpoints, try not to let the intersections fall extremely close to PSLG\n"
2693 printf( " points or each other.\n\n" );
2695 " If you have problems refining a triangulation not produced by Triangle:\n" );
2697 " Are you sure the triangulation is geometrically valid? Is it formatted\n" );
2699 " correctly for Triangle? Are the triangles all listed so the first three\n"
2701 printf( " points are their corners in counterclockwise order?\n\n" );
2702 printf( "Show Me:\n\n" );
2704 " Triangle comes with a separate program named `Show Me', whose primary\n" );
2706 " purpose is to draw meshes on your screen or in PostScript. Its secondary\n"
2709 " purpose is to check the validity of your input files, and do so more\n" );
2711 " thoroughly than Triangle does. Show Me requires that you have the X\n" );
2713 " Windows system. If you didn't receive Show Me with Triangle, complain to\n"
2715 printf( " whomever you obtained Triangle from, then send me mail.\n\n" );
2716 printf( "Triangle on the Web:\n\n" );
2718 " To see an illustrated, updated version of these instructions, check out\n" );
2720 printf( " http://www.cs.cmu.edu/~quake/triangle.html\n" );
2722 printf( "A Brief Plea:\n" );
2725 " If you use Triangle, and especially if you use it to accomplish real\n" );
2727 " work, I would like very much to hear from you. A short letter or email\n" );
2729 " (to jrs@cs.cmu.edu) describing how you use Triangle will mean a lot to\n" );
2731 " me. The more people I know are using this program, the more easily I can\n"
2734 " justify spending time on improvements and on the three-dimensional\n" );
2736 " successor to Triangle, which in turn will benefit you. Also, I can put\n" );
2738 " you on a list to receive email whenever a new version of Triangle is\n" );
2739 printf( " available.\n\n" );
2741 " If you use a mesh generated by Triangle in a publication, please include\n"
2743 printf( " an acknowledgment as well.\n\n" );
2744 printf( "Research credit:\n\n" );
2746 " Of course, I can take credit for only a fraction of the ideas that made\n" );
2748 " this mesh generator possible. Triangle owes its existence to the efforts\n"
2751 " of many fine computational geometers and other researchers, including\n" );
2753 " Marshall Bern, L. Paul Chew, Boris Delaunay, Rex A. Dwyer, David\n" );
2755 " Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.\n" );
2757 " Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,\n" );
2759 " Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher\n"
2762 " J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the\n" );
2763 printf( " beginning of the source code for references.\n\n" );
2767 #endif /* not TRILIBRARY */
2769 /*****************************************************************************/
2771 /* internalerror() Ask the user to send me the defective product. Exit. */
2773 /*****************************************************************************/
2775 void internalerror(){
2776 printf( " Please report this bug to jrs@cs.cmu.edu\n" );
2777 printf( " Include the message above, your input data set, and the exact\n" );
2778 printf( " command line you used to run Triangle.\n" );
2782 /*****************************************************************************/
2784 /* parsecommandline() Read the command line, identify switches, and set */
2785 /* up options and file names. */
2787 /* The effects of this routine are felt entirely through global variables. */
2789 /*****************************************************************************/
2791 void parsecommandline( argc, argv )
2799 #else /* not TRILIBRARY */
2804 #endif /* not TRILIBRARY */
2809 char workstring[FILENAMESIZE];
2812 poly = refine = quality = vararea = fixedarea = regionattrib = convex = 0;
2814 edgesout = voronoi = neighbors = geomview = 0;
2815 nobound = nopolywritten = nonodewritten = noelewritten = noiterationnum = 0;
2816 noholes = noexact = 0;
2817 incremental = sweepline = 0;
2826 quiet = verbose = 0;
2829 innodefilename[0] = '\0';
2830 #endif /* not TRILIBRARY */
2832 for ( i = STARTINDEX; i < argc; i++ ) {
2835 if ( argv[i][0] == '-' ) {
2836 #endif /* not TRILIBRARY */
2837 for ( j = STARTINDEX; argv[i][j] != '\0'; j++ ) {
2838 if ( argv[i][j] == 'p' ) {
2843 if ( argv[i][j] == 'r' ) {
2846 if ( argv[i][j] == 'q' ) {
2848 if ((( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' )) ||
2849 ( argv[i][j + 1] == '.' )) {
2851 while ((( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' )) ||
2852 ( argv[i][j + 1] == '.' )) {
2854 workstring[k] = argv[i][j];
2857 workstring[k] = '\0';
2858 minangle = (REAL) strtod( workstring, (char **) NULL );
2864 if ( argv[i][j] == 'a' ) {
2866 if ((( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' )) ||
2867 ( argv[i][j + 1] == '.' )) {
2870 while ((( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' )) ||
2871 ( argv[i][j + 1] == '.' )) {
2873 workstring[k] = argv[i][j];
2876 workstring[k] = '\0';
2877 maxarea = (REAL) strtod( workstring, (char **) NULL );
2878 if ( maxarea <= 0.0 ) {
2879 printf( "Error: Maximum area must be greater than zero.\n" );
2887 #endif /* not CDT_ONLY */
2888 if ( argv[i][j] == 'A' ) {
2891 if ( argv[i][j] == 'c' ) {
2894 if ( argv[i][j] == 'z' ) {
2897 if ( argv[i][j] == 'e' ) {
2900 if ( argv[i][j] == 'v' ) {
2903 if ( argv[i][j] == 'n' ) {
2906 if ( argv[i][j] == 'g' ) {
2909 if ( argv[i][j] == 'B' ) {
2912 if ( argv[i][j] == 'P' ) {
2915 if ( argv[i][j] == 'N' ) {
2918 if ( argv[i][j] == 'E' ) {
2923 if ( argv[i][j] == 'I' ) {
2926 #endif /* not TRILIBRARY */
2927 if ( argv[i][j] == 'O' ) {
2930 if ( argv[i][j] == 'X' ) {
2933 if ( argv[i][j] == 'o' ) {
2934 if ( argv[i][j + 1] == '2' ) {
2941 if ( argv[i][j] == 'Y' ) {
2944 if ( argv[i][j] == 'S' ) {
2946 while (( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' )) {
2948 steiner = steiner * 10 + (int) ( argv[i][j] - '0' );
2951 #endif /* not CDT_ONLY */
2954 if ( argv[i][j] == 'i' ) {
2957 if ( argv[i][j] == 'F' ) {
2960 #endif /* not REDUCED */
2961 if ( argv[i][j] == 'l' ) {
2968 if ( argv[i][j] == 's' ) {
2971 #endif /* not CDT_ONLY */
2972 if ( argv[i][j] == 'C' ) {
2975 #endif /* not REDUCED */
2976 if ( argv[i][j] == 'Q' ) {
2979 if ( argv[i][j] == 'V' ) {
2984 if (( argv[i][j] == 'h' ) || ( argv[i][j] == 'H' ) ||
2985 ( argv[i][j] == '?' )) {
2988 #endif /* not TRILIBRARY */
2993 strncpy( innodefilename, argv[i], FILENAMESIZE - 1 );
2994 innodefilename[FILENAMESIZE - 1] = '\0';
2996 #endif /* not TRILIBRARY */
3000 if ( innodefilename[0] == '\0' ) {
3003 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".node" )) {
3004 innodefilename[strlen( innodefilename ) - 5] = '\0';
3006 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".poly" )) {
3007 innodefilename[strlen( innodefilename ) - 5] = '\0';
3012 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 4], ".ele" )) {
3013 innodefilename[strlen( innodefilename ) - 4] = '\0';
3016 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".area" )) {
3017 innodefilename[strlen( innodefilename ) - 5] = '\0';
3022 #endif /* not CDT_ONLY */
3023 #endif /* not TRILIBRARY */
3024 steinerleft = steiner;
3025 useshelles = poly || refine || quality || convex;
3026 goodangle = (REAL)cos( minangle * PI / 180.0 );
3027 goodangle *= goodangle;
3028 if ( refine && noiterationnum ) {
3030 "Error: You cannot use the -I switch when refining a triangulation.\n" );
3033 /* Be careful not to allocate space for element area constraints that */
3034 /* will never be assigned any value (other than the default -1.0). */
3035 if ( !refine && !poly ) {
3038 /* Be careful not to add an extra attribute to each element unless the */
3039 /* input supports it (PSLG in, but not refining a preexisting mesh). */
3040 if ( refine || !poly ) {
3046 strcpy( inpolyfilename, innodefilename );
3047 strcpy( inelefilename, innodefilename );
3048 strcpy( areafilename, innodefilename );
3050 strcpy( workstring, innodefilename );
3052 while ( workstring[j] != '\0' ) {
3053 if (( workstring[j] == '.' ) && ( workstring[j + 1] != '\0' )) {
3059 if ( increment > 0 ) {
3062 if (( workstring[j] >= '0' ) && ( workstring[j] <= '9' )) {
3063 meshnumber = meshnumber * 10 + (int) ( workstring[j] - '0' );
3069 } while ( workstring[j] != '\0' );
3071 if ( noiterationnum ) {
3072 strcpy( outnodefilename, innodefilename );
3073 strcpy( outelefilename, innodefilename );
3074 strcpy( edgefilename, innodefilename );
3075 strcpy( vnodefilename, innodefilename );
3076 strcpy( vedgefilename, innodefilename );
3077 strcpy( neighborfilename, innodefilename );
3078 strcpy( offfilename, innodefilename );
3079 strcat( outnodefilename, ".node" );
3080 strcat( outelefilename, ".ele" );
3081 strcat( edgefilename, ".edge" );
3082 strcat( vnodefilename, ".v.node" );
3083 strcat( vedgefilename, ".v.edge" );
3084 strcat( neighborfilename, ".neigh" );
3085 strcat( offfilename, ".off" );
3087 else if ( increment == 0 ) {
3088 strcpy( outnodefilename, innodefilename );
3089 strcpy( outpolyfilename, innodefilename );
3090 strcpy( outelefilename, innodefilename );
3091 strcpy( edgefilename, innodefilename );
3092 strcpy( vnodefilename, innodefilename );
3093 strcpy( vedgefilename, innodefilename );
3094 strcpy( neighborfilename, innodefilename );
3095 strcpy( offfilename, innodefilename );
3096 strcat( outnodefilename, ".1.node" );
3097 strcat( outpolyfilename, ".1.poly" );
3098 strcat( outelefilename, ".1.ele" );
3099 strcat( edgefilename, ".1.edge" );
3100 strcat( vnodefilename, ".1.v.node" );
3101 strcat( vedgefilename, ".1.v.edge" );
3102 strcat( neighborfilename, ".1.neigh" );
3103 strcat( offfilename, ".1.off" );
3106 workstring[increment] = '%';
3107 workstring[increment + 1] = 'd';
3108 workstring[increment + 2] = '\0';
3109 sprintf( outnodefilename, workstring, meshnumber + 1 );
3110 strcpy( outpolyfilename, outnodefilename );
3111 strcpy( outelefilename, outnodefilename );
3112 strcpy( edgefilename, outnodefilename );
3113 strcpy( vnodefilename, outnodefilename );
3114 strcpy( vedgefilename, outnodefilename );
3115 strcpy( neighborfilename, outnodefilename );
3116 strcpy( offfilename, outnodefilename );
3117 strcat( outnodefilename, ".node" );
3118 strcat( outpolyfilename, ".poly" );
3119 strcat( outelefilename, ".ele" );
3120 strcat( edgefilename, ".edge" );
3121 strcat( vnodefilename, ".v.node" );
3122 strcat( vedgefilename, ".v.edge" );
3123 strcat( neighborfilename, ".neigh" );
3124 strcat( offfilename, ".off" );
3126 strcat( innodefilename, ".node" );
3127 strcat( inpolyfilename, ".poly" );
3128 strcat( inelefilename, ".ele" );
3129 strcat( areafilename, ".area" );
3130 #endif /* not TRILIBRARY */
3135 /********* User interaction routines begin here *********/
3137 /********* Debugging routines begin here *********/
3141 /*****************************************************************************/
3143 /* printtriangle() Print out the details of a triangle/edge handle. */
3145 /* I originally wrote this procedure to simplify debugging; it can be */
3146 /* called directly from the debugger, and presents information about a */
3147 /* triangle/edge handle in digestible form. It's also used when the */
3148 /* highest level of verbosity (`-VVV') is specified. */
3150 /*****************************************************************************/
3152 void printtriangle( t )
3155 struct triedge printtri;
3156 struct edge printsh;
3159 printf( "triangle x%lx with orientation %d:\n", (unsigned long) t->tri,
3161 decode( t->tri[0], printtri );
3162 if ( printtri.tri == dummytri ) {
3163 printf( " [0] = Outer space\n" );
3166 printf( " [0] = x%lx %d\n", (unsigned long) printtri.tri,
3169 decode( t->tri[1], printtri );
3170 if ( printtri.tri == dummytri ) {
3171 printf( " [1] = Outer space\n" );
3174 printf( " [1] = x%lx %d\n", (unsigned long) printtri.tri,
3177 decode( t->tri[2], printtri );
3178 if ( printtri.tri == dummytri ) {
3179 printf( " [2] = Outer space\n" );
3182 printf( " [2] = x%lx %d\n", (unsigned long) printtri.tri,
3185 org( *t, printpoint );
3186 if ( printpoint == (point) NULL ) {
3187 printf( " Origin[%d] = NULL\n", ( t->orient + 1 ) % 3 + 3 );
3190 printf( " Origin[%d] = x%lx (%.12g, %.12g)\n",
3191 ( t->orient + 1 ) % 3 + 3, (unsigned long) printpoint,
3192 printpoint[0], printpoint[1] );
3194 dest( *t, printpoint );
3195 if ( printpoint == (point) NULL ) {
3196 printf( " Dest [%d] = NULL\n", ( t->orient + 2 ) % 3 + 3 );
3199 printf( " Dest [%d] = x%lx (%.12g, %.12g)\n",
3200 ( t->orient + 2 ) % 3 + 3, (unsigned long) printpoint,
3201 printpoint[0], printpoint[1] );
3203 apex( *t, printpoint );
3204 if ( printpoint == (point) NULL ) {
3205 printf( " Apex [%d] = NULL\n", t->orient + 3 );
3208 printf( " Apex [%d] = x%lx (%.12g, %.12g)\n",
3209 t->orient + 3, (unsigned long) printpoint,
3210 printpoint[0], printpoint[1] );
3213 sdecode( t->tri[6], printsh );
3214 if ( printsh.sh != dummysh ) {
3215 printf( " [6] = x%lx %d\n", (unsigned long) printsh.sh,
3218 sdecode( t->tri[7], printsh );
3219 if ( printsh.sh != dummysh ) {
3220 printf( " [7] = x%lx %d\n", (unsigned long) printsh.sh,
3223 sdecode( t->tri[8], printsh );
3224 if ( printsh.sh != dummysh ) {
3225 printf( " [8] = x%lx %d\n", (unsigned long) printsh.sh,
3230 printf( " Area constraint: %.4g\n", areabound( *t ));
3234 /*****************************************************************************/
3236 /* printshelle() Print out the details of a shell edge handle. */
3238 /* I originally wrote this procedure to simplify debugging; it can be */
3239 /* called directly from the debugger, and presents information about a */
3240 /* shell edge handle in digestible form. It's also used when the highest */
3241 /* level of verbosity (`-VVV') is specified. */
3243 /*****************************************************************************/
3245 void printshelle( s )
3248 struct edge printsh;
3249 struct triedge printtri;
3252 printf( "shell edge x%lx with orientation %d and mark %d:\n",
3253 (unsigned long) s->sh, s->shorient, mark( *s ));
3254 sdecode( s->sh[0], printsh );
3255 if ( printsh.sh == dummysh ) {
3256 printf( " [0] = No shell\n" );
3259 printf( " [0] = x%lx %d\n", (unsigned long) printsh.sh,
3262 sdecode( s->sh[1], printsh );
3263 if ( printsh.sh == dummysh ) {
3264 printf( " [1] = No shell\n" );
3267 printf( " [1] = x%lx %d\n", (unsigned long) printsh.sh,
3270 sorg( *s, printpoint );
3271 if ( printpoint == (point) NULL ) {
3272 printf( " Origin[%d] = NULL\n", 2 + s->shorient );
3275 printf( " Origin[%d] = x%lx (%.12g, %.12g)\n",
3276 2 + s->shorient, (unsigned long) printpoint,
3277 printpoint[0], printpoint[1] );
3279 sdest( *s, printpoint );
3280 if ( printpoint == (point) NULL ) {
3281 printf( " Dest [%d] = NULL\n", 3 - s->shorient );
3284 printf( " Dest [%d] = x%lx (%.12g, %.12g)\n",
3285 3 - s->shorient, (unsigned long) printpoint,
3286 printpoint[0], printpoint[1] );
3288 decode( s->sh[4], printtri );
3289 if ( printtri.tri == dummytri ) {
3290 printf( " [4] = Outer space\n" );
3293 printf( " [4] = x%lx %d\n", (unsigned long) printtri.tri,
3296 decode( s->sh[5], printtri );
3297 if ( printtri.tri == dummytri ) {
3298 printf( " [5] = Outer space\n" );
3301 printf( " [5] = x%lx %d\n", (unsigned long) printtri.tri,
3308 /********* Debugging routines end here *********/
3310 /********* Memory management routines begin here *********/
3314 /*****************************************************************************/
3316 /* poolinit() Initialize a pool of memory for allocation of items. */
3318 /* This routine initializes the machinery for allocating items. A `pool' */
3319 /* is created whose records have size at least `bytecount'. Items will be */
3320 /* allocated in `itemcount'-item blocks. Each item is assumed to be a */
3321 /* collection of words, and either pointers or floating-point values are */
3322 /* assumed to be the "primary" word type. (The "primary" word type is used */
3323 /* to determine alignment of items.) If `alignment' isn't zero, all items */
3324 /* will be `alignment'-byte aligned in memory. `alignment' must be either */
3325 /* a multiple or a factor of the primary word size; powers of two are safe. */
3326 /* `alignment' is normally used to create a few unused bits at the bottom */
3327 /* of each item's pointer, in which information may be stored. */
3329 /* Don't change this routine unless you understand it. */
3331 /*****************************************************************************/
3333 void poolinit( pool, bytecount, itemcount, wtype, alignment )
3334 struct memorypool *pool;
3337 enum wordtype wtype;
3342 /* Initialize values in the pool. */
3343 pool->itemwordtype = wtype;
3344 wordsize = ( pool->itemwordtype == POINTER ) ? sizeof( VOID * ) : sizeof( REAL );
3345 /* Find the proper alignment, which must be at least as large as: */
3346 /* - The parameter `alignment'. */
3347 /* - The primary word type, to avoid unaligned accesses. */
3348 /* - sizeof(VOID *), so the stack of dead items can be maintained */
3349 /* without unaligned accesses. */
3350 if ( alignment > wordsize ) {
3351 pool->alignbytes = alignment;
3354 pool->alignbytes = wordsize;
3356 if ( sizeof( VOID * ) > pool->alignbytes ) {
3357 pool->alignbytes = sizeof( VOID * );
3359 pool->itemwords = (( bytecount + pool->alignbytes - 1 ) / pool->alignbytes )
3360 * ( pool->alignbytes / wordsize );
3361 pool->itembytes = pool->itemwords * wordsize;
3362 pool->itemsperblock = itemcount;
3364 /* Allocate a block of items. Space for `itemsperblock' items and one */
3365 /* pointer (to point to the next block) are allocated, as well as space */
3366 /* to ensure alignment of the items. */
3367 pool->firstblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes
3368 + sizeof( VOID * ) + pool->alignbytes );
3369 if ( pool->firstblock == (VOID **) NULL ) {
3370 printf( "Error: Out of memory.\n" );
3373 /* Set the next block pointer to NULL. */
3374 *( pool->firstblock ) = (VOID *) NULL;
3375 poolrestart( pool );
3378 /*****************************************************************************/
3380 /* poolrestart() Deallocate all items in a pool. */
3382 /* The pool is returned to its starting state, except that no memory is */
3383 /* freed to the operating system. Rather, the previously allocated blocks */
3384 /* are ready to be reused. */
3386 /*****************************************************************************/
3388 void poolrestart( pool )
3389 struct memorypool *pool;
3391 unsigned long alignptr;
3396 /* Set the currently active block. */
3397 pool->nowblock = pool->firstblock;
3398 /* Find the first item in the pool. Increment by the size of (VOID *). */
3399 alignptr = (unsigned long) ( pool->nowblock + 1 );
3400 /* Align the item on an `alignbytes'-byte boundary. */
3401 pool->nextitem = (VOID *)
3402 ( alignptr + (unsigned long) pool->alignbytes
3403 - ( alignptr % (unsigned long) pool->alignbytes ));
3404 /* There are lots of unallocated items left in this block. */
3405 pool->unallocateditems = pool->itemsperblock;
3406 /* The stack of deallocated items is empty. */
3407 pool->deaditemstack = (VOID *) NULL;
3410 /*****************************************************************************/
3412 /* pooldeinit() Free to the operating system all memory taken by a pool. */
3414 /*****************************************************************************/
3416 void pooldeinit( pool )
3417 struct memorypool *pool;
3419 while ( pool->firstblock != (VOID **) NULL ) {
3420 pool->nowblock = (VOID **) *( pool->firstblock );
3421 free( pool->firstblock );
3422 pool->firstblock = pool->nowblock;
3426 /*****************************************************************************/
3428 /* poolalloc() Allocate space for an item. */
3430 /*****************************************************************************/
3432 VOID *poolalloc( pool )
3433 struct memorypool *pool;
3437 unsigned long alignptr;
3439 /* First check the linked list of dead items. If the list is not */
3440 /* empty, allocate an item from the list rather than a fresh one. */
3441 if ( pool->deaditemstack != (VOID *) NULL ) {
3442 newitem = pool->deaditemstack; /* Take first item in list. */
3443 pool->deaditemstack = *(VOID **) pool->deaditemstack;
3446 /* Check if there are any free items left in the current block. */
3447 if ( pool->unallocateditems == 0 ) {
3448 /* Check if another block must be allocated. */
3449 if ( *( pool->nowblock ) == (VOID *) NULL ) {
3450 /* Allocate a new block of items, pointed to by the previous block. */
3451 newblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes
3452 + sizeof( VOID * ) + pool->alignbytes );
3453 if ( newblock == (VOID **) NULL ) {
3454 printf( "Error: Out of memory.\n" );
3457 *( pool->nowblock ) = (VOID *) newblock;
3458 /* The next block pointer is NULL. */
3459 *newblock = (VOID *) NULL;
3461 /* Move to the new block. */
3462 pool->nowblock = (VOID **) *( pool->nowblock );
3463 /* Find the first item in the block. */
3464 /* Increment by the size of (VOID *). */
3465 alignptr = (unsigned long) ( pool->nowblock + 1 );
3466 /* Align the item on an `alignbytes'-byte boundary. */
3467 pool->nextitem = (VOID *)
3468 ( alignptr + (unsigned long) pool->alignbytes
3469 - ( alignptr % (unsigned long) pool->alignbytes ));
3470 /* There are lots of unallocated items left in this block. */
3471 pool->unallocateditems = pool->itemsperblock;
3473 /* Allocate a new item. */
3474 newitem = pool->nextitem;
3475 /* Advance `nextitem' pointer to next free item in block. */
3476 if ( pool->itemwordtype == POINTER ) {
3477 pool->nextitem = (VOID *) ((VOID **) pool->nextitem + pool->itemwords );
3480 pool->nextitem = (VOID *) ((REAL *) pool->nextitem + pool->itemwords );
3482 pool->unallocateditems--;
3489 /*****************************************************************************/
3491 /* pooldealloc() Deallocate space for an item. */
3493 /* The deallocated space is stored in a queue for later reuse. */
3495 /*****************************************************************************/
3497 void pooldealloc( pool, dyingitem )
3498 struct memorypool *pool;
3501 /* Push freshly killed item onto stack. */
3502 *((VOID **) dyingitem ) = pool->deaditemstack;
3503 pool->deaditemstack = dyingitem;
3507 /*****************************************************************************/
3509 /* traversalinit() Prepare to traverse the entire list of items. */
3511 /* This routine is used in conjunction with traverse(). */
3513 /*****************************************************************************/
3515 void traversalinit( pool )
3516 struct memorypool *pool;
3518 unsigned long alignptr;
3520 /* Begin the traversal in the first block. */
3521 pool->pathblock = pool->firstblock;
3522 /* Find the first item in the block. Increment by the size of (VOID *). */
3523 alignptr = (unsigned long) ( pool->pathblock + 1 );
3524 /* Align with item on an `alignbytes'-byte boundary. */
3525 pool->pathitem = (VOID *)
3526 ( alignptr + (unsigned long) pool->alignbytes
3527 - ( alignptr % (unsigned long) pool->alignbytes ));
3528 /* Set the number of items left in the current block. */
3529 pool->pathitemsleft = pool->itemsperblock;
3532 /*****************************************************************************/
3534 /* traverse() Find the next item in the list. */
3536 /* This routine is used in conjunction with traversalinit(). Be forewarned */
3537 /* that this routine successively returns all items in the list, including */
3538 /* deallocated ones on the deaditemqueue. It's up to you to figure out */
3539 /* which ones are actually dead. Why? I don't want to allocate extra */
3540 /* space just to demarcate dead items. It can usually be done more */
3541 /* space-efficiently by a routine that knows something about the structure */
3544 /*****************************************************************************/
3546 VOID *traverse( pool )
3547 struct memorypool *pool;
3550 unsigned long alignptr;
3552 /* Stop upon exhausting the list of items. */
3553 if ( pool->pathitem == pool->nextitem ) {
3554 return (VOID *) NULL;
3556 /* Check whether any untraversed items remain in the current block. */
3557 if ( pool->pathitemsleft == 0 ) {
3558 /* Find the next block. */
3559 pool->pathblock = (VOID **) *( pool->pathblock );
3560 /* Find the first item in the block. Increment by the size of (VOID *). */
3561 alignptr = (unsigned long) ( pool->pathblock + 1 );
3562 /* Align with item on an `alignbytes'-byte boundary. */
3563 pool->pathitem = (VOID *)
3564 ( alignptr + (unsigned long) pool->alignbytes
3565 - ( alignptr % (unsigned long) pool->alignbytes ));
3566 /* Set the number of items left in the current block. */
3567 pool->pathitemsleft = pool->itemsperblock;
3569 newitem = pool->pathitem;
3570 /* Find the next item in the block. */
3571 if ( pool->itemwordtype == POINTER ) {
3572 pool->pathitem = (VOID *) ((VOID **) pool->pathitem + pool->itemwords );
3575 pool->pathitem = (VOID *) ((REAL *) pool->pathitem + pool->itemwords );
3577 pool->pathitemsleft--;
3581 /*****************************************************************************/
3583 /* dummyinit() Initialize the triangle that fills "outer space" and the */
3584 /* omnipresent shell edge. */
3586 /* The triangle that fills "outer space", called `dummytri', is pointed to */
3587 /* by every triangle and shell edge on a boundary (be it outer or inner) of */
3588 /* the triangulation. Also, `dummytri' points to one of the triangles on */
3589 /* the convex hull (until the holes and concavities are carved), making it */
3590 /* possible to find a starting triangle for point location. */
3592 /* The omnipresent shell edge, `dummysh', is pointed to by every triangle */
3593 /* or shell edge that doesn't have a full complement of real shell edges */
3596 /*****************************************************************************/
3598 void dummyinit( trianglewords, shellewords )
3602 unsigned long alignptr;
3604 /* `triwords' and `shwords' are used by the mesh manipulation primitives */
3605 /* to extract orientations of triangles and shell edges from pointers. */
3606 triwords = trianglewords; /* Initialize `triwords' once and for all. */
3607 shwords = shellewords; /* Initialize `shwords' once and for all. */
3609 /* Set up `dummytri', the `triangle' that occupies "outer space". */
3610 dummytribase = (triangle *) malloc( triwords * sizeof( triangle )
3611 + triangles.alignbytes );
3612 if ( dummytribase == (triangle *) NULL ) {
3613 printf( "Error: Out of memory.\n" );
3616 /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */
3617 alignptr = (unsigned long) dummytribase;
3618 dummytri = (triangle *)
3619 ( alignptr + (unsigned long) triangles.alignbytes
3620 - ( alignptr % (unsigned long) triangles.alignbytes ));
3621 /* Initialize the three adjoining triangles to be "outer space". These */
3622 /* will eventually be changed by various bonding operations, but their */
3623 /* values don't really matter, as long as they can legally be */
3625 dummytri[0] = (triangle) dummytri;
3626 dummytri[1] = (triangle) dummytri;
3627 dummytri[2] = (triangle) dummytri;
3628 /* Three NULL vertex points. */
3629 dummytri[3] = (triangle) NULL;
3630 dummytri[4] = (triangle) NULL;
3631 dummytri[5] = (triangle) NULL;
3634 /* Set up `dummysh', the omnipresent "shell edge" pointed to by any */
3635 /* triangle side or shell edge end that isn't attached to a real shell */
3637 dummyshbase = (shelle *) malloc( shwords * sizeof( shelle )
3638 + shelles.alignbytes );
3639 if ( dummyshbase == (shelle *) NULL ) {
3640 printf( "Error: Out of memory.\n" );
3643 /* Align `dummysh' on a `shelles.alignbytes'-byte boundary. */
3644 alignptr = (unsigned long) dummyshbase;
3645 dummysh = (shelle *)
3646 ( alignptr + (unsigned long) shelles.alignbytes
3647 - ( alignptr % (unsigned long) shelles.alignbytes ));
3648 /* Initialize the two adjoining shell edges to be the omnipresent shell */
3649 /* edge. These will eventually be changed by various bonding */
3650 /* operations, but their values don't really matter, as long as they */
3651 /* can legally be dereferenced. */
3652 dummysh[0] = (shelle) dummysh;
3653 dummysh[1] = (shelle) dummysh;
3654 /* Two NULL vertex points. */
3655 dummysh[2] = (shelle) NULL;
3656 dummysh[3] = (shelle) NULL;
3657 /* Initialize the two adjoining triangles to be "outer space". */
3658 dummysh[4] = (shelle) dummytri;
3659 dummysh[5] = (shelle) dummytri;
3660 /* Set the boundary marker to zero. */
3661 *(int *) ( dummysh + 6 ) = 0;
3663 /* Initialize the three adjoining shell edges of `dummytri' to be */
3664 /* the omnipresent shell edge. */
3665 dummytri[6] = (triangle) dummysh;
3666 dummytri[7] = (triangle) dummysh;
3667 dummytri[8] = (triangle) dummysh;
3671 /*****************************************************************************/
3673 /* initializepointpool() Calculate the size of the point data structure */
3674 /* and initialize its memory pool. */
3676 /* This routine also computes the `pointmarkindex' and `point2triindex' */
3677 /* indices used to find values within each point. */
3679 /*****************************************************************************/
3681 void initializepointpool(){
3684 /* The index within each point at which the boundary marker is found. */
3685 /* Ensure the point marker is aligned to a sizeof(int)-byte address. */
3686 pointmarkindex = (( mesh_dim + nextras ) * sizeof( REAL ) + sizeof( int ) - 1 )
3688 pointsize = ( pointmarkindex + 1 ) * sizeof( int );
3690 /* The index within each point at which a triangle pointer is found. */
3691 /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */
3692 point2triindex = ( pointsize + sizeof( triangle ) - 1 ) / sizeof( triangle );
3693 pointsize = ( point2triindex + 1 ) * sizeof( triangle );
3695 /* Initialize the pool of points. */
3696 poolinit( &points, pointsize, POINTPERBLOCK,
3697 ( sizeof( REAL ) >= sizeof( triangle )) ? FLOATINGPOINT : POINTER, 0 );
3700 /*****************************************************************************/
3702 /* initializetrisegpools() Calculate the sizes of the triangle and shell */
3703 /* edge data structures and initialize their */
3706 /* This routine also computes the `highorderindex', `elemattribindex', and */
3707 /* `areaboundindex' indices used to find values within each triangle. */
3709 /*****************************************************************************/
3711 void initializetrisegpools(){
3714 /* The index within each triangle at which the extra nodes (above three) */
3715 /* associated with high order elements are found. There are three */
3716 /* pointers to other triangles, three pointers to corners, and possibly */
3717 /* three pointers to shell edges before the extra nodes. */
3718 highorderindex = 6 + ( useshelles * 3 );
3719 /* The number of bytes occupied by a triangle. */
3720 trisize = (( order + 1 ) * ( order + 2 ) / 2 + ( highorderindex - 3 )) *
3722 /* The index within each triangle at which its attributes are found, */
3723 /* where the index is measured in REALs. */
3724 elemattribindex = ( trisize + sizeof( REAL ) - 1 ) / sizeof( REAL );
3725 /* The index within each triangle at which the maximum area constraint */
3726 /* is found, where the index is measured in REALs. Note that if the */
3727 /* `regionattrib' flag is set, an additional attribute will be added. */
3728 areaboundindex = elemattribindex + eextras + regionattrib;
3729 /* If triangle attributes or an area bound are needed, increase the number */
3730 /* of bytes occupied by a triangle. */
3732 trisize = ( areaboundindex + 1 ) * sizeof( REAL );
3734 else if ( eextras + regionattrib > 0 ) {
3735 trisize = areaboundindex * sizeof( REAL );
3737 /* If a Voronoi diagram or triangle neighbor graph is requested, make */
3738 /* sure there's room to store an integer index in each triangle. This */
3739 /* integer index can occupy the same space as the shell edges or */
3740 /* attributes or area constraint or extra nodes. */
3741 if (( voronoi || neighbors ) &&
3742 ( trisize < 6 * sizeof( triangle ) + sizeof( int ))) {
3743 trisize = 6 * sizeof( triangle ) + sizeof( int );
3745 /* Having determined the memory size of a triangle, initialize the pool. */
3746 poolinit( &triangles, trisize, TRIPERBLOCK, POINTER, 4 );
3749 /* Initialize the pool of shell edges. */
3750 poolinit( &shelles, 6 * sizeof( triangle ) + sizeof( int ), SHELLEPERBLOCK,
3753 /* Initialize the "outer space" triangle and omnipresent shell edge. */
3754 dummyinit( triangles.itemwords, shelles.itemwords );
3757 /* Initialize the "outer space" triangle. */
3758 dummyinit( triangles.itemwords, 0 );
3762 /*****************************************************************************/
3764 /* triangledealloc() Deallocate space for a triangle, marking it dead. */
3766 /*****************************************************************************/
3768 void triangledealloc( dyingtriangle )
3769 triangle * dyingtriangle;
3771 /* Set triangle's vertices to NULL. This makes it possible to */
3772 /* detect dead triangles when traversing the list of all triangles. */
3773 dyingtriangle[3] = (triangle) NULL;
3774 dyingtriangle[4] = (triangle) NULL;
3775 dyingtriangle[5] = (triangle) NULL;
3776 pooldealloc( &triangles, (VOID *) dyingtriangle );
3779 /*****************************************************************************/
3781 /* triangletraverse() Traverse the triangles, skipping dead ones. */
3783 /*****************************************************************************/
3785 triangle *triangletraverse(){
3786 triangle *newtriangle;
3789 newtriangle = (triangle *) traverse( &triangles );
3790 if ( newtriangle == (triangle *) NULL ) {
3791 return (triangle *) NULL;
3793 } while ( newtriangle[3] == (triangle) NULL ); /* Skip dead ones. */
3797 /*****************************************************************************/
3799 /* shelledealloc() Deallocate space for a shell edge, marking it dead. */
3801 /*****************************************************************************/
3803 void shelledealloc( dyingshelle )
3804 shelle * dyingshelle;
3806 /* Set shell edge's vertices to NULL. This makes it possible to */
3807 /* detect dead shells when traversing the list of all shells. */
3808 dyingshelle[2] = (shelle) NULL;
3809 dyingshelle[3] = (shelle) NULL;
3810 pooldealloc( &shelles, (VOID *) dyingshelle );
3813 /*****************************************************************************/
3815 /* shelletraverse() Traverse the shell edges, skipping dead ones. */
3817 /*****************************************************************************/
3819 shelle *shelletraverse(){
3823 newshelle = (shelle *) traverse( &shelles );
3824 if ( newshelle == (shelle *) NULL ) {
3825 return (shelle *) NULL;
3827 } while ( newshelle[2] == (shelle) NULL ); /* Skip dead ones. */
3831 /*****************************************************************************/
3833 /* pointdealloc() Deallocate space for a point, marking it dead. */
3835 /*****************************************************************************/
3837 void pointdealloc( dyingpoint )
3840 /* Mark the point as dead. This makes it possible to detect dead points */
3841 /* when traversing the list of all points. */
3842 setpointmark( dyingpoint, DEADPOINT );
3843 pooldealloc( &points, (VOID *) dyingpoint );
3846 /*****************************************************************************/
3848 /* pointtraverse() Traverse the points, skipping dead ones. */
3850 /*****************************************************************************/
3852 point pointtraverse(){
3856 newpoint = (point) traverse( &points );
3857 if ( newpoint == (point) NULL ) {
3858 return (point) NULL;
3860 } while ( pointmark( newpoint ) == DEADPOINT ); /* Skip dead ones. */
3864 /*****************************************************************************/
3866 /* badsegmentdealloc() Deallocate space for a bad segment, marking it */
3869 /*****************************************************************************/
3874 void badsegmentdealloc( dyingseg )
3875 struct edge *dyingseg;
3877 /* Set segment's orientation to -1. This makes it possible to */
3878 /* detect dead segments when traversing the list of all segments. */
3879 dyingseg->shorient = -1;
3880 pooldealloc( &badsegments, (VOID *) dyingseg );
3883 #endif /* not CDT_ONLY */
3885 /*****************************************************************************/
3887 /* badsegmenttraverse() Traverse the bad segments, skipping dead ones. */
3889 /*****************************************************************************/
3894 struct edge *badsegmenttraverse(){
3895 struct edge *newseg;
3898 newseg = (struct edge *) traverse( &badsegments );
3899 if ( newseg == (struct edge *) NULL ) {
3900 return (struct edge *) NULL;
3902 } while ( newseg->shorient == -1 ); /* Skip dead ones. */
3906 #endif /* not CDT_ONLY */
3908 /*****************************************************************************/
3910 /* getpoint() Get a specific point, by number, from the list. */
3912 /* The first point is number 'firstnumber'. */
3914 /* Note that this takes O(n) time (with a small constant, if POINTPERBLOCK */
3915 /* is large). I don't care to take the trouble to make it work in constant */
3918 /*****************************************************************************/
3920 point getpoint( number )
3925 unsigned long alignptr;
3928 getblock = points.firstblock;
3929 current = firstnumber;
3930 /* Find the right block. */
3931 while ( current + points.itemsperblock <= number ) {
3932 getblock = (VOID **) *getblock;
3933 current += points.itemsperblock;
3935 /* Now find the right point. */
3936 alignptr = (unsigned long) ( getblock + 1 );
3937 foundpoint = (point) ( alignptr + (unsigned long) points.alignbytes
3938 - ( alignptr % (unsigned long) points.alignbytes ));
3939 while ( current < number ) {
3940 foundpoint += points.itemwords;
3946 /*****************************************************************************/
3948 /* triangledeinit() Free all remaining allocated memory. */
3950 /*****************************************************************************/
3952 void triangledeinit(){
3953 pooldeinit( &triangles );
3954 free( dummytribase );
3956 pooldeinit( &shelles );
3957 free( dummyshbase );
3959 pooldeinit( &points );
3963 pooldeinit( &badsegments );
3964 if (( minangle > 0.0 ) || vararea || fixedarea ) {
3965 pooldeinit( &badtriangles );
3968 #endif /* not CDT_ONLY */
3973 /********* Memory management routines end here *********/
3975 /********* Constructors begin here *********/
3979 /*****************************************************************************/
3981 /* maketriangle() Create a new triangle with orientation zero. */
3983 /*****************************************************************************/
3985 void maketriangle( newtriedge )
3986 struct triedge *newtriedge;
3990 newtriedge->tri = (triangle *) poolalloc( &triangles );
3991 /* Initialize the three adjoining triangles to be "outer space". */
3992 newtriedge->tri[0] = (triangle) dummytri;
3993 newtriedge->tri[1] = (triangle) dummytri;
3994 newtriedge->tri[2] = (triangle) dummytri;
3995 /* Three NULL vertex points. */
3996 newtriedge->tri[3] = (triangle) NULL;
3997 newtriedge->tri[4] = (triangle) NULL;
3998 newtriedge->tri[5] = (triangle) NULL;
3999 /* Initialize the three adjoining shell edges to be the omnipresent */
4002 newtriedge->tri[6] = (triangle) dummysh;
4003 newtriedge->tri[7] = (triangle) dummysh;
4004 newtriedge->tri[8] = (triangle) dummysh;
4006 for ( i = 0; i < eextras; i++ ) {
4007 setelemattribute( *newtriedge, i, 0.0 );
4010 setareabound( *newtriedge, -1.0 );
4013 newtriedge->orient = 0;
4016 /*****************************************************************************/
4018 /* makeshelle() Create a new shell edge with orientation zero. */
4020 /*****************************************************************************/
4022 void makeshelle( newedge )
4023 struct edge *newedge;
4025 newedge->sh = (shelle *) poolalloc( &shelles );
4026 /* Initialize the two adjoining shell edges to be the omnipresent */
4028 newedge->sh[0] = (shelle) dummysh;
4029 newedge->sh[1] = (shelle) dummysh;
4030 /* Two NULL vertex points. */
4031 newedge->sh[2] = (shelle) NULL;
4032 newedge->sh[3] = (shelle) NULL;
4033 /* Initialize the two adjoining triangles to be "outer space". */
4034 newedge->sh[4] = (shelle) dummytri;
4035 newedge->sh[5] = (shelle) dummytri;
4036 /* Set the boundary marker to zero. */
4037 setmark( *newedge, 0 );
4039 newedge->shorient = 0;
4044 /********* Constructors end here *********/
4046 /********* Determinant evaluation routines begin here *********/
4050 /* The adaptive exact arithmetic geometric predicates implemented herein are */
4051 /* described in detail in my Technical Report CMU-CS-96-140. The complete */
4052 /* reference is given in the header. */
4054 /* Which of the following two methods of finding the absolute values is */
4055 /* fastest is compiler-dependent. A few compilers can inline and optimize */
4056 /* the fabs() call; but most will incur the overhead of a function call, */
4057 /* which is disastrously slow. A faster way on IEEE machines might be to */
4058 /* mask the appropriate bit, but that's difficult to do in C. */
4061 Absolute( a ) (( a ) >= 0.0 ? ( a ) : -( a ))
4062 /* #define Absolute(a) fabs(a) */
4064 /* Many of the operations are broken up into two pieces, a main part that */
4065 /* performs an approximate operation, and a "tail" that computes the */
4066 /* roundoff error of that operation. */
4068 /* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */
4069 /* Split(), and Two_Product() are all implemented as described in the */
4070 /* reference. Each of these macros requires certain variables to be */
4071 /* defined in the calling routine. The variables `bvirt', `c', `abig', */
4072 /* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */
4073 /* they store the result of an operation that may incur roundoff error. */
4074 /* The input parameter `x' (or the highest numbered `x_' parameter) must */
4075 /* also be declared `INEXACT'. */
4078 Fast_Two_Sum_Tail( a, b, x, y ) \
4083 Fast_Two_Sum( a, b, x, y ) \
4084 x = (REAL) ( a + b ); \
4085 Fast_Two_Sum_Tail( a, b, x, y )
4088 Two_Sum_Tail( a, b, x, y ) \
4089 bvirt = (REAL) ( x - a ); \
4090 avirt = x - bvirt; \
4091 bround = b - bvirt; \
4092 around = a - avirt; \
4096 Two_Sum( a, b, x, y ) \
4097 x = (REAL) ( a + b ); \
4098 Two_Sum_Tail( a, b, x, y )
4101 Two_Diff_Tail( a, b, x, y ) \
4102 bvirt = (REAL) ( a - x ); \
4103 avirt = x + bvirt; \
4104 bround = bvirt - b; \
4105 around = a - avirt; \
4109 Two_Diff( a, b, x, y ) \
4110 x = (REAL) ( a - b ); \
4111 Two_Diff_Tail( a, b, x, y )
4114 Split( a, ahi, alo ) \
4115 c = (REAL) ( splitter * a ); \
4116 abig = (REAL) ( c - a ); \
4117 ahi = (REAL)( c - abig ); \
4118 alo = (REAL)( a - ahi )
4121 Two_Product_Tail( a, b, x, y ) \
4122 Split( a, ahi, alo ); \
4123 Split( b, bhi, blo ); \
4124 err1 = x - ( ahi * bhi ); \
4125 err2 = err1 - ( alo * bhi ); \
4126 err3 = err2 - ( ahi * blo ); \
4127 y = ( alo * blo ) - err3
4130 Two_Product( a, b, x, y ) \
4131 x = (REAL) ( a * b ); \
4132 Two_Product_Tail( a, b, x, y )
4134 /* Two_Product_Presplit() is Two_Product() where one of the inputs has */
4135 /* already been split. Avoids redundant splitting. */
4138 Two_Product_Presplit( a, b, bhi, blo, x, y ) \
4139 x = (REAL) ( a * b ); \
4140 Split( a, ahi, alo ); \
4141 err1 = x - ( ahi * bhi ); \
4142 err2 = err1 - ( alo * bhi ); \
4143 err3 = err2 - ( ahi * blo ); \
4144 y = ( alo * blo ) - err3
4146 /* Square() can be done more quickly than Two_Product(). */
4149 Square_Tail( a, x, y ) \
4150 Split( a, ahi, alo ); \
4151 err1 = x - ( ahi * ahi ); \
4152 err3 = err1 - (( ahi + ahi ) * alo ); \
4153 y = ( alo * alo ) - err3
4157 x = (REAL) ( a * a ); \
4158 Square_Tail( a, x, y )
4160 /* Macros for summing expansions of various fixed lengths. These are all */
4161 /* unrolled versions of Expansion_Sum(). */
4164 Two_One_Sum( a1, a0, b, x2, x1, x0 ) \
4165 Two_Sum( a0, b, _i, x0 ); \
4166 Two_Sum( a1, _i, x2, x1 )
4169 Two_One_Diff( a1, a0, b, x2, x1, x0 ) \
4170 Two_Diff( a0, b, _i, x0 ); \
4171 Two_Sum( a1, _i, x2, x1 )
4174 Two_Two_Sum( a1, a0, b1, b0, x3, x2, x1, x0 ) \
4175 Two_One_Sum( a1, a0, b0, _j, _0, x0 ); \
4176 Two_One_Sum( _j, _0, b1, x3, x2, x1 )
4179 Two_Two_Diff( a1, a0, b1, b0, x3, x2, x1, x0 ) \
4180 Two_One_Diff( a1, a0, b0, _j, _0, x0 ); \
4181 Two_One_Diff( _j, _0, b1, x3, x2, x1 )
4183 /*****************************************************************************/
4185 /* exactinit() Initialize the variables used for exact arithmetic. */
4187 /* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */
4188 /* floating-point arithmetic. `epsilon' bounds the relative roundoff */
4189 /* error. It is used for floating-point error analysis. */
4191 /* `splitter' is used to split floating-point numbers into two half- */
4192 /* length significands for exact multiplication. */
4194 /* I imagine that a highly optimizing compiler might be too smart for its */
4195 /* own good, and somehow cause this routine to fail, if it pretends that */
4196 /* floating-point arithmetic is too much like real arithmetic. */
4198 /* Don't change this routine unless you fully understand it. */
4200 /*****************************************************************************/
4204 REAL check, lastcheck;
4212 /* Repeatedly divide `epsilon' by two until it is too small to add to */
4213 /* one without causing roundoff. (Also check if the sum is equal to */
4214 /* the previous sum, for machines that round up instead of using exact */
4215 /* rounding. Not that these routines will work on such machines anyway. */
4219 if ( every_other ) {
4222 every_other = !every_other;
4223 check = (REAL)( 1.0 + epsilon );
4224 } while (( check != 1.0 ) && ( check != lastcheck ));
4226 if ( verbose > 1 ) {
4227 printf( "Floating point roundoff is of magnitude %.17g\n", epsilon );
4228 printf( "Floating point splitter is %.17g\n", splitter );
4230 /* Error bounds for orientation and incircle tests. */
4231 resulterrbound = (REAL)(( 3.0 + 8.0 * epsilon ) * epsilon );
4232 ccwerrboundA = (REAL)(( 3.0 + 16.0 * epsilon ) * epsilon );
4233 ccwerrboundB = (REAL)(( 2.0 + 12.0 * epsilon ) * epsilon );
4234 ccwerrboundC = (REAL)(( 9.0 + 64.0 * epsilon ) * epsilon * epsilon );
4235 iccerrboundA = (REAL)(( 10.0 + 96.0 * epsilon ) * epsilon );
4236 iccerrboundB = (REAL)(( 4.0 + 48.0 * epsilon ) * epsilon );
4237 iccerrboundC = (REAL)(( 44.0 + 576.0 * epsilon ) * epsilon * epsilon );
4240 /*****************************************************************************/
4242 /* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */
4243 /* components from the output expansion. */
4245 /* Sets h = e + f. See my Robust Predicates paper for details. */
4247 /* If round-to-even is used (as with IEEE 754), maintains the strongly */
4248 /* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */
4249 /* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */
4252 /*****************************************************************************/
4254 int fast_expansion_sum_zeroelim( elen, e, flen, f, h ) /* h cannot be e or f. */
4265 REAL avirt, bround, around;
4266 int eindex, findex, hindex;
4271 eindex = findex = 0;
4272 if (( fnow > enow ) == ( fnow > -enow )) {
4281 if (( eindex < elen ) && ( findex < flen )) {
4282 if (( fnow > enow ) == ( fnow > -enow )) {
4283 Fast_Two_Sum( enow, Q, Qnew, hh );
4287 Fast_Two_Sum( fnow, Q, Qnew, hh );
4294 while (( eindex < elen ) && ( findex < flen )) {
4295 if (( fnow > enow ) == ( fnow > -enow )) {
4296 Two_Sum( Q, enow, Qnew, hh );
4300 Two_Sum( Q, fnow, Qnew, hh );
4309 while ( eindex < elen ) {
4310 Two_Sum( Q, enow, Qnew, hh );
4317 while ( findex < flen ) {
4318 Two_Sum( Q, fnow, Qnew, hh );
4325 if (( Q != 0.0 ) || ( hindex == 0 )) {
4331 /*****************************************************************************/
4333 /* scale_expansion_zeroelim() Multiply an expansion by a scalar, */
4334 /* eliminating zero components from the */
4335 /* output expansion. */
4337 /* Sets h = be. See my Robust Predicates paper for details. */
4339 /* Maintains the nonoverlapping property. If round-to-even is used (as */
4340 /* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
4341 /* properties as well. (That is, if e has one of these properties, so */
4344 /*****************************************************************************/
4346 int scale_expansion_zeroelim( elen, e, b, h ) /* e and h cannot be the same. */
4352 INEXACT REAL Q, sum;
4354 INEXACT REAL product1;
4359 REAL avirt, bround, around;
4362 REAL ahi, alo, bhi, blo;
4363 REAL err1, err2, err3;
4365 Split( b, bhi, blo );
4366 Two_Product_Presplit( e[0], b, bhi, blo, Q, hh );
4371 for ( eindex = 1; eindex < elen; eindex++ ) {
4373 Two_Product_Presplit( enow, b, bhi, blo, product1, product0 );
4374 Two_Sum( Q, product0, sum, hh );
4378 Fast_Two_Sum( product1, sum, Q, hh );
4383 if (( Q != 0.0 ) || ( hindex == 0 )) {
4389 /*****************************************************************************/
4391 /* estimate() Produce a one-word estimate of an expansion's value. */
4393 /* See my Robust Predicates paper for details. */
4395 /*****************************************************************************/
4397 REAL estimate( elen, e )
4405 for ( eindex = 1; eindex < elen; eindex++ ) {
4411 /*****************************************************************************/
4413 /* counterclockwise() Return a positive value if the points pa, pb, and */
4414 /* pc occur in counterclockwise order; a negative */
4415 /* value if they occur in clockwise order; and zero */
4416 /* if they are collinear. The result is also a rough */
4417 /* approximation of twice the signed area of the */
4418 /* triangle defined by the three points. */
4420 /* Uses exact arithmetic if necessary to ensure a correct answer. The */
4421 /* result returned is the determinant of a matrix. This determinant is */
4422 /* computed adaptively, in the sense that exact arithmetic is used only to */
4423 /* the degree it is needed to ensure that the returned value has the */
4424 /* correct sign. Hence, this function is usually quite fast, but will run */
4425 /* more slowly when the input points are collinear or nearly so. */
4427 /* See my Robust Predicates paper for details. */
4429 /*****************************************************************************/
4431 REAL counterclockwiseadapt( pa, pb, pc, detsum )
4437 INEXACT REAL acx, acy, bcx, bcy;
4438 REAL acxtail, acytail, bcxtail, bcytail;
4439 INEXACT REAL detleft, detright;
4440 REAL detlefttail, detrighttail;
4442 REAL B[4], C1[8], C2[12], D[16];
4444 int C1length, C2length, Dlength;
4447 INEXACT REAL s1, t1;
4451 REAL avirt, bround, around;
4454 REAL ahi, alo, bhi, blo;
4455 REAL err1, err2, err3;
4456 INEXACT REAL _i, _j;
4459 acx = (REAL) ( pa[0] - pc[0] );
4460 bcx = (REAL) ( pb[0] - pc[0] );
4461 acy = (REAL) ( pa[1] - pc[1] );
4462 bcy = (REAL) ( pb[1] - pc[1] );
4464 Two_Product( acx, bcy, detleft, detlefttail );
4465 Two_Product( acy, bcx, detright, detrighttail );
4467 Two_Two_Diff( detleft, detlefttail, detright, detrighttail,
4468 B3, B[2], B[1], B[0] );
4471 det = estimate( 4, B );
4472 errbound = (REAL)( ccwerrboundB * detsum );
4473 if (( det >= errbound ) || ( -det >= errbound )) {
4477 Two_Diff_Tail( pa[0], pc[0], acx, acxtail );
4478 Two_Diff_Tail( pb[0], pc[0], bcx, bcxtail );
4479 Two_Diff_Tail( pa[1], pc[1], acy, acytail );
4480 Two_Diff_Tail( pb[1], pc[1], bcy, bcytail );
4482 if (( acxtail == 0.0 ) && ( acytail == 0.0 )
4483 && ( bcxtail == 0.0 ) && ( bcytail == 0.0 )) {
4487 errbound = (REAL)( ccwerrboundC * detsum + resulterrbound * Absolute( det ));
4488 det += ( acx * bcytail + bcy * acxtail )
4489 - ( acy * bcxtail + bcx * acytail );
4490 if (( det >= errbound ) || ( -det >= errbound )) {
4494 Two_Product( acxtail, bcy, s1, s0 );
4495 Two_Product( acytail, bcx, t1, t0 );
4496 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4498 C1length = fast_expansion_sum_zeroelim( 4, B, 4, u, C1 );
4500 Two_Product( acx, bcytail, s1, s0 );
4501 Two_Product( acy, bcxtail, t1, t0 );
4502 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4504 C2length = fast_expansion_sum_zeroelim( C1length, C1, 4, u, C2 );
4506 Two_Product( acxtail, bcytail, s1, s0 );
4507 Two_Product( acytail, bcxtail, t1, t0 );
4508 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4510 Dlength = fast_expansion_sum_zeroelim( C2length, C2, 4, u, D );
4512 return( D[Dlength - 1] );
4515 REAL counterclockwise( pa, pb, pc )
4520 REAL detleft, detright, det;
4521 REAL detsum, errbound;
4523 counterclockcount++;
4525 detleft = ( pa[0] - pc[0] ) * ( pb[1] - pc[1] );
4526 detright = ( pa[1] - pc[1] ) * ( pb[0] - pc[0] );
4527 det = detleft - detright;
4533 if ( detleft > 0.0 ) {
4534 if ( detright <= 0.0 ) {
4538 detsum = detleft + detright;
4541 else if ( detleft < 0.0 ) {
4542 if ( detright >= 0.0 ) {
4546 detsum = -detleft - detright;
4553 errbound = ccwerrboundA * detsum;
4554 if (( det >= errbound ) || ( -det >= errbound )) {
4558 return counterclockwiseadapt( pa, pb, pc, detsum );
4561 /*****************************************************************************/
4563 /* incircle() Return a positive value if the point pd lies inside the */
4564 /* circle passing through pa, pb, and pc; a negative value if */
4565 /* it lies outside; and zero if the four points are cocircular.*/
4566 /* The points pa, pb, and pc must be in counterclockwise */
4567 /* order, or the sign of the result will be reversed. */
4569 /* Uses exact arithmetic if necessary to ensure a correct answer. The */
4570 /* result returned is the determinant of a matrix. This determinant is */
4571 /* computed adaptively, in the sense that exact arithmetic is used only to */
4572 /* the degree it is needed to ensure that the returned value has the */
4573 /* correct sign. Hence, this function is usually quite fast, but will run */
4574 /* more slowly when the input points are cocircular or nearly so. */
4576 /* See my Robust Predicates paper for details. */
4578 /*****************************************************************************/
4580 REAL incircleadapt( pa, pb, pc, pd, permanent )
4587 INEXACT REAL adx, bdx, cdx, ady, bdy, cdy;
4590 INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1;
4591 REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0;
4592 REAL bc[4], ca[4], ab[4];
4593 INEXACT REAL bc3, ca3, ab3;
4594 REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32];
4595 int axbclen, axxbclen, aybclen, ayybclen, alen;
4596 REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32];
4597 int bxcalen, bxxcalen, bycalen, byycalen, blen;
4598 REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32];
4599 int cxablen, cxxablen, cyablen, cyyablen, clen;
4602 REAL fin1[1152], fin2[1152];
4603 REAL *finnow, *finother, *finswap;
4606 REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail;
4607 INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1;
4608 REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0;
4609 REAL aa[4], bb[4], cc[4];
4610 INEXACT REAL aa3, bb3, cc3;
4611 INEXACT REAL ti1, tj1;
4614 INEXACT REAL u3, v3;
4615 REAL temp8[8], temp16a[16], temp16b[16], temp16c[16];
4616 REAL temp32a[32], temp32b[32], temp48[48], temp64[64];
4617 int temp8len, temp16alen, temp16blen, temp16clen;
4618 int temp32alen, temp32blen, temp48len, temp64len;
4619 REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8];
4620 int axtbblen, axtcclen, aytbblen, aytcclen;
4621 REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8];
4622 int bxtaalen, bxtcclen, bytaalen, bytcclen;
4623 REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8];
4624 int cxtaalen, cxtbblen, cytaalen, cytbblen;
4625 REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8];
4626 int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen;
4627 REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16];
4628 int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen;
4629 REAL axtbctt[8], aytbctt[8], bxtcatt[8];
4630 REAL bytcatt[8], cxtabtt[8], cytabtt[8];
4631 int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen;
4632 REAL abt[8], bct[8], cat[8];
4633 int abtlen, bctlen, catlen;
4634 REAL abtt[4], bctt[4], catt[4];
4635 int abttlen, bcttlen, cattlen;
4636 INEXACT REAL abtt3, bctt3, catt3;
4640 REAL avirt, bround, around;
4643 REAL ahi, alo, bhi, blo;
4644 REAL err1, err2, err3;
4645 INEXACT REAL _i, _j;
4648 adx = (REAL) ( pa[0] - pd[0] );
4649 bdx = (REAL) ( pb[0] - pd[0] );
4650 cdx = (REAL) ( pc[0] - pd[0] );
4651 ady = (REAL) ( pa[1] - pd[1] );
4652 bdy = (REAL) ( pb[1] - pd[1] );
4653 cdy = (REAL) ( pc[1] - pd[1] );
4655 Two_Product( bdx, cdy, bdxcdy1, bdxcdy0 );
4656 Two_Product( cdx, bdy, cdxbdy1, cdxbdy0 );
4657 Two_Two_Diff( bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0] );
4659 axbclen = scale_expansion_zeroelim( 4, bc, adx, axbc );
4660 axxbclen = scale_expansion_zeroelim( axbclen, axbc, adx, axxbc );
4661 aybclen = scale_expansion_zeroelim( 4, bc, ady, aybc );
4662 ayybclen = scale_expansion_zeroelim( aybclen, aybc, ady, ayybc );
4663 alen = fast_expansion_sum_zeroelim( axxbclen, axxbc, ayybclen, ayybc, adet );
4665 Two_Product( cdx, ady, cdxady1, cdxady0 );
4666 Two_Product( adx, cdy, adxcdy1, adxcdy0 );
4667 Two_Two_Diff( cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0] );
4669 bxcalen = scale_expansion_zeroelim( 4, ca, bdx, bxca );
4670 bxxcalen = scale_expansion_zeroelim( bxcalen, bxca, bdx, bxxca );
4671 bycalen = scale_expansion_zeroelim( 4, ca, bdy, byca );
4672 byycalen = scale_expansion_zeroelim( bycalen, byca, bdy, byyca );
4673 blen = fast_expansion_sum_zeroelim( bxxcalen, bxxca, byycalen, byyca, bdet );
4675 Two_Product( adx, bdy, adxbdy1, adxbdy0 );
4676 Two_Product( bdx, ady, bdxady1, bdxady0 );
4677 Two_Two_Diff( adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0] );
4679 cxablen = scale_expansion_zeroelim( 4, ab, cdx, cxab );
4680 cxxablen = scale_expansion_zeroelim( cxablen, cxab, cdx, cxxab );
4681 cyablen = scale_expansion_zeroelim( 4, ab, cdy, cyab );
4682 cyyablen = scale_expansion_zeroelim( cyablen, cyab, cdy, cyyab );
4683 clen = fast_expansion_sum_zeroelim( cxxablen, cxxab, cyyablen, cyyab, cdet );
4685 ablen = fast_expansion_sum_zeroelim( alen, adet, blen, bdet, abdet );
4686 finlength = fast_expansion_sum_zeroelim( ablen, abdet, clen, cdet, fin1 );
4688 det = estimate( finlength, fin1 );
4689 errbound = (REAL)( iccerrboundB * permanent );
4690 if (( det >= errbound ) || ( -det >= errbound )) {
4694 Two_Diff_Tail( pa[0], pd[0], adx, adxtail );
4695 Two_Diff_Tail( pa[1], pd[1], ady, adytail );
4696 Two_Diff_Tail( pb[0], pd[0], bdx, bdxtail );
4697 Two_Diff_Tail( pb[1], pd[1], bdy, bdytail );
4698 Two_Diff_Tail( pc[0], pd[0], cdx, cdxtail );
4699 Two_Diff_Tail( pc[1], pd[1], cdy, cdytail );
4700 if (( adxtail == 0.0 ) && ( bdxtail == 0.0 ) && ( cdxtail == 0.0 )
4701 && ( adytail == 0.0 ) && ( bdytail == 0.0 ) && ( cdytail == 0.0 )) {
4705 errbound = (REAL)( iccerrboundC * permanent + resulterrbound * Absolute( det ));
4706 det += (REAL)((( adx * adx + ady * ady ) * (( bdx * cdytail + cdy * bdxtail )
4707 - ( bdy * cdxtail + cdx * bdytail ))
4708 + 2.0 * ( adx * adxtail + ady * adytail ) * ( bdx * cdy - bdy * cdx ))
4709 + (( bdx * bdx + bdy * bdy ) * (( cdx * adytail + ady * cdxtail )
4710 - ( cdy * adxtail + adx * cdytail ))
4711 + 2.0 * ( bdx * bdxtail + bdy * bdytail ) * ( cdx * ady - cdy * adx ))
4712 + (( cdx * cdx + cdy * cdy ) * (( adx * bdytail + bdy * adxtail )
4713 - ( ady * bdxtail + bdx * adytail ))
4714 + 2.0 * ( cdx * cdxtail + cdy * cdytail ) * ( adx * bdy - ady * bdx )));
4715 if (( det >= errbound ) || ( -det >= errbound )) {
4722 if (( bdxtail != 0.0 ) || ( bdytail != 0.0 )
4723 || ( cdxtail != 0.0 ) || ( cdytail != 0.0 )) {
4724 Square( adx, adxadx1, adxadx0 );
4725 Square( ady, adyady1, adyady0 );
4726 Two_Two_Sum( adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0] );
4729 if (( cdxtail != 0.0 ) || ( cdytail != 0.0 )
4730 || ( adxtail != 0.0 ) || ( adytail != 0.0 )) {
4731 Square( bdx, bdxbdx1, bdxbdx0 );
4732 Square( bdy, bdybdy1, bdybdy0 );
4733 Two_Two_Sum( bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0] );
4736 if (( adxtail != 0.0 ) || ( adytail != 0.0 )
4737 || ( bdxtail != 0.0 ) || ( bdytail != 0.0 )) {
4738 Square( cdx, cdxcdx1, cdxcdx0 );
4739 Square( cdy, cdycdy1, cdycdy0 );
4740 Two_Two_Sum( cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0] );
4744 if ( adxtail != 0.0 ) {
4745 axtbclen = scale_expansion_zeroelim( 4, bc, adxtail, axtbc );
4746 temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, 2.0 * adx,
4749 axtcclen = scale_expansion_zeroelim( 4, cc, adxtail, axtcc );
4750 temp16blen = scale_expansion_zeroelim( axtcclen, axtcc, bdy, temp16b );
4752 axtbblen = scale_expansion_zeroelim( 4, bb, adxtail, axtbb );
4753 temp16clen = scale_expansion_zeroelim( axtbblen, axtbb, -cdy, temp16c );
4755 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4756 temp16blen, temp16b, temp32a );
4757 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4758 temp32alen, temp32a, temp48 );
4759 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4761 finswap = finnow; finnow = finother; finother = finswap;
4763 if ( adytail != 0.0 ) {
4764 aytbclen = scale_expansion_zeroelim( 4, bc, adytail, aytbc );
4765 temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, 2.0 * ady,
4768 aytbblen = scale_expansion_zeroelim( 4, bb, adytail, aytbb );
4769 temp16blen = scale_expansion_zeroelim( aytbblen, aytbb, cdx, temp16b );
4771 aytcclen = scale_expansion_zeroelim( 4, cc, adytail, aytcc );
4772 temp16clen = scale_expansion_zeroelim( aytcclen, aytcc, -bdx, temp16c );
4774 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4775 temp16blen, temp16b, temp32a );
4776 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4777 temp32alen, temp32a, temp48 );
4778 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4780 finswap = finnow; finnow = finother; finother = finswap;
4782 if ( bdxtail != 0.0 ) {
4783 bxtcalen = scale_expansion_zeroelim( 4, ca, bdxtail, bxtca );
4784 temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, 2.0 * bdx,
4787 bxtaalen = scale_expansion_zeroelim( 4, aa, bdxtail, bxtaa );
4788 temp16blen = scale_expansion_zeroelim( bxtaalen, bxtaa, cdy, temp16b );
4790 bxtcclen = scale_expansion_zeroelim( 4, cc, bdxtail, bxtcc );
4791 temp16clen = scale_expansion_zeroelim( bxtcclen, bxtcc, -ady, temp16c );
4793 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4794 temp16blen, temp16b, temp32a );
4795 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4796 temp32alen, temp32a, temp48 );
4797 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4799 finswap = finnow; finnow = finother; finother = finswap;
4801 if ( bdytail != 0.0 ) {
4802 bytcalen = scale_expansion_zeroelim( 4, ca, bdytail, bytca );
4803 temp16alen = scale_expansion_zeroelim( bytcalen, bytca, 2.0 * bdy,
4806 bytcclen = scale_expansion_zeroelim( 4, cc, bdytail, bytcc );
4807 temp16blen = scale_expansion_zeroelim( bytcclen, bytcc, adx, temp16b );
4809 bytaalen = scale_expansion_zeroelim( 4, aa, bdytail, bytaa );
4810 temp16clen = scale_expansion_zeroelim( bytaalen, bytaa, -cdx, temp16c );
4812 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4813 temp16blen, temp16b, temp32a );
4814 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4815 temp32alen, temp32a, temp48 );
4816 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4818 finswap = finnow; finnow = finother; finother = finswap;
4820 if ( cdxtail != 0.0 ) {
4821 cxtablen = scale_expansion_zeroelim( 4, ab, cdxtail, cxtab );
4822 temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, 2.0 * cdx,
4825 cxtbblen = scale_expansion_zeroelim( 4, bb, cdxtail, cxtbb );
4826 temp16blen = scale_expansion_zeroelim( cxtbblen, cxtbb, ady, temp16b );
4828 cxtaalen = scale_expansion_zeroelim( 4, aa, cdxtail, cxtaa );
4829 temp16clen = scale_expansion_zeroelim( cxtaalen, cxtaa, -bdy, temp16c );
4831 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4832 temp16blen, temp16b, temp32a );
4833 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4834 temp32alen, temp32a, temp48 );
4835 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4837 finswap = finnow; finnow = finother; finother = finswap;
4839 if ( cdytail != 0.0 ) {
4840 cytablen = scale_expansion_zeroelim( 4, ab, cdytail, cytab );
4841 temp16alen = scale_expansion_zeroelim( cytablen, cytab, 2.0 * cdy,
4844 cytaalen = scale_expansion_zeroelim( 4, aa, cdytail, cytaa );
4845 temp16blen = scale_expansion_zeroelim( cytaalen, cytaa, bdx, temp16b );
4847 cytbblen = scale_expansion_zeroelim( 4, bb, cdytail, cytbb );
4848 temp16clen = scale_expansion_zeroelim( cytbblen, cytbb, -adx, temp16c );
4850 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4851 temp16blen, temp16b, temp32a );
4852 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4853 temp32alen, temp32a, temp48 );
4854 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4856 finswap = finnow; finnow = finother; finother = finswap;
4859 if (( adxtail != 0.0 ) || ( adytail != 0.0 )) {
4860 if (( bdxtail != 0.0 ) || ( bdytail != 0.0 )
4861 || ( cdxtail != 0.0 ) || ( cdytail != 0.0 )) {
4862 Two_Product( bdxtail, cdy, ti1, ti0 );
4863 Two_Product( bdx, cdytail, tj1, tj0 );
4864 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
4867 Two_Product( cdxtail, negate, ti1, ti0 );
4869 Two_Product( cdx, negate, tj1, tj0 );
4870 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
4872 bctlen = fast_expansion_sum_zeroelim( 4, u, 4, v, bct );
4874 Two_Product( bdxtail, cdytail, ti1, ti0 );
4875 Two_Product( cdxtail, bdytail, tj1, tj0 );
4876 Two_Two_Diff( ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0] );
4887 if ( adxtail != 0.0 ) {
4888 temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, adxtail, temp16a );
4889 axtbctlen = scale_expansion_zeroelim( bctlen, bct, adxtail, axtbct );
4890 temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, 2.0 * adx,
4892 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4893 temp32alen, temp32a, temp48 );
4894 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4896 finswap = finnow; finnow = finother; finother = finswap;
4897 if ( bdytail != 0.0 ) {
4898 temp8len = scale_expansion_zeroelim( 4, cc, adxtail, temp8 );
4899 temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail,
4901 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4902 temp16a, finother );
4903 finswap = finnow; finnow = finother; finother = finswap;
4905 if ( cdytail != 0.0 ) {
4906 temp8len = scale_expansion_zeroelim( 4, bb, -adxtail, temp8 );
4907 temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail,
4909 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4910 temp16a, finother );
4911 finswap = finnow; finnow = finother; finother = finswap;
4914 temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, adxtail,
4916 axtbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adxtail, axtbctt );
4917 temp16alen = scale_expansion_zeroelim( axtbcttlen, axtbctt, 2.0 * adx,
4919 temp16blen = scale_expansion_zeroelim( axtbcttlen, axtbctt, adxtail,
4921 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4922 temp16blen, temp16b, temp32b );
4923 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4924 temp32blen, temp32b, temp64 );
4925 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4927 finswap = finnow; finnow = finother; finother = finswap;
4929 if ( adytail != 0.0 ) {
4930 temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, adytail, temp16a );
4931 aytbctlen = scale_expansion_zeroelim( bctlen, bct, adytail, aytbct );
4932 temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, 2.0 * ady,
4934 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4935 temp32alen, temp32a, temp48 );
4936 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4938 finswap = finnow; finnow = finother; finother = finswap;
4941 temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, adytail,
4943 aytbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adytail, aytbctt );
4944 temp16alen = scale_expansion_zeroelim( aytbcttlen, aytbctt, 2.0 * ady,
4946 temp16blen = scale_expansion_zeroelim( aytbcttlen, aytbctt, adytail,
4948 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4949 temp16blen, temp16b, temp32b );
4950 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4951 temp32blen, temp32b, temp64 );
4952 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4954 finswap = finnow; finnow = finother; finother = finswap;
4957 if (( bdxtail != 0.0 ) || ( bdytail != 0.0 )) {
4958 if (( cdxtail != 0.0 ) || ( cdytail != 0.0 )
4959 || ( adxtail != 0.0 ) || ( adytail != 0.0 )) {
4960 Two_Product( cdxtail, ady, ti1, ti0 );
4961 Two_Product( cdx, adytail, tj1, tj0 );
4962 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
4965 Two_Product( adxtail, negate, ti1, ti0 );
4967 Two_Product( adx, negate, tj1, tj0 );
4968 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
4970 catlen = fast_expansion_sum_zeroelim( 4, u, 4, v, cat );
4972 Two_Product( cdxtail, adytail, ti1, ti0 );
4973 Two_Product( adxtail, cdytail, tj1, tj0 );
4974 Two_Two_Diff( ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0] );
4985 if ( bdxtail != 0.0 ) {
4986 temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, bdxtail, temp16a );
4987 bxtcatlen = scale_expansion_zeroelim( catlen, cat, bdxtail, bxtcat );
4988 temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, 2.0 * bdx,
4990 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4991 temp32alen, temp32a, temp48 );
4992 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4994 finswap = finnow; finnow = finother; finother = finswap;
4995 if ( cdytail != 0.0 ) {
4996 temp8len = scale_expansion_zeroelim( 4, aa, bdxtail, temp8 );
4997 temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail,
4999 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
5000 temp16a, finother );
5001 finswap = finnow; finnow = finother; finother = finswap;
5003 if ( adytail != 0.0 ) {
5004 temp8len = scale_expansion_zeroelim( 4, cc, -bdxtail, temp8 );
5005 temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail,
5007 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
5008 temp16a, finother );
5009 finswap = finnow; finnow = finother; finother = finswap;
5012 temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, bdxtail,
5014 bxtcattlen = scale_expansion_zeroelim( cattlen, catt, bdxtail, bxtcatt );
5015 temp16alen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, 2.0 * bdx,
5017 temp16blen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, bdxtail,
5019 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5020 temp16blen, temp16b, temp32b );
5021 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
5022 temp32blen, temp32b, temp64 );
5023 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
5025 finswap = finnow; finnow = finother; finother = finswap;
5027 if ( bdytail != 0.0 ) {
5028 temp16alen = scale_expansion_zeroelim( bytcalen, bytca, bdytail, temp16a );
5029 bytcatlen = scale_expansion_zeroelim( catlen, cat, bdytail, bytcat );
5030 temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, 2.0 * bdy,
5032 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5033 temp32alen, temp32a, temp48 );
5034 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
5036 finswap = finnow; finnow = finother; finother = finswap;
5039 temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, bdytail,
5041 bytcattlen = scale_expansion_zeroelim( cattlen, catt, bdytail, bytcatt );
5042 temp16alen = scale_expansion_zeroelim( bytcattlen, bytcatt, 2.0 * bdy,
5044 temp16blen = scale_expansion_zeroelim( bytcattlen, bytcatt, bdytail,
5046 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5047 temp16blen, temp16b, temp32b );
5048 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
5049 temp32blen, temp32b, temp64 );
5050 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
5052 finswap = finnow; finnow = finother; finother = finswap;
5055 if (( cdxtail != 0.0 ) || ( cdytail != 0.0 )) {
5056 if (( adxtail != 0.0 ) || ( adytail != 0.0 )
5057 || ( bdxtail != 0.0 ) || ( bdytail != 0.0 )) {
5058 Two_Product( adxtail, bdy, ti1, ti0 );
5059 Two_Product( adx, bdytail, tj1, tj0 );
5060 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
5063 Two_Product( bdxtail, negate, ti1, ti0 );
5065 Two_Product( bdx, negate, tj1, tj0 );
5066 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
5068 abtlen = fast_expansion_sum_zeroelim( 4, u, 4, v, abt );
5070 Two_Product( adxtail, bdytail, ti1, ti0 );
5071 Two_Product( bdxtail, adytail, tj1, tj0 );
5072 Two_Two_Diff( ti1, ti0, tj1, tj0, abtt3, abtt[2], abtt[1], abtt[0] );
5083 if ( cdxtail != 0.0 ) {
5084 temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, cdxtail, temp16a );
5085 cxtabtlen = scale_expansion_zeroelim( abtlen, abt, cdxtail, cxtabt );
5086 temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, 2.0 * cdx,
5088 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5089 temp32alen, temp32a, temp48 );
5090 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
5092 finswap = finnow; finnow = finother; finother = finswap;
5093 if ( adytail != 0.0 ) {
5094 temp8len = scale_expansion_zeroelim( 4, bb, cdxtail, temp8 );
5095 temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail,
5097 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
5098 temp16a, finother );
5099 finswap = finnow; finnow = finother; finother = finswap;
5101 if ( bdytail != 0.0 ) {
5102 temp8len = scale_expansion_zeroelim( 4, aa, -cdxtail, temp8 );
5103 temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail,
5105 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
5106 temp16a, finother );
5107 finswap = finnow; finnow = finother; finother = finswap;
5110 temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, cdxtail,
5112 cxtabttlen = scale_expansion_zeroelim( abttlen, abtt, cdxtail, cxtabtt );
5113 temp16alen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, 2.0 * cdx,
5115 temp16blen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, cdxtail,
5117 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5118 temp16blen, temp16b, temp32b );
5119 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
5120 temp32blen, temp32b, temp64 );
5121 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
5123 finswap = finnow; finnow = finother; finother = finswap;
5125 if ( cdytail != 0.0 ) {
5126 temp16alen = scale_expansion_zeroelim( cytablen, cytab, cdytail, temp16a );
5127 cytabtlen = scale_expansion_zeroelim( abtlen, abt, cdytail, cytabt );
5128 temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, 2.0 * cdy,
5130 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5131 temp32alen, temp32a, temp48 );
5132 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
5134 finswap = finnow; finnow = finother; finother = finswap;
5137 temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, cdytail,
5139 cytabttlen = scale_expansion_zeroelim( abttlen, abtt, cdytail, cytabtt );
5140 temp16alen = scale_expansion_zeroelim( cytabttlen, cytabtt, 2.0 * cdy,
5142 temp16blen = scale_expansion_zeroelim( cytabttlen, cytabtt, cdytail,
5144 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
5145 temp16blen, temp16b, temp32b );
5146 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
5147 temp32blen, temp32b, temp64 );
5148 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
5150 finswap = finnow; finnow = finother; finother = finswap;
5154 return finnow[finlength - 1];
5157 REAL incircle( pa, pb, pc, pd )
5163 REAL adx, bdx, cdx, ady, bdy, cdy;
5164 REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady;
5165 REAL alift, blift, clift;
5167 REAL permanent, errbound;
5171 adx = pa[0] - pd[0];
5172 bdx = pb[0] - pd[0];
5173 cdx = pc[0] - pd[0];
5174 ady = pa[1] - pd[1];
5175 bdy = pb[1] - pd[1];
5176 cdy = pc[1] - pd[1];
5180 alift = adx * adx + ady * ady;
5184 blift = bdx * bdx + bdy * bdy;
5188 clift = cdx * cdx + cdy * cdy;
5190 det = alift * ( bdxcdy - cdxbdy )
5191 + blift * ( cdxady - adxcdy )
5192 + clift * ( adxbdy - bdxady );
5198 permanent = ( Absolute( bdxcdy ) + Absolute( cdxbdy )) * alift
5199 + ( Absolute( cdxady ) + Absolute( adxcdy )) * blift
5200 + ( Absolute( adxbdy ) + Absolute( bdxady )) * clift;
5201 errbound = iccerrboundA * permanent;
5202 if (( det > errbound ) || ( -det > errbound )) {
5206 return incircleadapt( pa, pb, pc, pd, permanent );
5211 /********* Determinant evaluation routines end here *********/
5213 /*****************************************************************************/
5215 /* triangleinit() Initialize some variables. */
5217 /*****************************************************************************/
5219 void triangleinit(){
5220 points.maxitems = triangles.maxitems = shelles.maxitems = viri.maxitems =
5221 badsegments.maxitems = badtriangles.maxitems = splaynodes.maxitems = 0l;
5222 points.itembytes = triangles.itembytes = shelles.itembytes = viri.itembytes =
5223 badsegments.itembytes = badtriangles.itembytes = splaynodes.itembytes = 0;
5224 recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */
5225 samples = 1; /* Point location should take at least one sample. */
5226 checksegments = 0; /* There are no segments in the triangulation yet. */
5227 incirclecount = counterclockcount = hyperbolacount = 0;
5228 circumcentercount = circletopcount = 0;
5231 exactinit(); /* Initialize exact arithmetic constants. */
5234 /*****************************************************************************/
5236 /* randomnation() Generate a random number between 0 and `choices' - 1. */
5238 /* This is a simple linear congruential random number generator. Hence, it */
5239 /* is a bad random number generator, but good enough for most randomized */
5240 /* geometric algorithms. */
5242 /*****************************************************************************/
5244 unsigned long randomnation( choices )
5245 unsigned int choices;
5247 randomseed = ( randomseed * 1366l + 150889l ) % 714025l;
5248 return randomseed / ( 714025l / choices + 1 );
5251 /********* Mesh quality testing routines begin here *********/
5255 /*****************************************************************************/
5257 /* checkmesh() Test the mesh for topological consistency. */
5259 /*****************************************************************************/
5265 struct triedge triangleloop;
5266 struct triedge oppotri, oppooppotri;
5267 point triorg, tridest, triapex;
5268 point oppoorg, oppodest;
5271 triangle ptr; /* Temporary variable used by sym(). */
5273 /* Temporarily turn on exact arithmetic if it's off. */
5274 saveexact = noexact;
5277 printf( " Checking consistency of mesh...\n" );
5280 /* Run through the list of triangles, checking each one. */
5281 traversalinit( &triangles );
5282 triangleloop.tri = triangletraverse();
5283 while ( triangleloop.tri != (triangle *) NULL ) {
5284 /* Check all three edges of the triangle. */
5285 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5286 triangleloop.orient++ ) {
5287 org( triangleloop, triorg );
5288 dest( triangleloop, tridest );
5289 if ( triangleloop.orient == 0 ) { /* Only test for inversion once. */
5290 /* Test if the triangle is flat or inverted. */
5291 apex( triangleloop, triapex );
5292 if ( counterclockwise( triorg, tridest, triapex ) <= 0.0 ) {
5293 printf( " !! !! Inverted " );
5294 printtriangle( &triangleloop );
5298 /* Find the neighboring triangle on this edge. */
5299 sym( triangleloop, oppotri );
5300 if ( oppotri.tri != dummytri ) {
5301 /* Check that the triangle's neighbor knows it's a neighbor. */
5302 sym( oppotri, oppooppotri );
5303 if (( triangleloop.tri != oppooppotri.tri )
5304 || ( triangleloop.orient != oppooppotri.orient )) {
5305 printf( " !! !! Asymmetric triangle-triangle bond:\n" );
5306 if ( triangleloop.tri == oppooppotri.tri ) {
5307 printf( " (Right triangle, wrong orientation)\n" );
5309 printf( " First " );
5310 printtriangle( &triangleloop );
5311 printf( " Second (nonreciprocating) " );
5312 printtriangle( &oppotri );
5315 /* Check that both triangles agree on the identities */
5316 /* of their shared vertices. */
5317 org( oppotri, oppoorg );
5318 dest( oppotri, oppodest );
5319 if (( triorg != oppodest ) || ( tridest != oppoorg )) {
5320 printf( " !! !! Mismatched edge coordinates between two triangles:\n"
5322 printf( " First mismatched " );
5323 printtriangle( &triangleloop );
5324 printf( " Second mismatched " );
5325 printtriangle( &oppotri );
5330 triangleloop.tri = triangletraverse();
5332 if ( horrors == 0 ) {
5334 printf( " In my studied opinion, the mesh appears to be consistent.\n" );
5337 else if ( horrors == 1 ) {
5338 printf( " !! !! !! !! Precisely one festering wound discovered.\n" );
5341 printf( " !! !! !! !! %d abominations witnessed.\n", horrors );
5343 /* Restore the status of exact arithmetic. */
5344 noexact = saveexact;
5347 #endif /* not REDUCED */
5349 /*****************************************************************************/
5351 /* checkdelaunay() Ensure that the mesh is (constrained) Delaunay. */
5353 /*****************************************************************************/
5358 void checkdelaunay(){
5359 struct triedge triangleloop;
5360 struct triedge oppotri;
5361 struct edge opposhelle;
5362 point triorg, tridest, triapex;
5364 int shouldbedelaunay;
5367 triangle ptr; /* Temporary variable used by sym(). */
5368 shelle sptr; /* Temporary variable used by tspivot(). */
5370 /* Temporarily turn on exact arithmetic if it's off. */
5371 saveexact = noexact;
5374 printf( " Checking Delaunay property of mesh...\n" );
5377 /* Run through the list of triangles, checking each one. */
5378 traversalinit( &triangles );
5379 triangleloop.tri = triangletraverse();
5380 while ( triangleloop.tri != (triangle *) NULL ) {
5381 /* Check all three edges of the triangle. */
5382 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5383 triangleloop.orient++ ) {
5384 org( triangleloop, triorg );
5385 dest( triangleloop, tridest );
5386 apex( triangleloop, triapex );
5387 sym( triangleloop, oppotri );
5388 apex( oppotri, oppoapex );
5389 /* Only test that the edge is locally Delaunay if there is an */
5390 /* adjoining triangle whose pointer is larger (to ensure that */
5391 /* each pair isn't tested twice). */
5392 shouldbedelaunay = ( oppotri.tri != dummytri )
5393 && ( triapex != (point) NULL ) && ( oppoapex != (point) NULL )
5394 && ( triangleloop.tri < oppotri.tri );
5395 if ( checksegments && shouldbedelaunay ) {
5396 /* If a shell edge separates the triangles, then the edge is */
5397 /* constrained, so no local Delaunay test should be done. */
5398 tspivot( triangleloop, opposhelle );
5399 if ( opposhelle.sh != dummysh ) {
5400 shouldbedelaunay = 0;
5403 if ( shouldbedelaunay ) {
5404 if ( incircle( triorg, tridest, triapex, oppoapex ) > 0.0 ) {
5405 printf( " !! !! Non-Delaunay pair of triangles:\n" );
5406 printf( " First non-Delaunay " );
5407 printtriangle( &triangleloop );
5408 printf( " Second non-Delaunay " );
5409 printtriangle( &oppotri );
5414 triangleloop.tri = triangletraverse();
5416 if ( horrors == 0 ) {
5419 " By virtue of my perceptive intelligence, I declare the mesh Delaunay.\n" );
5422 else if ( horrors == 1 ) {
5424 " !! !! !! !! Precisely one terrifying transgression identified.\n" );
5427 printf( " !! !! !! !! %d obscenities viewed with horror.\n", horrors );
5429 /* Restore the status of exact arithmetic. */
5430 noexact = saveexact;
5433 #endif /* not REDUCED */
5435 /*****************************************************************************/
5437 /* enqueuebadtri() Add a bad triangle to the end of a queue. */
5439 /* The queue is actually a set of 64 queues. I use multiple queues to give */
5440 /* priority to smaller angles. I originally implemented a heap, but the */
5441 /* queues are (to my surprise) much faster. */
5443 /*****************************************************************************/
5448 void enqueuebadtri( instri, angle, insapex, insorg, insdest )
5449 struct triedge *instri;
5455 struct badface *newface;
5458 if ( verbose > 2 ) {
5459 printf( " Queueing bad triangle:\n" );
5460 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", insorg[0],
5461 insorg[1], insdest[0], insdest[1], insapex[0], insapex[1] );
5463 /* Allocate space for the bad triangle. */
5464 newface = (struct badface *) poolalloc( &badtriangles );
5465 triedgecopy( *instri, newface->badfacetri );
5466 newface->key = angle;
5467 newface->faceapex = insapex;
5468 newface->faceorg = insorg;
5469 newface->facedest = insdest;
5470 newface->nextface = (struct badface *) NULL;
5471 /* Determine the appropriate queue to put the bad triangle into. */
5472 if ( angle > 0.6 ) {
5473 queuenumber = (int) ( 160.0 * ( angle - 0.6 ));
5474 if ( queuenumber > 63 ) {
5479 /* It's not a bad angle; put the triangle in the lowest-priority queue. */
5482 /* Add the triangle to the end of a queue. */
5483 *queuetail[queuenumber] = newface;
5484 /* Maintain a pointer to the NULL pointer at the end of the queue. */
5485 queuetail[queuenumber] = &newface->nextface;
5488 #endif /* not CDT_ONLY */
5490 /*****************************************************************************/
5492 /* dequeuebadtri() Remove a triangle from the front of the queue. */
5494 /*****************************************************************************/
5499 struct badface *dequeuebadtri(){
5500 struct badface *result;
5503 /* Look for a nonempty queue. */
5504 for ( queuenumber = 63; queuenumber >= 0; queuenumber-- ) {
5505 result = queuefront[queuenumber];
5506 if ( result != (struct badface *) NULL ) {
5507 /* Remove the triangle from the queue. */
5508 queuefront[queuenumber] = result->nextface;
5509 /* Maintain a pointer to the NULL pointer at the end of the queue. */
5510 if ( queuefront[queuenumber] == (struct badface *) NULL ) {
5511 queuetail[queuenumber] = &queuefront[queuenumber];
5516 return (struct badface *) NULL;
5519 #endif /* not CDT_ONLY */
5521 /*****************************************************************************/
5523 /* checkedge4encroach() Check a segment to see if it is encroached; add */
5524 /* it to the list if it is. */
5526 /* An encroached segment is an unflippable edge that has a point in its */
5527 /* diametral circle (that is, it faces an angle greater than 90 degrees). */
5528 /* This definition is due to Ruppert. */
5530 /* Returns a nonzero value if the edge is encroached. */
5532 /*****************************************************************************/
5537 int checkedge4encroach( testedge )
5538 struct edge *testedge;
5540 struct triedge neighbortri;
5541 struct edge testsym;
5542 struct edge *badedge;
5545 point eorg, edest, eapex;
5546 triangle ptr; /* Temporary variable used by stpivot(). */
5551 sorg( *testedge, eorg );
5552 sdest( *testedge, edest );
5553 /* Check one neighbor of the shell edge. */
5554 stpivot( *testedge, neighbortri );
5555 /* Does the neighbor exist, or is this a boundary edge? */
5556 if ( neighbortri.tri != dummytri ) {
5558 /* Find a vertex opposite this edge. */
5559 apex( neighbortri, eapex );
5560 /* Check whether the vertex is inside the diametral circle of the */
5561 /* shell edge. Pythagoras' Theorem is used to check whether the */
5562 /* angle at the vertex is greater than 90 degrees. */
5563 if ( eapex[0] * ( eorg[0] + edest[0] ) + eapex[1] * ( eorg[1] + edest[1] ) >
5564 eapex[0] * eapex[0] + eorg[0] * edest[0] +
5565 eapex[1] * eapex[1] + eorg[1] * edest[1] ) {
5569 /* Check the other neighbor of the shell edge. */
5570 ssym( *testedge, testsym );
5571 stpivot( testsym, neighbortri );
5572 /* Does the neighbor exist, or is this a boundary edge? */
5573 if ( neighbortri.tri != dummytri ) {
5575 /* Find the other vertex opposite this edge. */
5576 apex( neighbortri, eapex );
5577 /* Check whether the vertex is inside the diametral circle of the */
5578 /* shell edge. Pythagoras' Theorem is used to check whether the */
5579 /* angle at the vertex is greater than 90 degrees. */
5580 if ( eapex[0] * ( eorg[0] + edest[0] ) +
5581 eapex[1] * ( eorg[1] + edest[1] ) >
5582 eapex[0] * eapex[0] + eorg[0] * edest[0] +
5583 eapex[1] * eapex[1] + eorg[1] * edest[1] ) {
5588 if ( addtolist && ( !nobisect || (( nobisect == 1 ) && ( sides == 2 )))) {
5589 if ( verbose > 2 ) {
5590 printf( " Queueing encroached segment (%.12g, %.12g) (%.12g, %.12g).\n",
5591 eorg[0], eorg[1], edest[0], edest[1] );
5593 /* Add the shell edge to the list of encroached segments. */
5594 /* Be sure to get the orientation right. */
5595 badedge = (struct edge *) poolalloc( &badsegments );
5596 if ( addtolist == 1 ) {
5597 shellecopy( *testedge, *badedge );
5600 shellecopy( testsym, *badedge );
5606 #endif /* not CDT_ONLY */
5608 /*****************************************************************************/
5610 /* testtriangle() Test a face for quality measures. */
5612 /* Tests a triangle to see if it satisfies the minimum angle condition and */
5613 /* the maximum area condition. Triangles that aren't up to spec are added */
5614 /* to the bad triangle queue. */
5616 /*****************************************************************************/
5621 void testtriangle( testtri )
5622 struct triedge *testtri;
5624 struct triedge sametesttri;
5625 struct edge edge1, edge2;
5626 point torg, tdest, tapex;
5628 REAL dxod, dyod, dxda, dyda, dxao, dyao;
5629 REAL dxod2, dyod2, dxda2, dyda2, dxao2, dyao2;
5630 REAL apexlen, orglen, destlen;
5633 shelle sptr; /* Temporary variable used by tspivot(). */
5635 org( *testtri, torg );
5636 dest( *testtri, tdest );
5637 apex( *testtri, tapex );
5638 dxod = torg[0] - tdest[0];
5639 dyod = torg[1] - tdest[1];
5640 dxda = tdest[0] - tapex[0];
5641 dyda = tdest[1] - tapex[1];
5642 dxao = tapex[0] - torg[0];
5643 dyao = tapex[1] - torg[1];
5644 dxod2 = dxod * dxod;
5645 dyod2 = dyod * dyod;
5646 dxda2 = dxda * dxda;
5647 dyda2 = dyda * dyda;
5648 dxao2 = dxao * dxao;
5649 dyao2 = dyao * dyao;
5650 /* Find the lengths of the triangle's three edges. */
5651 apexlen = dxod2 + dyod2;
5652 orglen = dxda2 + dyda2;
5653 destlen = dxao2 + dyao2;
5654 if (( apexlen < orglen ) && ( apexlen < destlen )) {
5655 /* The edge opposite the apex is shortest. */
5656 /* Find the square of the cosine of the angle at the apex. */
5657 angle = dxda * dxao + dyda * dyao;
5658 angle = angle * angle / ( orglen * destlen );
5659 anglevertex = tapex;
5660 lnext( *testtri, sametesttri );
5661 tspivot( sametesttri, edge1 );
5662 lnextself( sametesttri );
5663 tspivot( sametesttri, edge2 );
5665 else if ( orglen < destlen ) {
5666 /* The edge opposite the origin is shortest. */
5667 /* Find the square of the cosine of the angle at the origin. */
5668 angle = dxod * dxao + dyod * dyao;
5669 angle = angle * angle / ( apexlen * destlen );
5671 tspivot( *testtri, edge1 );
5672 lprev( *testtri, sametesttri );
5673 tspivot( sametesttri, edge2 );
5676 /* The edge opposite the destination is shortest. */
5677 /* Find the square of the cosine of the angle at the destination. */
5678 angle = dxod * dxda + dyod * dyda;
5679 angle = angle * angle / ( apexlen * orglen );
5680 anglevertex = tdest;
5681 tspivot( *testtri, edge1 );
5682 lnext( *testtri, sametesttri );
5683 tspivot( sametesttri, edge2 );
5685 /* Check if both edges that form the angle are segments. */
5686 if (( edge1.sh != dummysh ) && ( edge2.sh != dummysh )) {
5687 /* The angle is a segment intersection. */
5688 if (( angle > 0.9924 ) && !quiet ) { /* Roughly 5 degrees. */
5689 if ( angle > 1.0 ) {
5690 /* Beware of a floating exception in acos(). */
5693 /* Find the actual angle in degrees, for printing. */
5694 angle = acos( sqrt( angle )) * ( 180.0 / PI );
5696 "Warning: Small angle (%.4g degrees) between segments at point\n",
5698 printf( " (%.12g, %.12g)\n", anglevertex[0], anglevertex[1] );
5700 /* Don't add this bad triangle to the list; there's nothing that */
5701 /* can be done about a small angle between two segments. */
5704 /* Check whether the angle is smaller than permitted. */
5705 if ( angle > goodangle ) {
5706 /* Add this triangle to the list of bad triangles. */
5707 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5710 if ( vararea || fixedarea ) {
5711 /* Check whether the area is larger than permitted. */
5712 area = 0.5 * ( dxod * dyda - dyod * dxda );
5713 if ( fixedarea && ( area > maxarea )) {
5714 /* Add this triangle to the list of bad triangles. */
5715 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5717 else if ( vararea ) {
5718 /* Nonpositive area constraints are treated as unconstrained. */
5719 if (( area > areabound( *testtri )) && ( areabound( *testtri ) > 0.0 )) {
5720 /* Add this triangle to the list of bad triangles. */
5721 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5727 #endif /* not CDT_ONLY */
5731 /********* Mesh quality testing routines end here *********/
5733 /********* Point location routines begin here *********/
5737 /*****************************************************************************/
5739 /* makepointmap() Construct a mapping from points to triangles to improve */
5740 /* the speed of point location for segment insertion. */
5742 /* Traverses all the triangles, and provides each corner of each triangle */
5743 /* with a pointer to that triangle. Of course, pointers will be */
5744 /* overwritten by other pointers because (almost) each point is a corner */
5745 /* of several triangles, but in the end every point will point to some */
5746 /* triangle that contains it. */
5748 /*****************************************************************************/
5750 void makepointmap(){
5751 struct triedge triangleloop;
5755 printf( " Constructing mapping from points to triangles.\n" );
5757 traversalinit( &triangles );
5758 triangleloop.tri = triangletraverse();
5759 while ( triangleloop.tri != (triangle *) NULL ) {
5760 /* Check all three points of the triangle. */
5761 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5762 triangleloop.orient++ ) {
5763 org( triangleloop, triorg );
5764 setpoint2tri( triorg, encode( triangleloop ));
5766 triangleloop.tri = triangletraverse();
5770 /*****************************************************************************/
5772 /* preciselocate() Find a triangle or edge containing a given point. */
5774 /* Begins its search from `searchtri'. It is important that `searchtri' */
5775 /* be a handle with the property that `searchpoint' is strictly to the left */
5776 /* of the edge denoted by `searchtri', or is collinear with that edge and */
5777 /* does not intersect that edge. (In particular, `searchpoint' should not */
5778 /* be the origin or destination of that edge.) */
5780 /* These conditions are imposed because preciselocate() is normally used in */
5781 /* one of two situations: */
5783 /* (1) To try to find the location to insert a new point. Normally, we */
5784 /* know an edge that the point is strictly to the left of. In the */
5785 /* incremental Delaunay algorithm, that edge is a bounding box edge. */
5786 /* In Ruppert's Delaunay refinement algorithm for quality meshing, */
5787 /* that edge is the shortest edge of the triangle whose circumcenter */
5788 /* is being inserted. */
5790 /* (2) To try to find an existing point. In this case, any edge on the */
5791 /* convex hull is a good starting edge. The possibility that the */
5792 /* vertex one seeks is an endpoint of the starting edge must be */
5793 /* screened out before preciselocate() is called. */
5795 /* On completion, `searchtri' is a triangle that contains `searchpoint'. */
5797 /* This implementation differs from that given by Guibas and Stolfi. It */
5798 /* walks from triangle to triangle, crossing an edge only if `searchpoint' */
5799 /* is on the other side of the line containing that edge. After entering */
5800 /* a triangle, there are two edges by which one can leave that triangle. */
5801 /* If both edges are valid (`searchpoint' is on the other side of both */
5802 /* edges), one of the two is chosen by drawing a line perpendicular to */
5803 /* the entry edge (whose endpoints are `forg' and `fdest') passing through */
5804 /* `fapex'. Depending on which side of this perpendicular `searchpoint' */
5805 /* falls on, an exit edge is chosen. */
5807 /* This implementation is empirically faster than the Guibas and Stolfi */
5808 /* point location routine (which I originally used), which tends to spiral */
5809 /* in toward its target. */
5811 /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */
5812 /* is a handle whose origin is the existing vertex. */
5814 /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */
5815 /* handle whose primary edge is the edge on which the point lies. */
5817 /* Returns INTRIANGLE if the point lies strictly within a triangle. */
5818 /* `searchtri' is a handle on the triangle that contains the point. */
5820 /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */
5821 /* handle whose primary edge the point is to the right of. This might */
5822 /* occur when the circumcenter of a triangle falls just slightly outside */
5823 /* the mesh due to floating-point roundoff error. It also occurs when */
5824 /* seeking a hole or region point that a foolish user has placed outside */
5827 /* WARNING: This routine is designed for convex triangulations, and will */
5828 /* not generally work after the holes and concavities have been carved. */
5829 /* However, it can still be used to find the circumcenter of a triangle, as */
5830 /* long as the search is begun from the triangle in question. */
5832 /*****************************************************************************/
5834 enum locateresult preciselocate( searchpoint, searchtri )
5836 struct triedge *searchtri;
5838 struct triedge backtracktri;
5839 point forg, fdest, fapex;
5841 REAL orgorient, destorient;
5843 triangle ptr; /* Temporary variable used by sym(). */
5845 if ( verbose > 2 ) {
5846 printf( " Searching for point (%.12g, %.12g).\n",
5847 searchpoint[0], searchpoint[1] );
5850 org( *searchtri, forg );
5851 dest( *searchtri, fdest );
5852 apex( *searchtri, fapex );
5854 if ( verbose > 2 ) {
5855 printf( " At (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
5856 forg[0], forg[1], fdest[0], fdest[1], fapex[0], fapex[1] );
5858 /* Check whether the apex is the point we seek. */
5859 if (( fapex[0] == searchpoint[0] ) && ( fapex[1] == searchpoint[1] )) {
5860 lprevself( *searchtri );
5863 /* Does the point lie on the other side of the line defined by the */
5864 /* triangle edge opposite the triangle's destination? */
5865 destorient = counterclockwise( forg, fapex, searchpoint );
5866 /* Does the point lie on the other side of the line defined by the */
5867 /* triangle edge opposite the triangle's origin? */
5868 orgorient = counterclockwise( fapex, fdest, searchpoint );
5869 if ( destorient > 0.0 ) {
5870 if ( orgorient > 0.0 ) {
5871 /* Move left if the inner product of (fapex - searchpoint) and */
5872 /* (fdest - forg) is positive. This is equivalent to drawing */
5873 /* a line perpendicular to the line (forg, fdest) passing */
5874 /* through `fapex', and determining which side of this line */
5875 /* `searchpoint' falls on. */
5876 moveleft = ( fapex[0] - searchpoint[0] ) * ( fdest[0] - forg[0] ) +
5877 ( fapex[1] - searchpoint[1] ) * ( fdest[1] - forg[1] ) > 0.0;
5884 if ( orgorient > 0.0 ) {
5888 /* The point we seek must be on the boundary of or inside this */
5890 if ( destorient == 0.0 ) {
5891 lprevself( *searchtri );
5894 if ( orgorient == 0.0 ) {
5895 lnextself( *searchtri );
5902 /* Move to another triangle. Leave a trace `backtracktri' in case */
5903 /* floating-point roundoff or some such bogey causes us to walk */
5904 /* off a boundary of the triangulation. We can just bounce off */
5905 /* the boundary as if it were an elastic band. */
5907 lprev( *searchtri, backtracktri );
5911 lnext( *searchtri, backtracktri );
5914 sym( backtracktri, *searchtri );
5916 /* Check for walking off the edge. */
5917 if ( searchtri->tri == dummytri ) {
5919 triedgecopy( backtracktri, *searchtri );
5923 apex( *searchtri, fapex );
5924 /* Check if the point really is beyond the triangulation boundary. */
5925 destorient = counterclockwise( forg, fapex, searchpoint );
5926 orgorient = counterclockwise( fapex, fdest, searchpoint );
5927 if (( orgorient < 0.0 ) && ( destorient < 0.0 )) {
5932 apex( *searchtri, fapex );
5937 /*****************************************************************************/
5939 /* locate() Find a triangle or edge containing a given point. */
5941 /* Searching begins from one of: the input `searchtri', a recently */
5942 /* encountered triangle `recenttri', or from a triangle chosen from a */
5943 /* random sample. The choice is made by determining which triangle's */
5944 /* origin is closest to the point we are searcing for. Normally, */
5945 /* `searchtri' should be a handle on the convex hull of the triangulation. */
5947 /* Details on the random sampling method can be found in the Mucke, Saias, */
5948 /* and Zhu paper cited in the header of this code. */
5950 /* On completion, `searchtri' is a triangle that contains `searchpoint'. */
5952 /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */
5953 /* is a handle whose origin is the existing vertex. */
5955 /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */
5956 /* handle whose primary edge is the edge on which the point lies. */
5958 /* Returns INTRIANGLE if the point lies strictly within a triangle. */
5959 /* `searchtri' is a handle on the triangle that contains the point. */
5961 /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */
5962 /* handle whose primary edge the point is to the right of. This might */
5963 /* occur when the circumcenter of a triangle falls just slightly outside */
5964 /* the mesh due to floating-point roundoff error. It also occurs when */
5965 /* seeking a hole or region point that a foolish user has placed outside */
5968 /* WARNING: This routine is designed for convex triangulations, and will */
5969 /* not generally work after the holes and concavities have been carved. */
5971 /*****************************************************************************/
5973 enum locateresult locate( searchpoint, searchtri )
5975 struct triedge *searchtri;
5979 struct triedge sampletri;
5981 unsigned long alignptr;
5982 REAL searchdist, dist;
5984 long sampleblocks, samplesperblock, samplenum;
5987 triangle ptr; /* Temporary variable used by sym(). */
5989 if ( verbose > 2 ) {
5990 printf( " Randomly sampling for a triangle near point (%.12g, %.12g).\n",
5991 searchpoint[0], searchpoint[1] );
5993 /* Record the distance from the suggested starting triangle to the */
5994 /* point we seek. */
5995 org( *searchtri, torg );
5996 searchdist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
5997 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
5998 if ( verbose > 2 ) {
5999 printf( " Boundary triangle has origin (%.12g, %.12g).\n",
6003 /* If a recently encountered triangle has been recorded and has not been */
6004 /* deallocated, test it as a good starting point. */
6005 if ( recenttri.tri != (triangle *) NULL ) {
6006 if ( recenttri.tri[3] != (triangle) NULL ) {
6007 org( recenttri, torg );
6008 if (( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] )) {
6009 triedgecopy( recenttri, *searchtri );
6012 dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
6013 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
6014 if ( dist < searchdist ) {
6015 triedgecopy( recenttri, *searchtri );
6017 if ( verbose > 2 ) {
6018 printf( " Choosing recent triangle with origin (%.12g, %.12g).\n",
6025 /* The number of random samples taken is proportional to the cube root of */
6026 /* the number of triangles in the mesh. The next bit of code assumes */
6027 /* that the number of triangles increases monotonically. */
6028 while ( SAMPLEFACTOR * samples * samples * samples < triangles.items ) {
6031 triblocks = ( triangles.maxitems + TRIPERBLOCK - 1 ) / TRIPERBLOCK;
6032 samplesperblock = 1 + ( samples / triblocks );
6033 sampleblocks = samples / samplesperblock;
6034 sampleblock = triangles.firstblock;
6035 sampletri.orient = 0;
6036 for ( i = 0; i < sampleblocks; i++ ) {
6037 alignptr = (unsigned long) ( sampleblock + 1 );
6038 firsttri = (triangle *) ( alignptr + (unsigned long) triangles.alignbytes
6039 - ( alignptr % (unsigned long) triangles.alignbytes ));
6040 for ( j = 0; j < samplesperblock; j++ ) {
6041 if ( i == triblocks - 1 ) {
6042 samplenum = randomnation((int)
6043 ( triangles.maxitems - ( i * TRIPERBLOCK )));
6046 samplenum = randomnation( TRIPERBLOCK );
6048 sampletri.tri = (triangle *)
6049 ( firsttri + ( samplenum * triangles.itemwords ));
6050 if ( sampletri.tri[3] != (triangle) NULL ) {
6051 org( sampletri, torg );
6052 dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
6053 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
6054 if ( dist < searchdist ) {
6055 triedgecopy( sampletri, *searchtri );
6057 if ( verbose > 2 ) {
6058 printf( " Choosing triangle with origin (%.12g, %.12g).\n",
6064 sampleblock = (VOID **) *sampleblock;
6067 org( *searchtri, torg );
6068 dest( *searchtri, tdest );
6069 /* Check the starting triangle's vertices. */
6070 if (( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] )) {
6073 if (( tdest[0] == searchpoint[0] ) && ( tdest[1] == searchpoint[1] )) {
6074 lnextself( *searchtri );
6077 /* Orient `searchtri' to fit the preconditions of calling preciselocate(). */
6078 ahead = counterclockwise( torg, tdest, searchpoint );
6079 if ( ahead < 0.0 ) {
6080 /* Turn around so that `searchpoint' is to the left of the */
6081 /* edge specified by `searchtri'. */
6082 symself( *searchtri );
6084 else if ( ahead == 0.0 ) {
6085 /* Check if `searchpoint' is between `torg' and `tdest'. */
6086 if ((( torg[0] < searchpoint[0] ) == ( searchpoint[0] < tdest[0] ))
6087 && (( torg[1] < searchpoint[1] ) == ( searchpoint[1] < tdest[1] ))) {
6091 return preciselocate( searchpoint, searchtri );
6096 /********* Point location routines end here *********/
6098 /********* Mesh transformation routines begin here *********/
6102 /*****************************************************************************/
6104 /* insertshelle() Create a new shell edge and insert it between two */
6107 /* The new shell edge is inserted at the edge described by the handle */
6108 /* `tri'. Its vertices are properly initialized. The marker `shellemark' */
6109 /* is applied to the shell edge and, if appropriate, its vertices. */
6111 /*****************************************************************************/
6113 void insertshelle( tri, shellemark )
6114 struct triedge *tri; /* Edge at which to insert the new shell edge. */
6115 int shellemark; /* Marker for the new shell edge. */
6117 struct triedge oppotri;
6118 struct edge newshelle;
6119 point triorg, tridest;
6120 triangle ptr; /* Temporary variable used by sym(). */
6121 shelle sptr; /* Temporary variable used by tspivot(). */
6123 /* Mark points if possible. */
6124 org( *tri, triorg );
6125 dest( *tri, tridest );
6126 if ( pointmark( triorg ) == 0 ) {
6127 setpointmark( triorg, shellemark );
6129 if ( pointmark( tridest ) == 0 ) {
6130 setpointmark( tridest, shellemark );
6132 /* Check if there's already a shell edge here. */
6133 tspivot( *tri, newshelle );
6134 if ( newshelle.sh == dummysh ) {
6135 /* Make new shell edge and initialize its vertices. */
6136 makeshelle( &newshelle );
6137 setsorg( newshelle, tridest );
6138 setsdest( newshelle, triorg );
6139 /* Bond new shell edge to the two triangles it is sandwiched between. */
6140 /* Note that the facing triangle `oppotri' might be equal to */
6141 /* `dummytri' (outer space), but the new shell edge is bonded to it */
6143 tsbond( *tri, newshelle );
6144 sym( *tri, oppotri );
6145 ssymself( newshelle );
6146 tsbond( oppotri, newshelle );
6147 setmark( newshelle, shellemark );
6148 if ( verbose > 2 ) {
6149 printf( " Inserting new " );
6150 printshelle( &newshelle );
6154 if ( mark( newshelle ) == 0 ) {
6155 setmark( newshelle, shellemark );
6160 /*****************************************************************************/
6164 /* A "local transformation" replaces a small set of triangles with another */
6165 /* set of triangles. This may or may not involve inserting or deleting a */
6168 /* The term "casing" is used to describe the set of triangles that are */
6169 /* attached to the triangles being transformed, but are not transformed */
6170 /* themselves. Think of the casing as a fixed hollow structure inside */
6171 /* which all the action happens. A "casing" is only defined relative to */
6172 /* a single transformation; each occurrence of a transformation will */
6173 /* involve a different casing. */
6175 /* A "shell" is similar to a "casing". The term "shell" describes the set */
6176 /* of shell edges (if any) that are attached to the triangles being */
6177 /* transformed. However, I sometimes use "shell" to refer to a single */
6178 /* shell edge, so don't get confused. */
6180 /*****************************************************************************/
6182 /*****************************************************************************/
6184 /* flip() Transform two triangles to two different triangles by flipping */
6185 /* an edge within a quadrilateral. */
6187 /* Imagine the original triangles, abc and bad, oriented so that the */
6188 /* shared edge ab lies in a horizontal plane, with the point b on the left */
6189 /* and the point a on the right. The point c lies below the edge, and the */
6190 /* point d lies above the edge. The `flipedge' handle holds the edge ab */
6191 /* of triangle abc, and is directed left, from vertex a to vertex b. */
6193 /* The triangles abc and bad are deleted and replaced by the triangles cdb */
6194 /* and dca. The triangles that represent abc and bad are NOT deallocated; */
6195 /* they are reused for dca and cdb, respectively. Hence, any handles that */
6196 /* may have held the original triangles are still valid, although not */
6197 /* directed as they were before. */
6199 /* Upon completion of this routine, the `flipedge' handle holds the edge */
6200 /* dc of triangle dca, and is directed down, from vertex d to vertex c. */
6201 /* (Hence, the two triangles have rotated counterclockwise.) */
6203 /* WARNING: This transformation is geometrically valid only if the */
6204 /* quadrilateral adbc is convex. Furthermore, this transformation is */
6205 /* valid only if there is not a shell edge between the triangles abc and */
6206 /* bad. This routine does not check either of these preconditions, and */
6207 /* it is the responsibility of the calling routine to ensure that they are */
6208 /* met. If they are not, the streets shall be filled with wailing and */
6209 /* gnashing of teeth. */
6211 /*****************************************************************************/
6213 void flip( flipedge )
6214 struct triedge *flipedge; /* Handle for the triangle abc. */
6216 struct triedge botleft, botright;
6217 struct triedge topleft, topright;
6219 struct triedge botlcasing, botrcasing;
6220 struct triedge toplcasing, toprcasing;
6221 struct edge botlshelle, botrshelle;
6222 struct edge toplshelle, toprshelle;
6223 point leftpoint, rightpoint, botpoint;
6225 triangle ptr; /* Temporary variable used by sym(). */
6226 shelle sptr; /* Temporary variable used by tspivot(). */
6228 /* Identify the vertices of the quadrilateral. */
6229 org( *flipedge, rightpoint );
6230 dest( *flipedge, leftpoint );
6231 apex( *flipedge, botpoint );
6232 sym( *flipedge, top );
6235 if ( top.tri == dummytri ) {
6236 printf( "Internal error in flip(): Attempt to flip on boundary.\n" );
6237 lnextself( *flipedge );
6240 if ( checksegments ) {
6241 tspivot( *flipedge, toplshelle );
6242 if ( toplshelle.sh != dummysh ) {
6243 printf( "Internal error in flip(): Attempt to flip a segment.\n" );
6244 lnextself( *flipedge );
6248 #endif /* SELF_CHECK */
6249 apex( top, farpoint );
6251 /* Identify the casing of the quadrilateral. */
6252 lprev( top, topleft );
6253 sym( topleft, toplcasing );
6254 lnext( top, topright );
6255 sym( topright, toprcasing );
6256 lnext( *flipedge, botleft );
6257 sym( botleft, botlcasing );
6258 lprev( *flipedge, botright );
6259 sym( botright, botrcasing );
6260 /* Rotate the quadrilateral one-quarter turn counterclockwise. */
6261 bond( topleft, botlcasing );
6262 bond( botleft, botrcasing );
6263 bond( botright, toprcasing );
6264 bond( topright, toplcasing );
6266 if ( checksegments ) {
6267 /* Check for shell edges and rebond them to the quadrilateral. */
6268 tspivot( topleft, toplshelle );
6269 tspivot( botleft, botlshelle );
6270 tspivot( botright, botrshelle );
6271 tspivot( topright, toprshelle );
6272 if ( toplshelle.sh == dummysh ) {
6273 tsdissolve( topright );
6276 tsbond( topright, toplshelle );
6278 if ( botlshelle.sh == dummysh ) {
6279 tsdissolve( topleft );
6282 tsbond( topleft, botlshelle );
6284 if ( botrshelle.sh == dummysh ) {
6285 tsdissolve( botleft );
6288 tsbond( botleft, botrshelle );
6290 if ( toprshelle.sh == dummysh ) {
6291 tsdissolve( botright );
6294 tsbond( botright, toprshelle );
6298 /* New point assignments for the rotated quadrilateral. */
6299 setorg( *flipedge, farpoint );
6300 setdest( *flipedge, botpoint );
6301 setapex( *flipedge, rightpoint );
6302 setorg( top, botpoint );
6303 setdest( top, farpoint );
6304 setapex( top, leftpoint );
6305 if ( verbose > 2 ) {
6306 printf( " Edge flip results in left " );
6307 lnextself( topleft );
6308 printtriangle( &topleft );
6309 printf( " and right " );
6310 printtriangle( flipedge );
6314 /*****************************************************************************/
6316 /* insertsite() Insert a vertex into a Delaunay triangulation, */
6317 /* performing flips as necessary to maintain the Delaunay */
6320 /* The point `insertpoint' is located. If `searchtri.tri' is not NULL, */
6321 /* the search for the containing triangle begins from `searchtri'. If */
6322 /* `searchtri.tri' is NULL, a full point location procedure is called. */
6323 /* If `insertpoint' is found inside a triangle, the triangle is split into */
6324 /* three; if `insertpoint' lies on an edge, the edge is split in two, */
6325 /* thereby splitting the two adjacent triangles into four. Edge flips are */
6326 /* used to restore the Delaunay property. If `insertpoint' lies on an */
6327 /* existing vertex, no action is taken, and the value DUPLICATEPOINT is */
6328 /* returned. On return, `searchtri' is set to a handle whose origin is the */
6329 /* existing vertex. */
6331 /* Normally, the parameter `splitedge' is set to NULL, implying that no */
6332 /* segment should be split. In this case, if `insertpoint' is found to */
6333 /* lie on a segment, no action is taken, and the value VIOLATINGPOINT is */
6334 /* returned. On return, `searchtri' is set to a handle whose primary edge */
6335 /* is the violated segment. */
6337 /* If the calling routine wishes to split a segment by inserting a point in */
6338 /* it, the parameter `splitedge' should be that segment. In this case, */
6339 /* `searchtri' MUST be the triangle handle reached by pivoting from that */
6340 /* segment; no point location is done. */
6342 /* `segmentflaws' and `triflaws' are flags that indicate whether or not */
6343 /* there should be checks for the creation of encroached segments or bad */
6344 /* quality faces. If a newly inserted point encroaches upon segments, */
6345 /* these segments are added to the list of segments to be split if */
6346 /* `segmentflaws' is set. If bad triangles are created, these are added */
6347 /* to the queue if `triflaws' is set. */
6349 /* If a duplicate point or violated segment does not prevent the point */
6350 /* from being inserted, the return value will be ENCROACHINGPOINT if the */
6351 /* point encroaches upon a segment (and checking is enabled), or */
6352 /* SUCCESSFULPOINT otherwise. In either case, `searchtri' is set to a */
6353 /* handle whose origin is the newly inserted vertex. */
6355 /* insertsite() does not use flip() for reasons of speed; some */
6356 /* information can be reused from edge flip to edge flip, like the */
6357 /* locations of shell edges. */
6359 /*****************************************************************************/
6361 enum insertsiteresult insertsite( insertpoint, searchtri, splitedge,
6362 segmentflaws, triflaws )
6364 struct triedge *searchtri;
6365 struct edge *splitedge;
6369 struct triedge horiz;
6371 struct triedge botleft, botright;
6372 struct triedge topleft, topright;
6373 struct triedge newbotleft, newbotright;
6374 struct triedge newtopright;
6375 struct triedge botlcasing, botrcasing;
6376 struct triedge toplcasing, toprcasing;
6377 struct triedge testtri;
6378 struct edge botlshelle, botrshelle;
6379 struct edge toplshelle, toprshelle;
6380 struct edge brokenshelle;
6381 struct edge checkshelle;
6382 struct edge rightedge;
6383 struct edge newedge;
6384 struct edge *encroached;
6386 point leftpoint, rightpoint, botpoint, toppoint, farpoint;
6389 enum insertsiteresult success;
6390 enum locateresult intersect;
6394 triangle ptr; /* Temporary variable used by sym(). */
6395 shelle sptr; /* Temporary variable used by spivot() and tspivot(). */
6397 if ( verbose > 1 ) {
6398 printf( " Inserting (%.12g, %.12g).\n", insertpoint[0], insertpoint[1] );
6400 if ( splitedge == (struct edge *) NULL ) {
6401 /* Find the location of the point to be inserted. Check if a good */
6402 /* starting triangle has already been provided by the caller. */
6403 if ( searchtri->tri == (triangle *) NULL ) {
6404 /* Find a boundary triangle. */
6405 horiz.tri = dummytri;
6408 /* Search for a triangle containing `insertpoint'. */
6409 intersect = locate( insertpoint, &horiz );
6412 /* Start searching from the triangle provided by the caller. */
6413 triedgecopy( *searchtri, horiz );
6414 intersect = preciselocate( insertpoint, &horiz );
6418 /* The calling routine provides the edge in which the point is inserted. */
6419 triedgecopy( *searchtri, horiz );
6422 if ( intersect == ONVERTEX ) {
6423 /* There's already a vertex there. Return in `searchtri' a triangle */
6424 /* whose origin is the existing vertex. */
6425 triedgecopy( horiz, *searchtri );
6426 triedgecopy( horiz, recenttri );
6427 return DUPLICATEPOINT;
6429 if (( intersect == ONEDGE ) || ( intersect == OUTSIDE )) {
6430 /* The vertex falls on an edge or boundary. */
6431 if ( checksegments && ( splitedge == (struct edge *) NULL )) {
6432 /* Check whether the vertex falls on a shell edge. */
6433 tspivot( horiz, brokenshelle );
6434 if ( brokenshelle.sh != dummysh ) {
6435 /* The vertex falls on a shell edge. */
6436 if ( segmentflaws ) {
6437 if ( nobisect == 0 ) {
6438 /* Add the shell edge to the list of encroached segments. */
6439 encroached = (struct edge *) poolalloc( &badsegments );
6440 shellecopy( brokenshelle, *encroached );
6442 else if (( nobisect == 1 ) && ( intersect == ONEDGE )) {
6443 /* This segment may be split only if it is an internal boundary. */
6444 sym( horiz, testtri );
6445 if ( testtri.tri != dummytri ) {
6446 /* Add the shell edge to the list of encroached segments. */
6447 encroached = (struct edge *) poolalloc( &badsegments );
6448 shellecopy( brokenshelle, *encroached );
6452 /* Return a handle whose primary edge contains the point, */
6453 /* which has not been inserted. */
6454 triedgecopy( horiz, *searchtri );
6455 triedgecopy( horiz, recenttri );
6456 return VIOLATINGPOINT;
6459 /* Insert the point on an edge, dividing one triangle into two (if */
6460 /* the edge lies on a boundary) or two triangles into four. */
6461 lprev( horiz, botright );
6462 sym( botright, botrcasing );
6463 sym( horiz, topright );
6464 /* Is there a second triangle? (Or does this edge lie on a boundary?) */
6465 mirrorflag = topright.tri != dummytri;
6467 lnextself( topright );
6468 sym( topright, toprcasing );
6469 maketriangle( &newtopright );
6472 /* Splitting the boundary edge increases the number of boundary edges. */
6475 maketriangle( &newbotright );
6477 /* Set the vertices of changed and new triangles. */
6478 org( horiz, rightpoint );
6479 dest( horiz, leftpoint );
6480 apex( horiz, botpoint );
6481 setorg( newbotright, botpoint );
6482 setdest( newbotright, rightpoint );
6483 setapex( newbotright, insertpoint );
6484 setorg( horiz, insertpoint );
6485 for ( i = 0; i < eextras; i++ ) {
6486 /* Set the element attributes of a new triangle. */
6487 setelemattribute( newbotright, i, elemattribute( botright, i ));
6490 /* Set the area constraint of a new triangle. */
6491 setareabound( newbotright, areabound( botright ));
6494 dest( topright, toppoint );
6495 setorg( newtopright, rightpoint );
6496 setdest( newtopright, toppoint );
6497 setapex( newtopright, insertpoint );
6498 setorg( topright, insertpoint );
6499 for ( i = 0; i < eextras; i++ ) {
6500 /* Set the element attributes of another new triangle. */
6501 setelemattribute( newtopright, i, elemattribute( topright, i ));
6504 /* Set the area constraint of another new triangle. */
6505 setareabound( newtopright, areabound( topright ));
6509 /* There may be shell edges that need to be bonded */
6510 /* to the new triangle(s). */
6511 if ( checksegments ) {
6512 tspivot( botright, botrshelle );
6513 if ( botrshelle.sh != dummysh ) {
6514 tsdissolve( botright );
6515 tsbond( newbotright, botrshelle );
6518 tspivot( topright, toprshelle );
6519 if ( toprshelle.sh != dummysh ) {
6520 tsdissolve( topright );
6521 tsbond( newtopright, toprshelle );
6526 /* Bond the new triangle(s) to the surrounding triangles. */
6527 bond( newbotright, botrcasing );
6528 lprevself( newbotright );
6529 bond( newbotright, botright );
6530 lprevself( newbotright );
6532 bond( newtopright, toprcasing );
6533 lnextself( newtopright );
6534 bond( newtopright, topright );
6535 lnextself( newtopright );
6536 bond( newtopright, newbotright );
6539 if ( splitedge != (struct edge *) NULL ) {
6540 /* Split the shell edge into two. */
6541 setsdest( *splitedge, insertpoint );
6542 ssymself( *splitedge );
6543 spivot( *splitedge, rightedge );
6544 insertshelle( &newbotright, mark( *splitedge ));
6545 tspivot( newbotright, newedge );
6546 sbond( *splitedge, newedge );
6547 ssymself( newedge );
6548 sbond( newedge, rightedge );
6549 ssymself( *splitedge );
6554 if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) {
6555 printf( "Internal error in insertsite():\n" );
6556 printf( " Clockwise triangle prior to edge point insertion (bottom).\n" );
6559 if ( counterclockwise( leftpoint, rightpoint, toppoint ) < 0.0 ) {
6560 printf( "Internal error in insertsite():\n" );
6561 printf( " Clockwise triangle prior to edge point insertion (top).\n" );
6563 if ( counterclockwise( rightpoint, toppoint, insertpoint ) < 0.0 ) {
6564 printf( "Internal error in insertsite():\n" );
6565 printf( " Clockwise triangle after edge point insertion (top right).\n"
6568 if ( counterclockwise( toppoint, leftpoint, insertpoint ) < 0.0 ) {
6569 printf( "Internal error in insertsite():\n" );
6570 printf( " Clockwise triangle after edge point insertion (top left).\n"
6574 if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) {
6575 printf( "Internal error in insertsite():\n" );
6576 printf( " Clockwise triangle after edge point insertion (bottom left).\n"
6579 if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) {
6580 printf( "Internal error in insertsite():\n" );
6582 " Clockwise triangle after edge point insertion (bottom right).\n" );
6584 #endif /* SELF_CHECK */
6585 if ( verbose > 2 ) {
6586 printf( " Updating bottom left " );
6587 printtriangle( &botright );
6589 printf( " Updating top left " );
6590 printtriangle( &topright );
6591 printf( " Creating top right " );
6592 printtriangle( &newtopright );
6594 printf( " Creating bottom right " );
6595 printtriangle( &newbotright );
6598 /* Position `horiz' on the first edge to check for */
6599 /* the Delaunay property. */
6603 /* Insert the point in a triangle, splitting it into three. */
6604 lnext( horiz, botleft );
6605 lprev( horiz, botright );
6606 sym( botleft, botlcasing );
6607 sym( botright, botrcasing );
6608 maketriangle( &newbotleft );
6609 maketriangle( &newbotright );
6611 /* Set the vertices of changed and new triangles. */
6612 org( horiz, rightpoint );
6613 dest( horiz, leftpoint );
6614 apex( horiz, botpoint );
6615 setorg( newbotleft, leftpoint );
6616 setdest( newbotleft, botpoint );
6617 setapex( newbotleft, insertpoint );
6618 setorg( newbotright, botpoint );
6619 setdest( newbotright, rightpoint );
6620 setapex( newbotright, insertpoint );
6621 setapex( horiz, insertpoint );
6622 for ( i = 0; i < eextras; i++ ) {
6623 /* Set the element attributes of the new triangles. */
6624 attrib = elemattribute( horiz, i );
6625 setelemattribute( newbotleft, i, attrib );
6626 setelemattribute( newbotright, i, attrib );
6629 /* Set the area constraint of the new triangles. */
6630 area = areabound( horiz );
6631 setareabound( newbotleft, area );
6632 setareabound( newbotright, area );
6635 /* There may be shell edges that need to be bonded */
6636 /* to the new triangles. */
6637 if ( checksegments ) {
6638 tspivot( botleft, botlshelle );
6639 if ( botlshelle.sh != dummysh ) {
6640 tsdissolve( botleft );
6641 tsbond( newbotleft, botlshelle );
6643 tspivot( botright, botrshelle );
6644 if ( botrshelle.sh != dummysh ) {
6645 tsdissolve( botright );
6646 tsbond( newbotright, botrshelle );
6650 /* Bond the new triangles to the surrounding triangles. */
6651 bond( newbotleft, botlcasing );
6652 bond( newbotright, botrcasing );
6653 lnextself( newbotleft );
6654 lprevself( newbotright );
6655 bond( newbotleft, newbotright );
6656 lnextself( newbotleft );
6657 bond( botleft, newbotleft );
6658 lprevself( newbotright );
6659 bond( botright, newbotright );
6663 if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) {
6664 printf( "Internal error in insertsite():\n" );
6665 printf( " Clockwise triangle prior to point insertion.\n" );
6667 if ( counterclockwise( rightpoint, leftpoint, insertpoint ) < 0.0 ) {
6668 printf( "Internal error in insertsite():\n" );
6669 printf( " Clockwise triangle after point insertion (top).\n" );
6671 if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) {
6672 printf( "Internal error in insertsite():\n" );
6673 printf( " Clockwise triangle after point insertion (left).\n" );
6675 if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) {
6676 printf( "Internal error in insertsite():\n" );
6677 printf( " Clockwise triangle after point insertion (right).\n" );
6679 #endif /* SELF_CHECK */
6680 if ( verbose > 2 ) {
6681 printf( " Updating top " );
6682 printtriangle( &horiz );
6683 printf( " Creating left " );
6684 printtriangle( &newbotleft );
6685 printf( " Creating right " );
6686 printtriangle( &newbotright );
6690 /* The insertion is successful by default, unless an encroached */
6691 /* edge is found. */
6692 success = SUCCESSFULPOINT;
6693 /* Circle around the newly inserted vertex, checking each edge opposite */
6694 /* it for the Delaunay property. Non-Delaunay edges are flipped. */
6695 /* `horiz' is always the edge being checked. `first' marks where to */
6696 /* stop circling. */
6697 org( horiz, first );
6699 dest( horiz, leftpoint );
6700 /* Circle until finished. */
6702 /* By default, the edge will be flipped. */
6704 if ( checksegments ) {
6705 /* Check for a segment, which cannot be flipped. */
6706 tspivot( horiz, checkshelle );
6707 if ( checkshelle.sh != dummysh ) {
6708 /* The edge is a segment and cannot be flipped. */
6712 if ( segmentflaws ) {
6713 /* Does the new point encroach upon this segment? */
6714 if ( checkedge4encroach( &checkshelle )) {
6715 success = ENCROACHINGPOINT;
6718 #endif /* not CDT_ONLY */
6722 /* Check if the edge is a boundary edge. */
6724 if ( top.tri == dummytri ) {
6725 /* The edge is a boundary edge and cannot be flipped. */
6729 /* Find the point on the other side of the edge. */
6730 apex( top, farpoint );
6731 /* In the incremental Delaunay triangulation algorithm, any of */
6732 /* `leftpoint', `rightpoint', and `farpoint' could be vertices */
6733 /* of the triangular bounding box. These vertices must be */
6734 /* treated as if they are infinitely distant, even though their */
6735 /* "coordinates" are not. */
6736 if (( leftpoint == infpoint1 ) || ( leftpoint == infpoint2 )
6737 || ( leftpoint == infpoint3 )) {
6738 /* `leftpoint' is infinitely distant. Check the convexity of */
6739 /* the boundary of the triangulation. 'farpoint' might be */
6740 /* infinite as well, but trust me, this same condition */
6741 /* should be applied. */
6742 doflip = counterclockwise( insertpoint, rightpoint, farpoint ) > 0.0;
6744 else if (( rightpoint == infpoint1 ) || ( rightpoint == infpoint2 )
6745 || ( rightpoint == infpoint3 )) {
6746 /* `rightpoint' is infinitely distant. Check the convexity of */
6747 /* the boundary of the triangulation. 'farpoint' might be */
6748 /* infinite as well, but trust me, this same condition */
6749 /* should be applied. */
6750 doflip = counterclockwise( farpoint, leftpoint, insertpoint ) > 0.0;
6752 else if (( farpoint == infpoint1 ) || ( farpoint == infpoint2 )
6753 || ( farpoint == infpoint3 )) {
6754 /* `farpoint' is infinitely distant and cannot be inside */
6755 /* the circumcircle of the triangle `horiz'. */
6759 /* Test whether the edge is locally Delaunay. */
6760 doflip = incircle( leftpoint, insertpoint, rightpoint, farpoint )
6764 /* We made it! Flip the edge `horiz' by rotating its containing */
6765 /* quadrilateral (the two triangles adjacent to `horiz'). */
6766 /* Identify the casing of the quadrilateral. */
6767 lprev( top, topleft );
6768 sym( topleft, toplcasing );
6769 lnext( top, topright );
6770 sym( topright, toprcasing );
6771 lnext( horiz, botleft );
6772 sym( botleft, botlcasing );
6773 lprev( horiz, botright );
6774 sym( botright, botrcasing );
6775 /* Rotate the quadrilateral one-quarter turn counterclockwise. */
6776 bond( topleft, botlcasing );
6777 bond( botleft, botrcasing );
6778 bond( botright, toprcasing );
6779 bond( topright, toplcasing );
6780 if ( checksegments ) {
6781 /* Check for shell edges and rebond them to the quadrilateral. */
6782 tspivot( topleft, toplshelle );
6783 tspivot( botleft, botlshelle );
6784 tspivot( botright, botrshelle );
6785 tspivot( topright, toprshelle );
6786 if ( toplshelle.sh == dummysh ) {
6787 tsdissolve( topright );
6790 tsbond( topright, toplshelle );
6792 if ( botlshelle.sh == dummysh ) {
6793 tsdissolve( topleft );
6796 tsbond( topleft, botlshelle );
6798 if ( botrshelle.sh == dummysh ) {
6799 tsdissolve( botleft );
6802 tsbond( botleft, botrshelle );
6804 if ( toprshelle.sh == dummysh ) {
6805 tsdissolve( botright );
6808 tsbond( botright, toprshelle );
6811 /* New point assignments for the rotated quadrilateral. */
6812 setorg( horiz, farpoint );
6813 setdest( horiz, insertpoint );
6814 setapex( horiz, rightpoint );
6815 setorg( top, insertpoint );
6816 setdest( top, farpoint );
6817 setapex( top, leftpoint );
6818 for ( i = 0; i < eextras; i++ ) {
6819 /* Take the average of the two triangles' attributes. */
6820 attrib = (REAL)( 0.5 * ( elemattribute( top, i ) + elemattribute( horiz, i )));
6821 setelemattribute( top, i, attrib );
6822 setelemattribute( horiz, i, attrib );
6825 if (( areabound( top ) <= 0.0 ) || ( areabound( horiz ) <= 0.0 )) {
6829 /* Take the average of the two triangles' area constraints. */
6830 /* This prevents small area constraints from migrating a */
6831 /* long, long way from their original location due to flips. */
6832 area = (REAL)( 0.5 * ( areabound( top ) + areabound( horiz )));
6834 setareabound( top, area );
6835 setareabound( horiz, area );
6839 if ( insertpoint != (point) NULL ) {
6840 if ( counterclockwise( leftpoint, insertpoint, rightpoint ) < 0.0 ) {
6841 printf( "Internal error in insertsite():\n" );
6842 printf( " Clockwise triangle prior to edge flip (bottom).\n" );
6844 /* The following test has been removed because constrainededge() */
6845 /* sometimes generates inverted triangles that insertsite() */
6848 if (counterclockwise(rightpoint, farpoint, leftpoint) < 0.0) {
6849 printf("Internal error in insertsite():\n");
6850 printf(" Clockwise triangle prior to edge flip (top).\n");
6853 if ( counterclockwise( farpoint, leftpoint, insertpoint ) < 0.0 ) {
6854 printf( "Internal error in insertsite():\n" );
6855 printf( " Clockwise triangle after edge flip (left).\n" );
6857 if ( counterclockwise( insertpoint, rightpoint, farpoint ) < 0.0 ) {
6858 printf( "Internal error in insertsite():\n" );
6859 printf( " Clockwise triangle after edge flip (right).\n" );
6862 #endif /* SELF_CHECK */
6863 if ( verbose > 2 ) {
6864 printf( " Edge flip results in left " );
6865 lnextself( topleft );
6866 printtriangle( &topleft );
6867 printf( " and right " );
6868 printtriangle( &horiz );
6870 /* On the next iterations, consider the two edges that were */
6871 /* exposed (this is, are now visible to the newly inserted */
6872 /* point) by the edge flip. */
6874 leftpoint = farpoint;
6879 /* The handle `horiz' is accepted as locally Delaunay. */
6883 /* Check the triangle `horiz' for quality. */
6884 testtriangle( &horiz );
6886 #endif /* not CDT_ONLY */
6887 /* Look for the next edge around the newly inserted point. */
6889 sym( horiz, testtri );
6890 /* Check for finishing a complete revolution about the new point, or */
6891 /* falling off the edge of the triangulation. The latter will */
6892 /* happen when a point is inserted at a boundary. */
6893 if (( leftpoint == first ) || ( testtri.tri == dummytri )) {
6894 /* We're done. Return a triangle whose origin is the new point. */
6895 lnext( horiz, *searchtri );
6896 lnext( horiz, recenttri );
6899 /* Finish finding the next edge around the newly inserted point. */
6900 lnext( testtri, horiz );
6901 rightpoint = leftpoint;
6902 dest( horiz, leftpoint );
6907 /*****************************************************************************/
6909 /* triangulatepolygon() Find the Delaunay triangulation of a polygon that */
6910 /* has a certain "nice" shape. This includes the */
6911 /* polygons that result from deletion of a point or */
6912 /* insertion of a segment. */
6914 /* This is a conceptually difficult routine. The starting assumption is */
6915 /* that we have a polygon with n sides. n - 1 of these sides are currently */
6916 /* represented as edges in the mesh. One side, called the "base", need not */
6919 /* Inside the polygon is a structure I call a "fan", consisting of n - 1 */
6920 /* triangles that share a common origin. For each of these triangles, the */
6921 /* edge opposite the origin is one of the sides of the polygon. The */
6922 /* primary edge of each triangle is the edge directed from the origin to */
6923 /* the destination; note that this is not the same edge that is a side of */
6924 /* the polygon. `firstedge' is the primary edge of the first triangle. */
6925 /* From there, the triangles follow in counterclockwise order about the */
6926 /* polygon, until `lastedge', the primary edge of the last triangle. */
6927 /* `firstedge' and `lastedge' are probably connected to other triangles */
6928 /* beyond the extremes of the fan, but their identity is not important, as */
6929 /* long as the fan remains connected to them. */
6931 /* Imagine the polygon oriented so that its base is at the bottom. This */
6932 /* puts `firstedge' on the far right, and `lastedge' on the far left. */
6933 /* The right vertex of the base is the destination of `firstedge', and the */
6934 /* left vertex of the base is the apex of `lastedge'. */
6936 /* The challenge now is to find the right sequence of edge flips to */
6937 /* transform the fan into a Delaunay triangulation of the polygon. Each */
6938 /* edge flip effectively removes one triangle from the fan, committing it */
6939 /* to the polygon. The resulting polygon has one fewer edge. If `doflip' */
6940 /* is set, the final flip will be performed, resulting in a fan of one */
6941 /* (useless?) triangle. If `doflip' is not set, the final flip is not */
6942 /* performed, resulting in a fan of two triangles, and an unfinished */
6943 /* triangular polygon that is not yet filled out with a single triangle. */
6944 /* On completion of the routine, `lastedge' is the last remaining triangle, */
6945 /* or the leftmost of the last two. */
6947 /* Although the flips are performed in the order described above, the */
6948 /* decisions about what flips to perform are made in precisely the reverse */
6949 /* order. The recursive triangulatepolygon() procedure makes a decision, */
6950 /* uses up to two recursive calls to triangulate the "subproblems" */
6951 /* (polygons with fewer edges), and then performs an edge flip. */
6953 /* The "decision" it makes is which vertex of the polygon should be */
6954 /* connected to the base. This decision is made by testing every possible */
6955 /* vertex. Once the best vertex is found, the two edges that connect this */
6956 /* vertex to the base become the bases for two smaller polygons. These */
6957 /* are triangulated recursively. Unfortunately, this approach can take */
6958 /* O(n^2) time not only in the worst case, but in many common cases. It's */
6959 /* rarely a big deal for point deletion, where n is rarely larger than ten, */
6960 /* but it could be a big deal for segment insertion, especially if there's */
6961 /* a lot of long segments that each cut many triangles. I ought to code */
6962 /* a faster algorithm some time. */
6964 /* The `edgecount' parameter is the number of sides of the polygon, */
6965 /* including its base. `triflaws' is a flag that determines whether the */
6966 /* new triangles should be tested for quality, and enqueued if they are */
6969 /*****************************************************************************/
6971 void triangulatepolygon( firstedge, lastedge, edgecount, doflip, triflaws )
6972 struct triedge *firstedge;
6973 struct triedge *lastedge;
6978 struct triedge testtri;
6979 struct triedge besttri;
6980 struct triedge tempedge;
6981 point leftbasepoint, rightbasepoint;
6986 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
6988 /* Identify the base vertices. */
6989 apex( *lastedge, leftbasepoint );
6990 dest( *firstedge, rightbasepoint );
6991 if ( verbose > 2 ) {
6992 printf( " Triangulating interior polygon at edge\n" );
6993 printf( " (%.12g, %.12g) (%.12g, %.12g)\n", leftbasepoint[0],
6994 leftbasepoint[1], rightbasepoint[0], rightbasepoint[1] );
6996 /* Find the best vertex to connect the base to. */
6997 onext( *firstedge, besttri );
6998 dest( besttri, bestpoint );
6999 triedgecopy( besttri, testtri );
7001 for ( i = 2; i <= edgecount - 2; i++ ) {
7002 onextself( testtri );
7003 dest( testtri, testpoint );
7004 /* Is this a better vertex? */
7005 if ( incircle( leftbasepoint, rightbasepoint, bestpoint, testpoint ) > 0.0 ) {
7006 triedgecopy( testtri, besttri );
7007 bestpoint = testpoint;
7011 if ( verbose > 2 ) {
7012 printf( " Connecting edge to (%.12g, %.12g)\n", bestpoint[0],
7015 if ( bestnumber > 1 ) {
7016 /* Recursively triangulate the smaller polygon on the right. */
7017 oprev( besttri, tempedge );
7018 triangulatepolygon( firstedge, &tempedge, bestnumber + 1, 1, triflaws );
7020 if ( bestnumber < edgecount - 2 ) {
7021 /* Recursively triangulate the smaller polygon on the left. */
7022 sym( besttri, tempedge );
7023 triangulatepolygon( &besttri, lastedge, edgecount - bestnumber, 1,
7025 /* Find `besttri' again; it may have been lost to edge flips. */
7026 sym( tempedge, besttri );
7029 /* Do one final edge flip. */
7034 /* Check the quality of the newly committed triangle. */
7035 sym( besttri, testtri );
7036 testtriangle( &testtri );
7038 #endif /* not CDT_ONLY */
7040 /* Return the base triangle. */
7041 triedgecopy( besttri, *lastedge );
7044 /*****************************************************************************/
7046 /* deletesite() Delete a vertex from a Delaunay triangulation, ensuring */
7047 /* that the triangulation remains Delaunay. */
7049 /* The origin of `deltri' is deleted. The union of the triangles adjacent */
7050 /* to this point is a polygon, for which the Delaunay triangulation is */
7051 /* found. Two triangles are removed from the mesh. */
7053 /* Only interior points that do not lie on segments (shell edges) or */
7054 /* boundaries may be deleted. */
7056 /*****************************************************************************/
7061 void deletesite( deltri )
7062 struct triedge *deltri;
7064 struct triedge countingtri;
7065 struct triedge firstedge, lastedge;
7066 struct triedge deltriright;
7067 struct triedge lefttri, righttri;
7068 struct triedge leftcasing, rightcasing;
7069 struct edge leftshelle, rightshelle;
7073 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
7074 shelle sptr; /* Temporary variable used by tspivot(). */
7076 org( *deltri, delpoint );
7077 if ( verbose > 1 ) {
7078 printf( " Deleting (%.12g, %.12g).\n", delpoint[0], delpoint[1] );
7080 pointdealloc( delpoint );
7082 /* Count the degree of the point being deleted. */
7083 onext( *deltri, countingtri );
7085 while ( !triedgeequal( *deltri, countingtri )) {
7088 if ( countingtri.tri == dummytri ) {
7089 printf( "Internal error in deletesite():\n" );
7090 printf( " Attempt to delete boundary point.\n" );
7093 #endif /* SELF_CHECK */
7095 onextself( countingtri );
7100 if ( edgecount < 3 ) {
7101 printf( "Internal error in deletesite():\n Point has degree %d.\n",
7105 #endif /* SELF_CHECK */
7106 if ( edgecount > 3 ) {
7107 /* Triangulate the polygon defined by the union of all triangles */
7108 /* adjacent to the point being deleted. Check the quality of */
7109 /* the resulting triangles. */
7110 onext( *deltri, firstedge );
7111 oprev( *deltri, lastedge );
7112 triangulatepolygon( &firstedge, &lastedge, edgecount, 0, !nobisect );
7114 /* Splice out two triangles. */
7115 lprev( *deltri, deltriright );
7116 dnext( *deltri, lefttri );
7117 sym( lefttri, leftcasing );
7118 oprev( deltriright, righttri );
7119 sym( righttri, rightcasing );
7120 bond( *deltri, leftcasing );
7121 bond( deltriright, rightcasing );
7122 tspivot( lefttri, leftshelle );
7123 if ( leftshelle.sh != dummysh ) {
7124 tsbond( *deltri, leftshelle );
7126 tspivot( righttri, rightshelle );
7127 if ( rightshelle.sh != dummysh ) {
7128 tsbond( deltriright, rightshelle );
7131 /* Set the new origin of `deltri' and check its quality. */
7132 org( lefttri, neworg );
7133 setorg( *deltri, neworg );
7135 testtriangle( deltri );
7138 /* Delete the two spliced-out triangles. */
7139 triangledealloc( lefttri.tri );
7140 triangledealloc( righttri.tri );
7143 #endif /* not CDT_ONLY */
7147 /********* Mesh transformation routines end here *********/
7149 /********* Divide-and-conquer Delaunay triangulation begins here *********/
7153 /*****************************************************************************/
7155 /* The divide-and-conquer bounding box */
7157 /* I originally implemented the divide-and-conquer and incremental Delaunay */
7158 /* triangulations using the edge-based data structure presented by Guibas */
7159 /* and Stolfi. Switching to a triangle-based data structure doubled the */
7160 /* speed. However, I had to think of a few extra tricks to maintain the */
7161 /* elegance of the original algorithms. */
7163 /* The "bounding box" used by my variant of the divide-and-conquer */
7164 /* algorithm uses one triangle for each edge of the convex hull of the */
7165 /* triangulation. These bounding triangles all share a common apical */
7166 /* vertex, which is represented by NULL and which represents nothing. */
7167 /* The bounding triangles are linked in a circular fan about this NULL */
7168 /* vertex, and the edges on the convex hull of the triangulation appear */
7169 /* opposite the NULL vertex. You might find it easiest to imagine that */
7170 /* the NULL vertex is a point in 3D space behind the center of the */
7171 /* triangulation, and that the bounding triangles form a sort of cone. */
7173 /* This bounding box makes it easy to represent degenerate cases. For */
7174 /* instance, the triangulation of two vertices is a single edge. This edge */
7175 /* is represented by two bounding box triangles, one on each "side" of the */
7176 /* edge. These triangles are also linked together in a fan about the NULL */
7179 /* The bounding box also makes it easy to traverse the convex hull, as the */
7180 /* divide-and-conquer algorithm needs to do. */
7182 /*****************************************************************************/
7184 /*****************************************************************************/
7186 /* pointsort() Sort an array of points by x-coordinate, using the */
7187 /* y-coordinate as a secondary key. */
7189 /* Uses quicksort. Randomized O(n log n) time. No, I did not make any of */
7190 /* the usual quicksort mistakes. */
7192 /*****************************************************************************/
7194 void pointsort( sortarray, arraysize )
7200 REAL pivotx, pivoty;
7203 if ( arraysize == 2 ) {
7204 /* Recursive base case. */
7205 if (( sortarray[0][0] > sortarray[1][0] ) ||
7206 (( sortarray[0][0] == sortarray[1][0] ) &&
7207 ( sortarray[0][1] > sortarray[1][1] ))) {
7208 temp = sortarray[1];
7209 sortarray[1] = sortarray[0];
7210 sortarray[0] = temp;
7214 /* Choose a random pivot to split the array. */
7215 pivot = (int) randomnation( arraysize );
7216 pivotx = sortarray[pivot][0];
7217 pivoty = sortarray[pivot][1];
7218 /* Split the array. */
7221 while ( left < right ) {
7222 /* Search for a point whose x-coordinate is too large for the left. */
7225 } while (( left <= right ) && (( sortarray[left][0] < pivotx ) ||
7226 (( sortarray[left][0] == pivotx ) &&
7227 ( sortarray[left][1] < pivoty ))));
7228 /* Search for a point whose x-coordinate is too small for the right. */
7231 } while (( left <= right ) && (( sortarray[right][0] > pivotx ) ||
7232 (( sortarray[right][0] == pivotx ) &&
7233 ( sortarray[right][1] > pivoty ))));
7234 if ( left < right ) {
7235 /* Swap the left and right points. */
7236 temp = sortarray[left];
7237 sortarray[left] = sortarray[right];
7238 sortarray[right] = temp;
7242 /* Recursively sort the left subset. */
7243 pointsort( sortarray, left );
7245 if ( right < arraysize - 2 ) {
7246 /* Recursively sort the right subset. */
7247 pointsort( &sortarray[right + 1], arraysize - right - 1 );
7251 /*****************************************************************************/
7253 /* pointmedian() An order statistic algorithm, almost. Shuffles an array */
7254 /* of points so that the first `median' points occur */
7255 /* lexicographically before the remaining points. */
7257 /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */
7258 /* if axis == 1. Very similar to the pointsort() procedure, but runs in */
7259 /* randomized linear time. */
7261 /*****************************************************************************/
7263 void pointmedian( sortarray, arraysize, median, axis )
7271 REAL pivot1, pivot2;
7274 if ( arraysize == 2 ) {
7275 /* Recursive base case. */
7276 if (( sortarray[0][axis] > sortarray[1][axis] ) ||
7277 (( sortarray[0][axis] == sortarray[1][axis] ) &&
7278 ( sortarray[0][1 - axis] > sortarray[1][1 - axis] ))) {
7279 temp = sortarray[1];
7280 sortarray[1] = sortarray[0];
7281 sortarray[0] = temp;
7285 /* Choose a random pivot to split the array. */
7286 pivot = (int) randomnation( arraysize );
7287 pivot1 = sortarray[pivot][axis];
7288 pivot2 = sortarray[pivot][1 - axis];
7289 /* Split the array. */
7292 while ( left < right ) {
7293 /* Search for a point whose x-coordinate is too large for the left. */
7296 } while (( left <= right ) && (( sortarray[left][axis] < pivot1 ) ||
7297 (( sortarray[left][axis] == pivot1 ) &&
7298 ( sortarray[left][1 - axis] < pivot2 ))));
7299 /* Search for a point whose x-coordinate is too small for the right. */
7302 } while (( left <= right ) && (( sortarray[right][axis] > pivot1 ) ||
7303 (( sortarray[right][axis] == pivot1 ) &&
7304 ( sortarray[right][1 - axis] > pivot2 ))));
7305 if ( left < right ) {
7306 /* Swap the left and right points. */
7307 temp = sortarray[left];
7308 sortarray[left] = sortarray[right];
7309 sortarray[right] = temp;
7312 /* Unlike in pointsort(), at most one of the following */
7313 /* conditionals is true. */
7314 if ( left > median ) {
7315 /* Recursively shuffle the left subset. */
7316 pointmedian( sortarray, left, median, axis );
7318 if ( right < median - 1 ) {
7319 /* Recursively shuffle the right subset. */
7320 pointmedian( &sortarray[right + 1], arraysize - right - 1,
7321 median - right - 1, axis );
7325 /*****************************************************************************/
7327 /* alternateaxes() Sorts the points as appropriate for the divide-and- */
7328 /* conquer algorithm with alternating cuts. */
7330 /* Partitions by x-coordinate if axis == 0; by y-coordinate if axis == 1. */
7331 /* For the base case, subsets containing only two or three points are */
7332 /* always sorted by x-coordinate. */
7334 /*****************************************************************************/
7336 void alternateaxes( sortarray, arraysize, axis )
7343 divider = arraysize >> 1;
7344 if ( arraysize <= 3 ) {
7345 /* Recursive base case: subsets of two or three points will be */
7346 /* handled specially, and should always be sorted by x-coordinate. */
7349 /* Partition with a horizontal or vertical cut. */
7350 pointmedian( sortarray, arraysize, divider, axis );
7351 /* Recursively partition the subsets with a cross cut. */
7352 if ( arraysize - divider >= 2 ) {
7353 if ( divider >= 2 ) {
7354 alternateaxes( sortarray, divider, 1 - axis );
7356 alternateaxes( &sortarray[divider], arraysize - divider, 1 - axis );
7360 /*****************************************************************************/
7362 /* mergehulls() Merge two adjacent Delaunay triangulations into a */
7363 /* single Delaunay triangulation. */
7365 /* This is similar to the algorithm given by Guibas and Stolfi, but uses */
7366 /* a triangle-based, rather than edge-based, data structure. */
7368 /* The algorithm walks up the gap between the two triangulations, knitting */
7369 /* them together. As they are merged, some of their bounding triangles */
7370 /* are converted into real triangles of the triangulation. The procedure */
7371 /* pulls each hull's bounding triangles apart, then knits them together */
7372 /* like the teeth of two gears. The Delaunay property determines, at each */
7373 /* step, whether the next "tooth" is a bounding triangle of the left hull */
7374 /* or the right. When a bounding triangle becomes real, its apex is */
7375 /* changed from NULL to a real point. */
7377 /* Only two new triangles need to be allocated. These become new bounding */
7378 /* triangles at the top and bottom of the seam. They are used to connect */
7379 /* the remaining bounding triangles (those that have not been converted */
7380 /* into real triangles) into a single fan. */
7382 /* On entry, `farleft' and `innerleft' are bounding triangles of the left */
7383 /* triangulation. The origin of `farleft' is the leftmost vertex, and */
7384 /* the destination of `innerleft' is the rightmost vertex of the */
7385 /* triangulation. Similarly, `innerright' and `farright' are bounding */
7386 /* triangles of the right triangulation. The origin of `innerright' and */
7387 /* destination of `farright' are the leftmost and rightmost vertices. */
7389 /* On completion, the origin of `farleft' is the leftmost vertex of the */
7390 /* merged triangulation, and the destination of `farright' is the rightmost */
7393 /*****************************************************************************/
7395 void mergehulls( farleft, innerleft, innerright, farright, axis )
7396 struct triedge *farleft;
7397 struct triedge *innerleft;
7398 struct triedge *innerright;
7399 struct triedge *farright;
7402 struct triedge leftcand, rightcand;
7403 struct triedge baseedge;
7404 struct triedge nextedge;
7405 struct triedge sidecasing, topcasing, outercasing;
7406 struct triedge checkedge;
7407 point innerleftdest;
7408 point innerrightorg;
7409 point innerleftapex, innerrightapex;
7410 point farleftpt, farrightpt;
7411 point farleftapex, farrightapex;
7412 point lowerleft, lowerright;
7413 point upperleft, upperright;
7418 int leftfinished, rightfinished;
7419 triangle ptr; /* Temporary variable used by sym(). */
7421 dest( *innerleft, innerleftdest );
7422 apex( *innerleft, innerleftapex );
7423 org( *innerright, innerrightorg );
7424 apex( *innerright, innerrightapex );
7425 /* Special treatment for horizontal cuts. */
7426 if ( dwyer && ( axis == 1 )) {
7427 org( *farleft, farleftpt );
7428 apex( *farleft, farleftapex );
7429 dest( *farright, farrightpt );
7430 apex( *farright, farrightapex );
7431 /* The pointers to the extremal points are shifted to point to the */
7432 /* topmost and bottommost point of each hull, rather than the */
7433 /* leftmost and rightmost points. */
7434 while ( farleftapex[1] < farleftpt[1] ) {
7435 lnextself( *farleft );
7436 symself( *farleft );
7437 farleftpt = farleftapex;
7438 apex( *farleft, farleftapex );
7440 sym( *innerleft, checkedge );
7441 apex( checkedge, checkvertex );
7442 while ( checkvertex[1] > innerleftdest[1] ) {
7443 lnext( checkedge, *innerleft );
7444 innerleftapex = innerleftdest;
7445 innerleftdest = checkvertex;
7446 sym( *innerleft, checkedge );
7447 apex( checkedge, checkvertex );
7449 while ( innerrightapex[1] < innerrightorg[1] ) {
7450 lnextself( *innerright );
7451 symself( *innerright );
7452 innerrightorg = innerrightapex;
7453 apex( *innerright, innerrightapex );
7455 sym( *farright, checkedge );
7456 apex( checkedge, checkvertex );
7457 while ( checkvertex[1] > farrightpt[1] ) {
7458 lnext( checkedge, *farright );
7459 farrightapex = farrightpt;
7460 farrightpt = checkvertex;
7461 sym( *farright, checkedge );
7462 apex( checkedge, checkvertex );
7465 /* Find a line tangent to and below both hulls. */
7468 /* Make innerleftdest the "bottommost" point of the left hull. */
7469 if ( counterclockwise( innerleftdest, innerleftapex, innerrightorg ) > 0.0 ) {
7470 lprevself( *innerleft );
7471 symself( *innerleft );
7472 innerleftdest = innerleftapex;
7473 apex( *innerleft, innerleftapex );
7476 /* Make innerrightorg the "bottommost" point of the right hull. */
7477 if ( counterclockwise( innerrightapex, innerrightorg, innerleftdest ) > 0.0 ) {
7478 lnextself( *innerright );
7479 symself( *innerright );
7480 innerrightorg = innerrightapex;
7481 apex( *innerright, innerrightapex );
7484 } while ( changemade );
7485 /* Find the two candidates to be the next "gear tooth". */
7486 sym( *innerleft, leftcand );
7487 sym( *innerright, rightcand );
7488 /* Create the bottom new bounding triangle. */
7489 maketriangle( &baseedge );
7490 /* Connect it to the bounding boxes of the left and right triangulations. */
7491 bond( baseedge, *innerleft );
7492 lnextself( baseedge );
7493 bond( baseedge, *innerright );
7494 lnextself( baseedge );
7495 setorg( baseedge, innerrightorg );
7496 setdest( baseedge, innerleftdest );
7497 /* Apex is intentionally left NULL. */
7498 if ( verbose > 2 ) {
7499 printf( " Creating base bounding " );
7500 printtriangle( &baseedge );
7502 /* Fix the extreme triangles if necessary. */
7503 org( *farleft, farleftpt );
7504 if ( innerleftdest == farleftpt ) {
7505 lnext( baseedge, *farleft );
7507 dest( *farright, farrightpt );
7508 if ( innerrightorg == farrightpt ) {
7509 lprev( baseedge, *farright );
7511 /* The vertices of the current knitting edge. */
7512 lowerleft = innerleftdest;
7513 lowerright = innerrightorg;
7514 /* The candidate vertices for knitting. */
7515 apex( leftcand, upperleft );
7516 apex( rightcand, upperright );
7517 /* Walk up the gap between the two triangulations, knitting them together. */
7519 /* Have we reached the top? (This isn't quite the right question, */
7520 /* because even though the left triangulation might seem finished now, */
7521 /* moving up on the right triangulation might reveal a new point of */
7522 /* the left triangulation. And vice-versa.) */
7523 leftfinished = counterclockwise( upperleft, lowerleft, lowerright ) <= 0.0;
7524 rightfinished = counterclockwise( upperright, lowerleft, lowerright ) <= 0.0;
7525 if ( leftfinished && rightfinished ) {
7526 /* Create the top new bounding triangle. */
7527 maketriangle( &nextedge );
7528 setorg( nextedge, lowerleft );
7529 setdest( nextedge, lowerright );
7530 /* Apex is intentionally left NULL. */
7531 /* Connect it to the bounding boxes of the two triangulations. */
7532 bond( nextedge, baseedge );
7533 lnextself( nextedge );
7534 bond( nextedge, rightcand );
7535 lnextself( nextedge );
7536 bond( nextedge, leftcand );
7537 if ( verbose > 2 ) {
7538 printf( " Creating top bounding " );
7539 printtriangle( &baseedge );
7541 /* Special treatment for horizontal cuts. */
7542 if ( dwyer && ( axis == 1 )) {
7543 org( *farleft, farleftpt );
7544 apex( *farleft, farleftapex );
7545 dest( *farright, farrightpt );
7546 apex( *farright, farrightapex );
7547 sym( *farleft, checkedge );
7548 apex( checkedge, checkvertex );
7549 /* The pointers to the extremal points are restored to the leftmost */
7550 /* and rightmost points (rather than topmost and bottommost). */
7551 while ( checkvertex[0] < farleftpt[0] ) {
7552 lprev( checkedge, *farleft );
7553 farleftapex = farleftpt;
7554 farleftpt = checkvertex;
7555 sym( *farleft, checkedge );
7556 apex( checkedge, checkvertex );
7558 while ( farrightapex[0] > farrightpt[0] ) {
7559 lprevself( *farright );
7560 symself( *farright );
7561 farrightpt = farrightapex;
7562 apex( *farright, farrightapex );
7567 /* Consider eliminating edges from the left triangulation. */
7568 if ( !leftfinished ) {
7569 /* What vertex would be exposed if an edge were deleted? */
7570 lprev( leftcand, nextedge );
7571 symself( nextedge );
7572 apex( nextedge, nextapex );
7573 /* If nextapex is NULL, then no vertex would be exposed; the */
7574 /* triangulation would have been eaten right through. */
7575 if ( nextapex != (point) NULL ) {
7576 /* Check whether the edge is Delaunay. */
7577 badedge = incircle( lowerleft, lowerright, upperleft, nextapex ) > 0.0;
7579 /* Eliminate the edge with an edge flip. As a result, the */
7580 /* left triangulation will have one more boundary triangle. */
7581 lnextself( nextedge );
7582 sym( nextedge, topcasing );
7583 lnextself( nextedge );
7584 sym( nextedge, sidecasing );
7585 bond( nextedge, topcasing );
7586 bond( leftcand, sidecasing );
7587 lnextself( leftcand );
7588 sym( leftcand, outercasing );
7589 lprevself( nextedge );
7590 bond( nextedge, outercasing );
7591 /* Correct the vertices to reflect the edge flip. */
7592 setorg( leftcand, lowerleft );
7593 setdest( leftcand, NULL );
7594 setapex( leftcand, nextapex );
7595 setorg( nextedge, NULL );
7596 setdest( nextedge, upperleft );
7597 setapex( nextedge, nextapex );
7598 /* Consider the newly exposed vertex. */
7599 upperleft = nextapex;
7600 /* What vertex would be exposed if another edge were deleted? */
7601 triedgecopy( sidecasing, nextedge );
7602 apex( nextedge, nextapex );
7603 if ( nextapex != (point) NULL ) {
7604 /* Check whether the edge is Delaunay. */
7605 badedge = incircle( lowerleft, lowerright, upperleft, nextapex )
7609 /* Avoid eating right through the triangulation. */
7615 /* Consider eliminating edges from the right triangulation. */
7616 if ( !rightfinished ) {
7617 /* What vertex would be exposed if an edge were deleted? */
7618 lnext( rightcand, nextedge );
7619 symself( nextedge );
7620 apex( nextedge, nextapex );
7621 /* If nextapex is NULL, then no vertex would be exposed; the */
7622 /* triangulation would have been eaten right through. */
7623 if ( nextapex != (point) NULL ) {
7624 /* Check whether the edge is Delaunay. */
7625 badedge = incircle( lowerleft, lowerright, upperright, nextapex ) > 0.0;
7627 /* Eliminate the edge with an edge flip. As a result, the */
7628 /* right triangulation will have one more boundary triangle. */
7629 lprevself( nextedge );
7630 sym( nextedge, topcasing );
7631 lprevself( nextedge );
7632 sym( nextedge, sidecasing );
7633 bond( nextedge, topcasing );
7634 bond( rightcand, sidecasing );
7635 lprevself( rightcand );
7636 sym( rightcand, outercasing );
7637 lnextself( nextedge );
7638 bond( nextedge, outercasing );
7639 /* Correct the vertices to reflect the edge flip. */
7640 setorg( rightcand, NULL );
7641 setdest( rightcand, lowerright );
7642 setapex( rightcand, nextapex );
7643 setorg( nextedge, upperright );
7644 setdest( nextedge, NULL );
7645 setapex( nextedge, nextapex );
7646 /* Consider the newly exposed vertex. */
7647 upperright = nextapex;
7648 /* What vertex would be exposed if another edge were deleted? */
7649 triedgecopy( sidecasing, nextedge );
7650 apex( nextedge, nextapex );
7651 if ( nextapex != (point) NULL ) {
7652 /* Check whether the edge is Delaunay. */
7653 badedge = incircle( lowerleft, lowerright, upperright, nextapex )
7657 /* Avoid eating right through the triangulation. */
7663 if ( leftfinished || ( !rightfinished &&
7664 ( incircle( upperleft, lowerleft, lowerright, upperright ) > 0.0 ))) {
7665 /* Knit the triangulations, adding an edge from `lowerleft' */
7666 /* to `upperright'. */
7667 bond( baseedge, rightcand );
7668 lprev( rightcand, baseedge );
7669 setdest( baseedge, lowerleft );
7670 lowerright = upperright;
7671 sym( baseedge, rightcand );
7672 apex( rightcand, upperright );
7675 /* Knit the triangulations, adding an edge from `upperleft' */
7676 /* to `lowerright'. */
7677 bond( baseedge, leftcand );
7678 lnext( leftcand, baseedge );
7679 setorg( baseedge, lowerright );
7680 lowerleft = upperleft;
7681 sym( baseedge, leftcand );
7682 apex( leftcand, upperleft );
7684 if ( verbose > 2 ) {
7685 printf( " Connecting " );
7686 printtriangle( &baseedge );
7691 /*****************************************************************************/
7693 /* divconqrecurse() Recursively form a Delaunay triangulation by the */
7694 /* divide-and-conquer method. */
7696 /* Recursively breaks down the problem into smaller pieces, which are */
7697 /* knitted together by mergehulls(). The base cases (problems of two or */
7698 /* three points) are handled specially here. */
7700 /* On completion, `farleft' and `farright' are bounding triangles such that */
7701 /* the origin of `farleft' is the leftmost vertex (breaking ties by */
7702 /* choosing the highest leftmost vertex), and the destination of */
7703 /* `farright' is the rightmost vertex (breaking ties by choosing the */
7704 /* lowest rightmost vertex). */
7706 /*****************************************************************************/
7708 void divconqrecurse( sortarray, vertices, axis, farleft, farright )
7712 struct triedge *farleft;
7713 struct triedge *farright;
7715 struct triedge midtri, tri1, tri2, tri3;
7716 struct triedge innerleft, innerright;
7720 if ( verbose > 2 ) {
7721 printf( " Triangulating %d points.\n", vertices );
7723 if ( vertices == 2 ) {
7724 /* The triangulation of two vertices is an edge. An edge is */
7725 /* represented by two bounding triangles. */
7726 maketriangle( farleft );
7727 setorg( *farleft, sortarray[0] );
7728 setdest( *farleft, sortarray[1] );
7729 /* The apex is intentionally left NULL. */
7730 maketriangle( farright );
7731 setorg( *farright, sortarray[1] );
7732 setdest( *farright, sortarray[0] );
7733 /* The apex is intentionally left NULL. */
7734 bond( *farleft, *farright );
7735 lprevself( *farleft );
7736 lnextself( *farright );
7737 bond( *farleft, *farright );
7738 lprevself( *farleft );
7739 lnextself( *farright );
7740 bond( *farleft, *farright );
7741 if ( verbose > 2 ) {
7742 printf( " Creating " );
7743 printtriangle( farleft );
7744 printf( " Creating " );
7745 printtriangle( farright );
7747 /* Ensure that the origin of `farleft' is sortarray[0]. */
7748 lprev( *farright, *farleft );
7751 else if ( vertices == 3 ) {
7752 /* The triangulation of three vertices is either a triangle (with */
7753 /* three bounding triangles) or two edges (with four bounding */
7754 /* triangles). In either case, four triangles are created. */
7755 maketriangle( &midtri );
7756 maketriangle( &tri1 );
7757 maketriangle( &tri2 );
7758 maketriangle( &tri3 );
7759 area = counterclockwise( sortarray[0], sortarray[1], sortarray[2] );
7760 if ( area == 0.0 ) {
7761 /* Three collinear points; the triangulation is two edges. */
7762 setorg( midtri, sortarray[0] );
7763 setdest( midtri, sortarray[1] );
7764 setorg( tri1, sortarray[1] );
7765 setdest( tri1, sortarray[0] );
7766 setorg( tri2, sortarray[2] );
7767 setdest( tri2, sortarray[1] );
7768 setorg( tri3, sortarray[1] );
7769 setdest( tri3, sortarray[2] );
7770 /* All apices are intentionally left NULL. */
7771 bond( midtri, tri1 );
7773 lnextself( midtri );
7777 bond( midtri, tri3 );
7779 lnextself( midtri );
7783 bond( midtri, tri1 );
7785 /* Ensure that the origin of `farleft' is sortarray[0]. */
7786 triedgecopy( tri1, *farleft );
7787 /* Ensure that the destination of `farright' is sortarray[2]. */
7788 triedgecopy( tri2, *farright );
7791 /* The three points are not collinear; the triangulation is one */
7792 /* triangle, namely `midtri'. */
7793 setorg( midtri, sortarray[0] );
7794 setdest( tri1, sortarray[0] );
7795 setorg( tri3, sortarray[0] );
7796 /* Apices of tri1, tri2, and tri3 are left NULL. */
7798 /* The vertices are in counterclockwise order. */
7799 setdest( midtri, sortarray[1] );
7800 setorg( tri1, sortarray[1] );
7801 setdest( tri2, sortarray[1] );
7802 setapex( midtri, sortarray[2] );
7803 setorg( tri2, sortarray[2] );
7804 setdest( tri3, sortarray[2] );
7807 /* The vertices are in clockwise order. */
7808 setdest( midtri, sortarray[2] );
7809 setorg( tri1, sortarray[2] );
7810 setdest( tri2, sortarray[2] );
7811 setapex( midtri, sortarray[1] );
7812 setorg( tri2, sortarray[1] );
7813 setdest( tri3, sortarray[1] );
7815 /* The topology does not depend on how the vertices are ordered. */
7816 bond( midtri, tri1 );
7817 lnextself( midtri );
7818 bond( midtri, tri2 );
7819 lnextself( midtri );
7820 bond( midtri, tri3 );
7830 /* Ensure that the origin of `farleft' is sortarray[0]. */
7831 triedgecopy( tri1, *farleft );
7832 /* Ensure that the destination of `farright' is sortarray[2]. */
7834 triedgecopy( tri2, *farright );
7837 lnext( *farleft, *farright );
7840 if ( verbose > 2 ) {
7841 printf( " Creating " );
7842 printtriangle( &midtri );
7843 printf( " Creating " );
7844 printtriangle( &tri1 );
7845 printf( " Creating " );
7846 printtriangle( &tri2 );
7847 printf( " Creating " );
7848 printtriangle( &tri3 );
7853 /* Split the vertices in half. */
7854 divider = vertices >> 1;
7855 /* Recursively triangulate each half. */
7856 divconqrecurse( sortarray, divider, 1 - axis, farleft, &innerleft );
7857 divconqrecurse( &sortarray[divider], vertices - divider, 1 - axis,
7858 &innerright, farright );
7859 if ( verbose > 1 ) {
7860 printf( " Joining triangulations with %d and %d vertices.\n", divider,
7861 vertices - divider );
7863 /* Merge the two triangulations into one. */
7864 mergehulls( farleft, &innerleft, &innerright, farright, axis );
7868 long removeghosts( startghost )
7869 struct triedge *startghost;
7871 struct triedge searchedge;
7872 struct triedge dissolveedge;
7873 struct triedge deadtri;
7876 triangle ptr; /* Temporary variable used by sym(). */
7879 printf( " Removing ghost triangles.\n" );
7881 /* Find an edge on the convex hull to start point location from. */
7882 lprev( *startghost, searchedge );
7883 symself( searchedge );
7884 dummytri[0] = encode( searchedge );
7885 /* Remove the bounding box and count the convex hull edges. */
7886 triedgecopy( *startghost, dissolveedge );
7890 lnext( dissolveedge, deadtri );
7891 lprevself( dissolveedge );
7892 symself( dissolveedge );
7893 /* If no PSLG is involved, set the boundary markers of all the points */
7894 /* on the convex hull. If a PSLG is used, this step is done later. */
7896 /* Watch out for the case where all the input points are collinear. */
7897 if ( dissolveedge.tri != dummytri ) {
7898 org( dissolveedge, markorg );
7899 if ( pointmark( markorg ) == 0 ) {
7900 setpointmark( markorg, 1 );
7904 /* Remove a bounding triangle from a convex hull triangle. */
7905 dissolve( dissolveedge );
7906 /* Find the next bounding triangle. */
7907 sym( deadtri, dissolveedge );
7908 /* Delete the bounding triangle. */
7909 triangledealloc( deadtri.tri );
7910 } while ( !triedgeequal( dissolveedge, *startghost ));
7914 /*****************************************************************************/
7916 /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */
7917 /* conquer method. */
7919 /* Sorts the points, calls a recursive procedure to triangulate them, and */
7920 /* removes the bounding box, setting boundary markers as appropriate. */
7922 /*****************************************************************************/
7924 long divconqdelaunay(){
7926 struct triedge hullleft, hullright;
7930 /* Allocate an array of pointers to points for sorting. */
7931 sortarray = (point *) malloc( inpoints * sizeof( point ));
7932 if ( sortarray == (point *) NULL ) {
7933 printf( "Error: Out of memory.\n" );
7936 traversalinit( &points );
7937 for ( i = 0; i < inpoints; i++ ) {
7938 sortarray[i] = pointtraverse();
7941 printf( " Sorting points.\n" );
7943 /* Sort the points. */
7944 pointsort( sortarray, inpoints );
7945 /* Discard duplicate points, which can really mess up the algorithm. */
7947 for ( j = 1; j < inpoints; j++ ) {
7948 if (( sortarray[i][0] == sortarray[j][0] )
7949 && ( sortarray[i][1] == sortarray[j][1] )) {
7952 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
7953 sortarray[j][0], sortarray[j][1] );
7955 /* Commented out - would eliminate point from output .node file, but causes
7956 a failure if some segment has this point as an endpoint.
7957 setpointmark(sortarray[j], DEADPOINT);
7962 sortarray[i] = sortarray[j];
7967 /* Re-sort the array of points to accommodate alternating cuts. */
7969 if ( i - divider >= 2 ) {
7970 if ( divider >= 2 ) {
7971 alternateaxes( sortarray, divider, 1 );
7973 alternateaxes( &sortarray[divider], i - divider, 1 );
7977 printf( " Forming triangulation.\n" );
7979 /* Form the Delaunay triangulation. */
7980 divconqrecurse( sortarray, i, 0, &hullleft, &hullright );
7983 return removeghosts( &hullleft );
7988 /********* Divide-and-conquer Delaunay triangulation ends here *********/
7990 /********* Incremental Delaunay triangulation begins here *********/
7994 /*****************************************************************************/
7996 /* boundingbox() Form an "infinite" bounding triangle to insert points */
7999 /* The points at "infinity" are assigned finite coordinates, which are used */
8000 /* by the point location routines, but (mostly) ignored by the Delaunay */
8001 /* edge flip routines. */
8003 /*****************************************************************************/
8009 struct triedge inftri; /* Handle for the triangular bounding box. */
8013 printf( " Creating triangular bounding box.\n" );
8015 /* Find the width (or height, whichever is larger) of the triangulation. */
8016 width = xmax - xmin;
8017 if ( ymax - ymin > width ) {
8018 width = ymax - ymin;
8020 if ( width == 0.0 ) {
8023 /* Create the vertices of the bounding box. */
8024 infpoint1 = (point) malloc( points.itembytes );
8025 infpoint2 = (point) malloc( points.itembytes );
8026 infpoint3 = (point) malloc( points.itembytes );
8027 if (( infpoint1 == (point) NULL ) || ( infpoint2 == (point) NULL )
8028 || ( infpoint3 == (point) NULL )) {
8029 printf( "Error: Out of memory.\n" );
8032 infpoint1[0] = xmin - 50.0 * width;
8033 infpoint1[1] = ymin - 40.0 * width;
8034 infpoint2[0] = xmax + 50.0 * width;
8035 infpoint2[1] = ymin - 40.0 * width;
8036 infpoint3[0] = 0.5 * ( xmin + xmax );
8037 infpoint3[1] = ymax + 60.0 * width;
8039 /* Create the bounding box. */
8040 maketriangle( &inftri );
8041 setorg( inftri, infpoint1 );
8042 setdest( inftri, infpoint2 );
8043 setapex( inftri, infpoint3 );
8044 /* Link dummytri to the bounding box so we can always find an */
8045 /* edge to begin searching (point location) from. */
8046 dummytri[0] = (triangle) inftri.tri;
8047 if ( verbose > 2 ) {
8048 printf( " Creating " );
8049 printtriangle( &inftri );
8053 #endif /* not REDUCED */
8055 /*****************************************************************************/
8057 /* removebox() Remove the "infinite" bounding triangle, setting boundary */
8058 /* markers as appropriate. */
8060 /* The triangular bounding box has three boundary triangles (one for each */
8061 /* side of the bounding box), and a bunch of triangles fanning out from */
8062 /* the three bounding box vertices (one triangle for each edge of the */
8063 /* convex hull of the inner mesh). This routine removes these triangles. */
8065 /*****************************************************************************/
8071 struct triedge deadtri;
8072 struct triedge searchedge;
8073 struct triedge checkedge;
8074 struct triedge nextedge, finaledge, dissolveedge;
8077 triangle ptr; /* Temporary variable used by sym(). */
8080 printf( " Removing triangular bounding box.\n" );
8082 /* Find a boundary triangle. */
8083 nextedge.tri = dummytri;
8084 nextedge.orient = 0;
8085 symself( nextedge );
8086 /* Mark a place to stop. */
8087 lprev( nextedge, finaledge );
8088 lnextself( nextedge );
8089 symself( nextedge );
8090 /* Find a triangle (on the boundary of the point set) that isn't */
8091 /* a bounding box triangle. */
8092 lprev( nextedge, searchedge );
8093 symself( searchedge );
8094 /* Check whether nextedge is another boundary triangle */
8095 /* adjacent to the first one. */
8096 lnext( nextedge, checkedge );
8097 symself( checkedge );
8098 if ( checkedge.tri == dummytri ) {
8099 /* Go on to the next triangle. There are only three boundary */
8100 /* triangles, and this next triangle cannot be the third one, */
8101 /* so it's safe to stop here. */
8102 lprevself( searchedge );
8103 symself( searchedge );
8105 /* Find a new boundary edge to search from, as the current search */
8106 /* edge lies on a bounding box triangle and will be deleted. */
8107 dummytri[0] = encode( searchedge );
8109 while ( !triedgeequal( nextedge, finaledge )) {
8111 lprev( nextedge, dissolveedge );
8112 symself( dissolveedge );
8113 /* If not using a PSLG, the vertices should be marked now. */
8114 /* (If using a PSLG, markhull() will do the job.) */
8116 /* Be careful! One must check for the case where all the input */
8117 /* points are collinear, and thus all the triangles are part of */
8118 /* the bounding box. Otherwise, the setpointmark() call below */
8119 /* will cause a bad pointer reference. */
8120 if ( dissolveedge.tri != dummytri ) {
8121 org( dissolveedge, markorg );
8122 if ( pointmark( markorg ) == 0 ) {
8123 setpointmark( markorg, 1 );
8127 /* Disconnect the bounding box triangle from the mesh triangle. */
8128 dissolve( dissolveedge );
8129 lnext( nextedge, deadtri );
8130 sym( deadtri, nextedge );
8131 /* Get rid of the bounding box triangle. */
8132 triangledealloc( deadtri.tri );
8133 /* Do we need to turn the corner? */
8134 if ( nextedge.tri == dummytri ) {
8135 /* Turn the corner. */
8136 triedgecopy( dissolveedge, nextedge );
8139 triangledealloc( finaledge.tri );
8141 free( infpoint1 ); /* Deallocate the bounding box vertices. */
8148 #endif /* not REDUCED */
8150 /*****************************************************************************/
8152 /* incrementaldelaunay() Form a Delaunay triangulation by incrementally */
8153 /* adding vertices. */
8155 /*****************************************************************************/
8160 long incrementaldelaunay(){
8161 struct triedge starttri;
8165 /* Create a triangular bounding box. */
8168 printf( " Incrementally inserting points.\n" );
8170 traversalinit( &points );
8171 pointloop = pointtraverse();
8173 while ( pointloop != (point) NULL ) {
8174 /* Find a boundary triangle to search from. */
8175 starttri.tri = (triangle *) NULL;
8176 if ( insertsite( pointloop, &starttri, (struct edge *) NULL, 0, 0 ) ==
8180 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8181 pointloop[0], pointloop[1] );
8183 /* Commented out - would eliminate point from output .node file.
8184 setpointmark(pointloop, DEADPOINT);
8187 pointloop = pointtraverse();
8190 /* Remove the bounding box. */
8194 #endif /* not REDUCED */
8198 /********* Incremental Delaunay triangulation ends here *********/
8200 /********* Sweepline Delaunay triangulation begins here *********/
8207 void eventheapinsert( heap, heapsize, newevent )
8208 struct event **heap;
8210 struct event *newevent;
8212 REAL eventx, eventy;
8217 eventx = newevent->xkey;
8218 eventy = newevent->ykey;
8219 eventnum = heapsize;
8220 notdone = eventnum > 0;
8222 parent = ( eventnum - 1 ) >> 1;
8223 if (( heap[parent]->ykey < eventy ) ||
8224 (( heap[parent]->ykey == eventy )
8225 && ( heap[parent]->xkey <= eventx ))) {
8229 heap[eventnum] = heap[parent];
8230 heap[eventnum]->heapposition = eventnum;
8233 notdone = eventnum > 0;
8236 heap[eventnum] = newevent;
8237 newevent->heapposition = eventnum;
8240 #endif /* not REDUCED */
8245 void eventheapify( heap, heapsize, eventnum )
8246 struct event **heap;
8250 struct event *thisevent;
8251 REAL eventx, eventy;
8252 int leftchild, rightchild;
8256 thisevent = heap[eventnum];
8257 eventx = thisevent->xkey;
8258 eventy = thisevent->ykey;
8259 leftchild = 2 * eventnum + 1;
8260 notdone = leftchild < heapsize;
8262 if (( heap[leftchild]->ykey < eventy ) ||
8263 (( heap[leftchild]->ykey == eventy )
8264 && ( heap[leftchild]->xkey < eventx ))) {
8265 smallest = leftchild;
8268 smallest = eventnum;
8270 rightchild = leftchild + 1;
8271 if ( rightchild < heapsize ) {
8272 if (( heap[rightchild]->ykey < heap[smallest]->ykey ) ||
8273 (( heap[rightchild]->ykey == heap[smallest]->ykey )
8274 && ( heap[rightchild]->xkey < heap[smallest]->xkey ))) {
8275 smallest = rightchild;
8278 if ( smallest == eventnum ) {
8282 heap[eventnum] = heap[smallest];
8283 heap[eventnum]->heapposition = eventnum;
8284 heap[smallest] = thisevent;
8285 thisevent->heapposition = smallest;
8287 eventnum = smallest;
8288 leftchild = 2 * eventnum + 1;
8289 notdone = leftchild < heapsize;
8294 #endif /* not REDUCED */
8299 void eventheapdelete( heap, heapsize, eventnum )
8300 struct event **heap;
8304 struct event *moveevent;
8305 REAL eventx, eventy;
8309 moveevent = heap[heapsize - 1];
8310 if ( eventnum > 0 ) {
8311 eventx = moveevent->xkey;
8312 eventy = moveevent->ykey;
8314 parent = ( eventnum - 1 ) >> 1;
8315 if (( heap[parent]->ykey < eventy ) ||
8316 (( heap[parent]->ykey == eventy )
8317 && ( heap[parent]->xkey <= eventx ))) {
8321 heap[eventnum] = heap[parent];
8322 heap[eventnum]->heapposition = eventnum;
8325 notdone = eventnum > 0;
8327 } while ( notdone );
8329 heap[eventnum] = moveevent;
8330 moveevent->heapposition = eventnum;
8331 eventheapify( heap, heapsize - 1, eventnum );
8334 #endif /* not REDUCED */
8339 void createeventheap( eventheap, events, freeevents )
8340 struct event ***eventheap;
8341 struct event **events;
8342 struct event **freeevents;
8348 maxevents = ( 3 * inpoints ) / 2;
8349 *eventheap = (struct event **) malloc( maxevents * sizeof( struct event * ));
8350 if ( *eventheap == (struct event **) NULL ) {
8351 printf( "Error: Out of memory.\n" );
8354 *events = (struct event *) malloc( maxevents * sizeof( struct event ));
8355 if ( *events == (struct event *) NULL ) {
8356 printf( "Error: Out of memory.\n" );
8359 traversalinit( &points );
8360 for ( i = 0; i < inpoints; i++ ) {
8361 thispoint = pointtraverse();
8362 ( *events )[i].eventptr = (VOID *) thispoint;
8363 ( *events )[i].xkey = thispoint[0];
8364 ( *events )[i].ykey = thispoint[1];
8365 eventheapinsert( *eventheap, i, *events + i );
8367 *freeevents = (struct event *) NULL;
8368 for ( i = maxevents - 1; i >= inpoints; i-- ) {
8369 ( *events )[i].eventptr = (VOID *) *freeevents;
8370 *freeevents = *events + i;
8374 #endif /* not REDUCED */
8379 int rightofhyperbola( fronttri, newsite )
8380 struct triedge *fronttri;
8383 point leftpoint, rightpoint;
8384 REAL dxa, dya, dxb, dyb;
8388 dest( *fronttri, leftpoint );
8389 apex( *fronttri, rightpoint );
8390 if (( leftpoint[1] < rightpoint[1] )
8391 || (( leftpoint[1] == rightpoint[1] ) && ( leftpoint[0] < rightpoint[0] ))) {
8392 if ( newsite[0] >= rightpoint[0] ) {
8397 if ( newsite[0] <= leftpoint[0] ) {
8401 dxa = leftpoint[0] - newsite[0];
8402 dya = leftpoint[1] - newsite[1];
8403 dxb = rightpoint[0] - newsite[0];
8404 dyb = rightpoint[1] - newsite[1];
8405 return dya * ( dxb * dxb + dyb * dyb ) > dyb * ( dxa * dxa + dya * dya );
8408 #endif /* not REDUCED */
8413 REAL circletop( pa, pb, pc, ccwabc )
8419 REAL xac, yac, xbc, ybc, xab, yab;
8420 REAL aclen2, bclen2, ablen2;
8424 xac = pa[0] - pc[0];
8425 yac = pa[1] - pc[1];
8426 xbc = pb[0] - pc[0];
8427 ybc = pb[1] - pc[1];
8428 xab = pa[0] - pb[0];
8429 yab = pa[1] - pb[1];
8430 aclen2 = xac * xac + yac * yac;
8431 bclen2 = xbc * xbc + ybc * ybc;
8432 ablen2 = xab * xab + yab * yab;
8433 return pc[1] + ( xac * bclen2 - xbc * aclen2 + sqrt( aclen2 * bclen2 * ablen2 ))
8437 #endif /* not REDUCED */
8442 void check4deadevent( checktri, freeevents, eventheap, heapsize )
8443 struct triedge *checktri;
8444 struct event **freeevents;
8445 struct event **eventheap;
8448 struct event *deadevent;
8452 org( *checktri, eventpoint );
8453 if ( eventpoint != (point) NULL ) {
8454 deadevent = (struct event *) eventpoint;
8455 eventnum = deadevent->heapposition;
8456 deadevent->eventptr = (VOID *) *freeevents;
8457 *freeevents = deadevent;
8458 eventheapdelete( eventheap, *heapsize, eventnum );
8460 setorg( *checktri, NULL );
8464 #endif /* not REDUCED */
8469 struct splaynode *splay( splaytree, searchpoint, searchtri )
8470 struct splaynode *splaytree;
8472 struct triedge *searchtri;
8474 struct splaynode *child, *grandchild;
8475 struct splaynode *lefttree, *righttree;
8476 struct splaynode *leftright;
8478 int rightofroot, rightofchild;
8480 if ( splaytree == (struct splaynode *) NULL ) {
8481 return (struct splaynode *) NULL;
8483 dest( splaytree->keyedge, checkpoint );
8484 if ( checkpoint == splaytree->keydest ) {
8485 rightofroot = rightofhyperbola( &splaytree->keyedge, searchpoint );
8486 if ( rightofroot ) {
8487 triedgecopy( splaytree->keyedge, *searchtri );
8488 child = splaytree->rchild;
8491 child = splaytree->lchild;
8493 if ( child == (struct splaynode *) NULL ) {
8496 dest( child->keyedge, checkpoint );
8497 if ( checkpoint != child->keydest ) {
8498 child = splay( child, searchpoint, searchtri );
8499 if ( child == (struct splaynode *) NULL ) {
8500 if ( rightofroot ) {
8501 splaytree->rchild = (struct splaynode *) NULL;
8504 splaytree->lchild = (struct splaynode *) NULL;
8509 rightofchild = rightofhyperbola( &child->keyedge, searchpoint );
8510 if ( rightofchild ) {
8511 triedgecopy( child->keyedge, *searchtri );
8512 grandchild = splay( child->rchild, searchpoint, searchtri );
8513 child->rchild = grandchild;
8516 grandchild = splay( child->lchild, searchpoint, searchtri );
8517 child->lchild = grandchild;
8519 if ( grandchild == (struct splaynode *) NULL ) {
8520 if ( rightofroot ) {
8521 splaytree->rchild = child->lchild;
8522 child->lchild = splaytree;
8525 splaytree->lchild = child->rchild;
8526 child->rchild = splaytree;
8530 if ( rightofchild ) {
8531 if ( rightofroot ) {
8532 splaytree->rchild = child->lchild;
8533 child->lchild = splaytree;
8536 splaytree->lchild = grandchild->rchild;
8537 grandchild->rchild = splaytree;
8539 child->rchild = grandchild->lchild;
8540 grandchild->lchild = child;
8543 if ( rightofroot ) {
8544 splaytree->rchild = grandchild->lchild;
8545 grandchild->lchild = splaytree;
8548 splaytree->lchild = child->rchild;
8549 child->rchild = splaytree;
8551 child->lchild = grandchild->rchild;
8552 grandchild->rchild = child;
8557 lefttree = splay( splaytree->lchild, searchpoint, searchtri );
8558 righttree = splay( splaytree->rchild, searchpoint, searchtri );
8560 pooldealloc( &splaynodes, (VOID *) splaytree );
8561 if ( lefttree == (struct splaynode *) NULL ) {
8564 else if ( righttree == (struct splaynode *) NULL ) {
8567 else if ( lefttree->rchild == (struct splaynode *) NULL ) {
8568 lefttree->rchild = righttree->lchild;
8569 righttree->lchild = lefttree;
8572 else if ( righttree->lchild == (struct splaynode *) NULL ) {
8573 righttree->lchild = lefttree->rchild;
8574 lefttree->rchild = righttree;
8578 /* printf("Holy Toledo!!!\n"); */
8579 leftright = lefttree->rchild;
8580 while ( leftright->rchild != (struct splaynode *) NULL ) {
8581 leftright = leftright->rchild;
8583 leftright->rchild = righttree;
8589 #endif /* not REDUCED */
8594 struct splaynode *splayinsert( splayroot, newkey, searchpoint )
8595 struct splaynode *splayroot;
8596 struct triedge *newkey;
8599 struct splaynode *newsplaynode;
8601 newsplaynode = (struct splaynode *) poolalloc( &splaynodes );
8602 triedgecopy( *newkey, newsplaynode->keyedge );
8603 dest( *newkey, newsplaynode->keydest );
8604 if ( splayroot == (struct splaynode *) NULL ) {
8605 newsplaynode->lchild = (struct splaynode *) NULL;
8606 newsplaynode->rchild = (struct splaynode *) NULL;
8608 else if ( rightofhyperbola( &splayroot->keyedge, searchpoint )) {
8609 newsplaynode->lchild = splayroot;
8610 newsplaynode->rchild = splayroot->rchild;
8611 splayroot->rchild = (struct splaynode *) NULL;
8614 newsplaynode->lchild = splayroot->lchild;
8615 newsplaynode->rchild = splayroot;
8616 splayroot->lchild = (struct splaynode *) NULL;
8618 return newsplaynode;
8621 #endif /* not REDUCED */
8626 struct splaynode *circletopinsert( splayroot, newkey, pa, pb, pc, topy )
8627 struct splaynode *splayroot;
8628 struct triedge *newkey;
8635 REAL xac, yac, xbc, ybc;
8636 REAL aclen2, bclen2;
8637 REAL searchpoint[2];
8638 struct triedge dummytri;
8640 ccwabc = counterclockwise( pa, pb, pc );
8641 xac = pa[0] - pc[0];
8642 yac = pa[1] - pc[1];
8643 xbc = pb[0] - pc[0];
8644 ybc = pb[1] - pc[1];
8645 aclen2 = xac * xac + yac * yac;
8646 bclen2 = xbc * xbc + ybc * ybc;
8647 searchpoint[0] = pc[0] - ( yac * bclen2 - ybc * aclen2 ) / ( 2.0 * ccwabc );
8648 searchpoint[1] = topy;
8649 return splayinsert( splay( splayroot, (point) searchpoint, &dummytri ), newkey,
8650 (point) searchpoint );
8653 #endif /* not REDUCED */
8658 struct splaynode *frontlocate( splayroot, bottommost, searchpoint, searchtri,
8660 struct splaynode *splayroot;
8661 struct triedge *bottommost;
8663 struct triedge *searchtri;
8667 triangle ptr; /* Temporary variable used by onext(). */
8669 triedgecopy( *bottommost, *searchtri );
8670 splayroot = splay( splayroot, searchpoint, searchtri );
8673 while ( !farrightflag && rightofhyperbola( searchtri, searchpoint )) {
8674 onextself( *searchtri );
8675 farrightflag = triedgeequal( *searchtri, *bottommost );
8677 *farright = farrightflag;
8681 #endif /* not REDUCED */
8686 long sweeplinedelaunay(){
8687 struct event **eventheap;
8688 struct event *events;
8689 struct event *freeevents;
8690 struct event *nextevent;
8691 struct event *newevent;
8692 struct splaynode *splayroot;
8693 struct triedge bottommost;
8694 struct triedge searchtri;
8695 struct triedge fliptri;
8696 struct triedge lefttri, righttri, farlefttri, farrighttri;
8697 struct triedge inserttri;
8698 point firstpoint, secondpoint;
8699 point nextpoint, lastpoint;
8701 point leftpoint, midpoint, rightpoint;
8702 REAL lefttest, righttest;
8704 int check4events, farrightflag;
8705 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
8707 poolinit( &splaynodes, sizeof( struct splaynode ), SPLAYNODEPERBLOCK, POINTER,
8709 splayroot = (struct splaynode *) NULL;
8712 printf( " Placing points in event heap.\n" );
8714 createeventheap( &eventheap, &events, &freeevents );
8715 heapsize = inpoints;
8718 printf( " Forming triangulation.\n" );
8720 maketriangle( &lefttri );
8721 maketriangle( &righttri );
8722 bond( lefttri, righttri );
8723 lnextself( lefttri );
8724 lprevself( righttri );
8725 bond( lefttri, righttri );
8726 lnextself( lefttri );
8727 lprevself( righttri );
8728 bond( lefttri, righttri );
8729 firstpoint = (point) eventheap[0]->eventptr;
8730 eventheap[0]->eventptr = (VOID *) freeevents;
8731 freeevents = eventheap[0];
8732 eventheapdelete( eventheap, heapsize, 0 );
8735 if ( heapsize == 0 ) {
8736 printf( "Error: Input points are all identical.\n" );
8739 secondpoint = (point) eventheap[0]->eventptr;
8740 eventheap[0]->eventptr = (VOID *) freeevents;
8741 freeevents = eventheap[0];
8742 eventheapdelete( eventheap, heapsize, 0 );
8744 if (( firstpoint[0] == secondpoint[0] )
8745 && ( firstpoint[1] == secondpoint[1] )) {
8747 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8748 secondpoint[0], secondpoint[1] );
8749 /* Commented out - would eliminate point from output .node file.
8750 setpointmark(secondpoint, DEADPOINT);
8753 } while (( firstpoint[0] == secondpoint[0] )
8754 && ( firstpoint[1] == secondpoint[1] ));
8755 setorg( lefttri, firstpoint );
8756 setdest( lefttri, secondpoint );
8757 setorg( righttri, secondpoint );
8758 setdest( righttri, firstpoint );
8759 lprev( lefttri, bottommost );
8760 lastpoint = secondpoint;
8761 while ( heapsize > 0 ) {
8762 nextevent = eventheap[0];
8763 eventheapdelete( eventheap, heapsize, 0 );
8766 if ( nextevent->xkey < xmin ) {
8767 decode( nextevent->eventptr, fliptri );
8768 oprev( fliptri, farlefttri );
8769 check4deadevent( &farlefttri, &freeevents, eventheap, &heapsize );
8770 onext( fliptri, farrighttri );
8771 check4deadevent( &farrighttri, &freeevents, eventheap, &heapsize );
8773 if ( triedgeequal( farlefttri, bottommost )) {
8774 lprev( fliptri, bottommost );
8777 setapex( fliptri, NULL );
8778 lprev( fliptri, lefttri );
8779 lnext( fliptri, righttri );
8780 sym( lefttri, farlefttri );
8782 if ( randomnation( SAMPLERATE ) == 0 ) {
8784 dest( fliptri, leftpoint );
8785 apex( fliptri, midpoint );
8786 org( fliptri, rightpoint );
8787 splayroot = circletopinsert( splayroot, &lefttri, leftpoint, midpoint,
8788 rightpoint, nextevent->ykey );
8792 nextpoint = (point) nextevent->eventptr;
8793 if (( nextpoint[0] == lastpoint[0] ) && ( nextpoint[1] == lastpoint[1] )) {
8795 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8796 nextpoint[0], nextpoint[1] );
8797 /* Commented out - would eliminate point from output .node file.
8798 setpointmark(nextpoint, DEADPOINT);
8803 lastpoint = nextpoint;
8805 splayroot = frontlocate( splayroot, &bottommost, nextpoint, &searchtri,
8808 triedgecopy(bottommost, searchtri);
8810 while (!farrightflag && rightofhyperbola(&searchtri, nextpoint)) {
8811 onextself(searchtri);
8812 farrightflag = triedgeequal(searchtri, bottommost);
8816 check4deadevent( &searchtri, &freeevents, eventheap, &heapsize );
8818 triedgecopy( searchtri, farrighttri );
8819 sym( searchtri, farlefttri );
8820 maketriangle( &lefttri );
8821 maketriangle( &righttri );
8822 dest( farrighttri, connectpoint );
8823 setorg( lefttri, connectpoint );
8824 setdest( lefttri, nextpoint );
8825 setorg( righttri, nextpoint );
8826 setdest( righttri, connectpoint );
8827 bond( lefttri, righttri );
8828 lnextself( lefttri );
8829 lprevself( righttri );
8830 bond( lefttri, righttri );
8831 lnextself( lefttri );
8832 lprevself( righttri );
8833 bond( lefttri, farlefttri );
8834 bond( righttri, farrighttri );
8835 if ( !farrightflag && triedgeequal( farrighttri, bottommost )) {
8836 triedgecopy( lefttri, bottommost );
8839 if ( randomnation( SAMPLERATE ) == 0 ) {
8840 splayroot = splayinsert( splayroot, &lefttri, nextpoint );
8842 else if ( randomnation( SAMPLERATE ) == 0 ) {
8843 lnext( righttri, inserttri );
8844 splayroot = splayinsert( splayroot, &inserttri, nextpoint );
8848 nextevent->eventptr = (VOID *) freeevents;
8849 freeevents = nextevent;
8851 if ( check4events ) {
8852 apex( farlefttri, leftpoint );
8853 dest( lefttri, midpoint );
8854 apex( lefttri, rightpoint );
8855 lefttest = counterclockwise( leftpoint, midpoint, rightpoint );
8856 if ( lefttest > 0.0 ) {
8857 newevent = freeevents;
8858 freeevents = (struct event *) freeevents->eventptr;
8859 newevent->xkey = xminextreme;
8860 newevent->ykey = circletop( leftpoint, midpoint, rightpoint,
8862 newevent->eventptr = (VOID *) encode( lefttri );
8863 eventheapinsert( eventheap, heapsize, newevent );
8865 setorg( lefttri, newevent );
8867 apex( righttri, leftpoint );
8868 org( righttri, midpoint );
8869 apex( farrighttri, rightpoint );
8870 righttest = counterclockwise( leftpoint, midpoint, rightpoint );
8871 if ( righttest > 0.0 ) {
8872 newevent = freeevents;
8873 freeevents = (struct event *) freeevents->eventptr;
8874 newevent->xkey = xminextreme;
8875 newevent->ykey = circletop( leftpoint, midpoint, rightpoint,
8877 newevent->eventptr = (VOID *) encode( farrighttri );
8878 eventheapinsert( eventheap, heapsize, newevent );
8880 setorg( farrighttri, newevent );
8885 pooldeinit( &splaynodes );
8886 lprevself( bottommost );
8887 return removeghosts( &bottommost );
8890 #endif /* not REDUCED */
8894 /********* Sweepline Delaunay triangulation ends here *********/
8896 /********* General mesh construction routines begin here *********/
8900 /*****************************************************************************/
8902 /* delaunay() Form a Delaunay triangulation. */
8904 /*****************************************************************************/
8908 initializetrisegpools();
8914 "Constructing Delaunay triangulation by divide-and-conquer method.\n" );
8916 return divconqdelaunay();
8917 #else /* not REDUCED */
8919 printf( "Constructing Delaunay triangulation " );
8920 if ( incremental ) {
8921 printf( "by incremental method.\n" );
8923 else if ( sweepline ) {
8924 printf( "by sweepline method.\n" );
8927 printf( "by divide-and-conquer method.\n" );
8930 if ( incremental ) {
8931 return incrementaldelaunay();
8933 else if ( sweepline ) {
8934 return sweeplinedelaunay();
8937 return divconqdelaunay();
8939 #endif /* not REDUCED */
8942 /*****************************************************************************/
8944 /* reconstruct() Reconstruct a triangulation from its .ele (and possibly */
8945 /* .poly) file. Used when the -r switch is used. */
8947 /* Reads an .ele file and reconstructs the original mesh. If the -p switch */
8948 /* is used, this procedure will also read a .poly file and reconstruct the */
8949 /* shell edges of the original mesh. If the -a switch is used, this */
8950 /* procedure will also read an .area file and set a maximum area constraint */
8951 /* on each triangle. */
8953 /* Points that are not corners of triangles, such as nodes on edges of */
8954 /* subparametric elements, are discarded. */
8956 /* This routine finds the adjacencies between triangles (and shell edges) */
8957 /* by forming one stack of triangles for each vertex. Each triangle is on */
8958 /* three different stacks simultaneously. Each triangle's shell edge */
8959 /* pointers are used to link the items in each stack. This memory-saving */
8960 /* feature makes the code harder to read. The most important thing to keep */
8961 /* in mind is that each triangle is removed from a stack precisely when */
8962 /* the corresponding pointer is adjusted to refer to a shell edge rather */
8963 /* than the next triangle of the stack. */
8965 /*****************************************************************************/
8973 int reconstruct( trianglelist, triangleattriblist, trianglearealist, elements,
8974 corners, attribs, segmentlist, segmentmarkerlist,
8977 REAL *triangleattriblist;
8978 REAL *trianglearealist;
8983 int *segmentmarkerlist;
8984 int numberofsegments;
8986 #else /* not TRILIBRARY */
8988 long reconstruct( elefilename, areafilename, polyfilename, polyfile )
8994 #endif /* not TRILIBRARY */
9001 #else /* not TRILIBRARY */
9004 char inputline[INPUTLINESIZE];
9007 #endif /* not TRILIBRARY */
9008 struct triedge triangleloop;
9009 struct triedge triangleleft;
9010 struct triedge checktri;
9011 struct triedge checkleft;
9012 struct triedge checkneighbor;
9013 struct edge shelleloop;
9014 triangle *vertexarray;
9018 point checkdest, checkapex;
9031 int elementnumber, segmentnumber;
9033 triangle ptr; /* Temporary variable used by sym(). */
9037 inelements = elements;
9038 incorners = corners;
9039 if ( incorners < 3 ) {
9040 printf( "Error: Triangles must have at least 3 points.\n" );
9044 #else /* not TRILIBRARY */
9045 /* Read the triangles from an .ele file. */
9047 printf( "Opening %s.\n", elefilename );
9049 elefile = fopen( elefilename, "r" );
9050 if ( elefile == (FILE *) NULL ) {
9051 printf( " Error: Cannot access file %s.\n", elefilename );
9054 /* Read number of triangles, number of points per triangle, and */
9055 /* number of triangle attributes from .ele file. */
9056 stringptr = readline( inputline, elefile, elefilename );
9057 inelements = (int) strtol( stringptr, &stringptr, 0 );
9058 stringptr = findfield( stringptr );
9059 if ( *stringptr == '\0' ) {
9063 incorners = (int) strtol( stringptr, &stringptr, 0 );
9064 if ( incorners < 3 ) {
9065 printf( "Error: Triangles in %s must have at least 3 points.\n",
9070 stringptr = findfield( stringptr );
9071 if ( *stringptr == '\0' ) {
9075 eextras = (int) strtol( stringptr, &stringptr, 0 );
9077 #endif /* not TRILIBRARY */
9079 initializetrisegpools();
9081 /* Create the triangles. */
9082 for ( elementnumber = 1; elementnumber <= inelements; elementnumber++ ) {
9083 maketriangle( &triangleloop );
9084 /* Mark the triangle as living. */
9085 triangleloop.tri[3] = (triangle) triangleloop.tri;
9091 insegments = numberofsegments;
9092 segmentmarkers = segmentmarkerlist != (int *) NULL;
9093 #else /* not TRILIBRARY */
9094 /* Read number of segments and number of segment */
9095 /* boundary markers from .poly file. */
9096 stringptr = readline( inputline, polyfile, inpolyfilename );
9097 insegments = (int) strtol( stringptr, &stringptr, 0 );
9098 stringptr = findfield( stringptr );
9099 if ( *stringptr == '\0' ) {
9103 segmentmarkers = (int) strtol( stringptr, &stringptr, 0 );
9105 #endif /* not TRILIBRARY */
9107 /* Create the shell edges. */
9108 for ( segmentnumber = 1; segmentnumber <= insegments; segmentnumber++ ) {
9109 makeshelle( &shelleloop );
9110 /* Mark the shell edge as living. */
9111 shelleloop.sh[2] = (shelle) shelleloop.sh;
9119 #else /* not TRILIBRARY */
9121 /* Open an .area file, check for consistency with the .ele file. */
9123 printf( "Opening %s.\n", areafilename );
9125 areafile = fopen( areafilename, "r" );
9126 if ( areafile == (FILE *) NULL ) {
9127 printf( " Error: Cannot access file %s.\n", areafilename );
9130 stringptr = readline( inputline, areafile, areafilename );
9131 areaelements = (int) strtol( stringptr, &stringptr, 0 );
9132 if ( areaelements != inelements ) {
9133 printf( "Error: %s and %s disagree on number of triangles.\n",
9134 elefilename, areafilename );
9138 #endif /* not TRILIBRARY */
9141 printf( "Reconstructing mesh.\n" );
9143 /* Allocate a temporary array that maps each point to some adjacent */
9144 /* triangle. I took care to allocate all the permanent memory for */
9145 /* triangles and shell edges first. */
9146 vertexarray = (triangle *) malloc( points.items * sizeof( triangle ));
9147 if ( vertexarray == (triangle *) NULL ) {
9148 printf( "Error: Out of memory.\n" );
9151 /* Each point is initially unrepresented. */
9152 for ( i = 0; i < points.items; i++ ) {
9153 vertexarray[i] = (triangle) dummytri;
9157 printf( " Assembling triangles.\n" );
9159 /* Read the triangles from the .ele file, and link */
9160 /* together those that share an edge. */
9161 traversalinit( &triangles );
9162 triangleloop.tri = triangletraverse();
9163 elementnumber = firstnumber;
9164 while ( triangleloop.tri != (triangle *) NULL ) {
9167 /* Copy the triangle's three corners. */
9168 for ( j = 0; j < 3; j++ ) {
9169 corner[j] = trianglelist[pointindex++];
9170 if (( corner[j] < firstnumber ) || ( corner[j] >= firstnumber + inpoints )) {
9171 printf( "Error: Triangle %d has an invalid vertex index.\n",
9176 #else /* not TRILIBRARY */
9177 /* Read triangle number and the triangle's three corners. */
9178 stringptr = readline( inputline, elefile, elefilename );
9179 for ( j = 0; j < 3; j++ ) {
9180 stringptr = findfield( stringptr );
9181 if ( *stringptr == '\0' ) {
9182 printf( "Error: Triangle %d is missing point %d in %s.\n",
9183 elementnumber, j + 1, elefilename );
9187 corner[j] = (int) strtol( stringptr, &stringptr, 0 );
9188 if (( corner[j] < firstnumber ) ||
9189 ( corner[j] >= firstnumber + inpoints )) {
9190 printf( "Error: Triangle %d has an invalid vertex index.\n",
9196 #endif /* not TRILIBRARY */
9198 /* Find out about (and throw away) extra nodes. */
9199 for ( j = 3; j < incorners; j++ ) {
9202 killpointindex = trianglelist[pointindex++];
9203 #else /* not TRILIBRARY */
9204 stringptr = findfield( stringptr );
9205 if ( *stringptr != '\0' ) {
9206 killpointindex = (int) strtol( stringptr, &stringptr, 0 );
9207 #endif /* not TRILIBRARY */
9208 if (( killpointindex >= firstnumber ) &&
9209 ( killpointindex < firstnumber + inpoints )) {
9210 /* Delete the non-corner point if it's not already deleted. */
9211 killpoint = getpoint( killpointindex );
9212 if ( pointmark( killpoint ) != DEADPOINT ) {
9213 pointdealloc( killpoint );
9219 #endif /* not TRILIBRARY */
9222 /* Read the triangle's attributes. */
9223 for ( j = 0; j < eextras; j++ ) {
9226 setelemattribute( triangleloop, j, triangleattriblist[attribindex++] );
9227 #else /* not TRILIBRARY */
9228 stringptr = findfield( stringptr );
9229 if ( *stringptr == '\0' ) {
9230 setelemattribute( triangleloop, j, 0 );
9233 setelemattribute( triangleloop, j,
9234 (REAL) strtod( stringptr, &stringptr ));
9236 #endif /* not TRILIBRARY */
9242 area = trianglearealist[elementnumber - firstnumber];
9243 #else /* not TRILIBRARY */
9244 /* Read an area constraint from the .area file. */
9245 stringptr = readline( inputline, areafile, areafilename );
9246 stringptr = findfield( stringptr );
9247 if ( *stringptr == '\0' ) {
9248 area = -1.0; /* No constraint on this triangle. */
9251 area = (REAL) strtod( stringptr, &stringptr );
9253 #endif /* not TRILIBRARY */
9254 setareabound( triangleloop, area );
9257 /* Set the triangle's vertices. */
9258 triangleloop.orient = 0;
9259 setorg( triangleloop, getpoint( corner[0] ));
9260 setdest( triangleloop, getpoint( corner[1] ));
9261 setapex( triangleloop, getpoint( corner[2] ));
9262 /* Try linking the triangle to others that share these vertices. */
9263 for ( triangleloop.orient = 0; triangleloop.orient < 3;
9264 triangleloop.orient++ ) {
9265 /* Take the number for the origin of triangleloop. */
9266 aroundpoint = corner[triangleloop.orient];
9267 /* Look for other triangles having this vertex. */
9268 nexttri = vertexarray[aroundpoint - firstnumber];
9269 /* Link the current triangle to the next one in the stack. */
9270 triangleloop.tri[6 + triangleloop.orient] = nexttri;
9271 /* Push the current triangle onto the stack. */
9272 vertexarray[aroundpoint - firstnumber] = encode( triangleloop );
9273 decode( nexttri, checktri );
9274 if ( checktri.tri != dummytri ) {
9275 dest( triangleloop, tdest );
9276 apex( triangleloop, tapex );
9277 /* Look for other triangles that share an edge. */
9279 dest( checktri, checkdest );
9280 apex( checktri, checkapex );
9281 if ( tapex == checkdest ) {
9282 /* The two triangles share an edge; bond them together. */
9283 lprev( triangleloop, triangleleft );
9284 bond( triangleleft, checktri );
9286 if ( tdest == checkapex ) {
9287 /* The two triangles share an edge; bond them together. */
9288 lprev( checktri, checkleft );
9289 bond( triangleloop, checkleft );
9291 /* Find the next triangle in the stack. */
9292 nexttri = checktri.tri[6 + checktri.orient];
9293 decode( nexttri, checktri );
9294 } while ( checktri.tri != dummytri );
9297 triangleloop.tri = triangletraverse();
9304 #else /* not TRILIBRARY */
9309 #endif /* not TRILIBRARY */
9311 hullsize = 0; /* Prepare to count the boundary edges. */
9314 printf( " Marking segments in triangulation.\n" );
9316 /* Read the segments from the .poly file, and link them */
9317 /* to their neighboring triangles. */
9319 traversalinit( &shelles );
9320 shelleloop.sh = shelletraverse();
9321 segmentnumber = firstnumber;
9322 while ( shelleloop.sh != (shelle *) NULL ) {
9325 end[0] = segmentlist[pointindex++];
9326 end[1] = segmentlist[pointindex++];
9327 if ( segmentmarkers ) {
9328 boundmarker = segmentmarkerlist[segmentnumber - firstnumber];
9330 #else /* not TRILIBRARY */
9331 /* Read the endpoints of each segment, and possibly a boundary marker. */
9332 stringptr = readline( inputline, polyfile, inpolyfilename );
9333 /* Skip the first (segment number) field. */
9334 stringptr = findfield( stringptr );
9335 if ( *stringptr == '\0' ) {
9336 printf( "Error: Segment %d has no endpoints in %s.\n", segmentnumber,
9341 end[0] = (int) strtol( stringptr, &stringptr, 0 );
9343 stringptr = findfield( stringptr );
9344 if ( *stringptr == '\0' ) {
9345 printf( "Error: Segment %d is missing its second endpoint in %s.\n",
9346 segmentnumber, polyfilename );
9350 end[1] = (int) strtol( stringptr, &stringptr, 0 );
9352 if ( segmentmarkers ) {
9353 stringptr = findfield( stringptr );
9354 if ( *stringptr == '\0' ) {
9358 boundmarker = (int) strtol( stringptr, &stringptr, 0 );
9361 #endif /* not TRILIBRARY */
9362 for ( j = 0; j < 2; j++ ) {
9363 if (( end[j] < firstnumber ) || ( end[j] >= firstnumber + inpoints )) {
9364 printf( "Error: Segment %d has an invalid vertex index.\n",
9370 /* set the shell edge's vertices. */
9371 shelleloop.shorient = 0;
9372 setsorg( shelleloop, getpoint( end[0] ));
9373 setsdest( shelleloop, getpoint( end[1] ));
9374 setmark( shelleloop, boundmarker );
9375 /* Try linking the shell edge to triangles that share these vertices. */
9376 for ( shelleloop.shorient = 0; shelleloop.shorient < 2;
9377 shelleloop.shorient++ ) {
9378 /* Take the number for the destination of shelleloop. */
9379 aroundpoint = end[1 - shelleloop.shorient];
9380 /* Look for triangles having this vertex. */
9381 prevlink = &vertexarray[aroundpoint - firstnumber];
9382 nexttri = vertexarray[aroundpoint - firstnumber];
9383 decode( nexttri, checktri );
9384 sorg( shelleloop, shorg );
9386 /* Look for triangles having this edge. Note that I'm only */
9387 /* comparing each triangle's destination with the shell edge; */
9388 /* each triangle's apex is handled through a different vertex. */
9389 /* Because each triangle appears on three vertices' lists, each */
9390 /* occurrence of a triangle on a list can (and does) represent */
9391 /* an edge. In this way, most edges are represented twice, and */
9392 /* every triangle-segment bond is represented once. */
9393 while ( notfound && ( checktri.tri != dummytri )) {
9394 dest( checktri, checkdest );
9395 if ( shorg == checkdest ) {
9396 /* We have a match. Remove this triangle from the list. */
9397 *prevlink = checktri.tri[6 + checktri.orient];
9398 /* Bond the shell edge to the triangle. */
9399 tsbond( checktri, shelleloop );
9400 /* Check if this is a boundary edge. */
9401 sym( checktri, checkneighbor );
9402 if ( checkneighbor.tri == dummytri ) {
9403 /* The next line doesn't insert a shell edge (because there's */
9404 /* already one there), but it sets the boundary markers of */
9405 /* the existing shell edge and its vertices. */
9406 insertshelle( &checktri, 1 );
9411 /* Find the next triangle in the stack. */
9412 prevlink = &checktri.tri[6 + checktri.orient];
9413 nexttri = checktri.tri[6 + checktri.orient];
9414 decode( nexttri, checktri );
9417 shelleloop.sh = shelletraverse();
9422 /* Mark the remaining edges as not being attached to any shell edge. */
9423 /* Also, count the (yet uncounted) boundary edges. */
9424 for ( i = 0; i < points.items; i++ ) {
9425 /* Search the stack of triangles adjacent to a point. */
9426 nexttri = vertexarray[i];
9427 decode( nexttri, checktri );
9428 while ( checktri.tri != dummytri ) {
9429 /* Find the next triangle in the stack before this */
9430 /* information gets overwritten. */
9431 nexttri = checktri.tri[6 + checktri.orient];
9432 /* No adjacent shell edge. (This overwrites the stack info.) */
9433 tsdissolve( checktri );
9434 sym( checktri, checkneighbor );
9435 if ( checkneighbor.tri == dummytri ) {
9436 insertshelle( &checktri, 1 );
9439 decode( nexttri, checktri );
9443 free( vertexarray );
9447 #endif /* not CDT_ONLY */
9451 /********* General mesh construction routines end here *********/
9453 /********* Segment (shell edge) insertion begins here *********/
9457 /*****************************************************************************/
9459 /* finddirection() Find the first triangle on the path from one point */
9462 /* Finds the triangle that intersects a line segment drawn from the */
9463 /* origin of `searchtri' to the point `endpoint', and returns the result */
9464 /* in `searchtri'. The origin of `searchtri' does not change, even though */
9465 /* the triangle returned may differ from the one passed in. This routine */
9466 /* is used to find the direction to move in to get from one point to */
9469 /* The return value notes whether the destination or apex of the found */
9470 /* triangle is collinear with the two points in question. */
9472 /*****************************************************************************/
9474 enum finddirectionresult finddirection( searchtri, endpoint )
9475 struct triedge *searchtri;
9478 struct triedge checktri;
9480 point leftpoint, rightpoint;
9481 REAL leftccw, rightccw;
9482 int leftflag, rightflag;
9483 triangle ptr; /* Temporary variable used by onext() and oprev(). */
9485 org( *searchtri, startpoint );
9486 dest( *searchtri, rightpoint );
9487 apex( *searchtri, leftpoint );
9488 /* Is `endpoint' to the left? */
9489 leftccw = counterclockwise( endpoint, startpoint, leftpoint );
9490 leftflag = leftccw > 0.0;
9491 /* Is `endpoint' to the right? */
9492 rightccw = counterclockwise( startpoint, endpoint, rightpoint );
9493 rightflag = rightccw > 0.0;
9494 if ( leftflag && rightflag ) {
9495 /* `searchtri' faces directly away from `endpoint'. We could go */
9496 /* left or right. Ask whether it's a triangle or a boundary */
9498 onext( *searchtri, checktri );
9499 if ( checktri.tri == dummytri ) {
9506 while ( leftflag ) {
9507 /* Turn left until satisfied. */
9508 onextself( *searchtri );
9509 if ( searchtri->tri == dummytri ) {
9510 printf( "Internal error in finddirection(): Unable to find a\n" );
9511 printf( " triangle leading from (%.12g, %.12g) to", startpoint[0],
9513 printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] );
9516 apex( *searchtri, leftpoint );
9518 leftccw = counterclockwise( endpoint, startpoint, leftpoint );
9519 leftflag = leftccw > 0.0;
9521 while ( rightflag ) {
9522 /* Turn right until satisfied. */
9523 oprevself( *searchtri );
9524 if ( searchtri->tri == dummytri ) {
9525 printf( "Internal error in finddirection(): Unable to find a\n" );
9526 printf( " triangle leading from (%.12g, %.12g) to", startpoint[0],
9528 printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] );
9531 dest( *searchtri, rightpoint );
9533 rightccw = counterclockwise( startpoint, endpoint, rightpoint );
9534 rightflag = rightccw > 0.0;
9536 if ( leftccw == 0.0 ) {
9537 return LEFTCOLLINEAR;
9539 else if ( rightccw == 0.0 ) {
9540 return RIGHTCOLLINEAR;
9547 /*****************************************************************************/
9549 /* segmentintersection() Find the intersection of an existing segment */
9550 /* and a segment that is being inserted. Insert */
9551 /* a point at the intersection, splitting an */
9552 /* existing shell edge. */
9554 /* The segment being inserted connects the apex of splittri to endpoint2. */
9555 /* splitshelle is the shell edge being split, and MUST be opposite */
9556 /* splittri. Hence, the edge being split connects the origin and */
9557 /* destination of splittri. */
9559 /* On completion, splittri is a handle having the newly inserted */
9560 /* intersection point as its origin, and endpoint1 as its destination. */
9562 /*****************************************************************************/
9564 void segmentintersection( splittri, splitshelle, endpoint2 )
9565 struct triedge *splittri;
9566 struct edge *splitshelle;
9571 point leftpoint, rightpoint;
9573 enum insertsiteresult success;
9574 enum finddirectionresult collinear;
9580 triangle ptr; /* Temporary variable used by onext(). */
9582 /* Find the other three segment endpoints. */
9583 apex( *splittri, endpoint1 );
9584 org( *splittri, torg );
9585 dest( *splittri, tdest );
9586 /* Segment intersection formulae; see the Antonio reference. */
9587 tx = tdest[0] - torg[0];
9588 ty = tdest[1] - torg[1];
9589 ex = endpoint2[0] - endpoint1[0];
9590 ey = endpoint2[1] - endpoint1[1];
9591 etx = torg[0] - endpoint2[0];
9592 ety = torg[1] - endpoint2[1];
9593 denom = ty * ex - tx * ey;
9594 if ( denom == 0.0 ) {
9595 printf( "Internal error in segmentintersection():" );
9596 printf( " Attempt to find intersection of parallel segments.\n" );
9599 split = ( ey * etx - ex * ety ) / denom;
9600 /* Create the new point. */
9601 newpoint = (point) poolalloc( &points );
9602 /* Interpolate its coordinate and attributes. */
9603 for ( i = 0; i < 2 + nextras; i++ ) {
9604 newpoint[i] = torg[i] + split * ( tdest[i] - torg[i] );
9606 setpointmark( newpoint, mark( *splitshelle ));
9607 if ( verbose > 1 ) {
9609 " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n",
9610 torg[0], torg[1], tdest[0], tdest[1], newpoint[0], newpoint[1] );
9612 /* Insert the intersection point. This should always succeed. */
9613 success = insertsite( newpoint, splittri, splitshelle, 0, 0 );
9614 if ( success != SUCCESSFULPOINT ) {
9615 printf( "Internal error in segmentintersection():\n" );
9616 printf( " Failure to split a segment.\n" );
9619 if ( steinerleft > 0 ) {
9622 /* Inserting the point may have caused edge flips. We wish to rediscover */
9623 /* the edge connecting endpoint1 to the new intersection point. */
9624 collinear = finddirection( splittri, endpoint1 );
9625 dest( *splittri, rightpoint );
9626 apex( *splittri, leftpoint );
9627 if (( leftpoint[0] == endpoint1[0] ) && ( leftpoint[1] == endpoint1[1] )) {
9628 onextself( *splittri );
9630 else if (( rightpoint[0] != endpoint1[0] ) ||
9631 ( rightpoint[1] != endpoint1[1] )) {
9632 printf( "Internal error in segmentintersection():\n" );
9633 printf( " Topological inconsistency after splitting a segment.\n" );
9636 /* `splittri' should have destination endpoint1. */
9639 /*****************************************************************************/
9641 /* scoutsegment() Scout the first triangle on the path from one endpoint */
9642 /* to another, and check for completion (reaching the */
9643 /* second endpoint), a collinear point, and the */
9644 /* intersection of two segments. */
9646 /* Returns one if the entire segment is successfully inserted, and zero if */
9647 /* the job must be finished by conformingedge() or constrainededge(). */
9649 /* If the first triangle on the path has the second endpoint as its */
9650 /* destination or apex, a shell edge is inserted and the job is done. */
9652 /* If the first triangle on the path has a destination or apex that lies on */
9653 /* the segment, a shell edge is inserted connecting the first endpoint to */
9654 /* the collinear point, and the search is continued from the collinear */
9657 /* If the first triangle on the path has a shell edge opposite its origin, */
9658 /* then there is a segment that intersects the segment being inserted. */
9659 /* Their intersection point is inserted, splitting the shell edge. */
9661 /* Otherwise, return zero. */
9663 /*****************************************************************************/
9665 int scoutsegment( searchtri, endpoint2, newmark )
9666 struct triedge *searchtri;
9670 struct triedge crosstri;
9671 struct edge crossedge;
9672 point leftpoint, rightpoint;
9674 enum finddirectionresult collinear;
9675 shelle sptr; /* Temporary variable used by tspivot(). */
9677 collinear = finddirection( searchtri, endpoint2 );
9678 dest( *searchtri, rightpoint );
9679 apex( *searchtri, leftpoint );
9680 if ((( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] )) ||
9681 (( rightpoint[0] == endpoint2[0] ) && ( rightpoint[1] == endpoint2[1] ))) {
9682 /* The segment is already an edge in the mesh. */
9683 if (( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] )) {
9684 lprevself( *searchtri );
9686 /* Insert a shell edge, if there isn't already one there. */
9687 insertshelle( searchtri, newmark );
9690 else if ( collinear == LEFTCOLLINEAR ) {
9691 /* We've collided with a point between the segment's endpoints. */
9692 /* Make the collinear point be the triangle's origin. */
9693 lprevself( *searchtri );
9694 insertshelle( searchtri, newmark );
9695 /* Insert the remainder of the segment. */
9696 return scoutsegment( searchtri, endpoint2, newmark );
9698 else if ( collinear == RIGHTCOLLINEAR ) {
9699 /* We've collided with a point between the segment's endpoints. */
9700 insertshelle( searchtri, newmark );
9701 /* Make the collinear point be the triangle's origin. */
9702 lnextself( *searchtri );
9703 /* Insert the remainder of the segment. */
9704 return scoutsegment( searchtri, endpoint2, newmark );
9707 lnext( *searchtri, crosstri );
9708 tspivot( crosstri, crossedge );
9709 /* Check for a crossing segment. */
9710 if ( crossedge.sh == dummysh ) {
9714 org( *searchtri, endpoint1 );
9715 /* Insert a point at the intersection. */
9716 segmentintersection( &crosstri, &crossedge, endpoint2 );
9717 triedgecopy( crosstri, *searchtri );
9718 insertshelle( searchtri, newmark );
9719 /* Insert the remainder of the segment. */
9720 return scoutsegment( searchtri, endpoint2, newmark );
9725 /*****************************************************************************/
9727 /* conformingedge() Force a segment into a conforming Delaunay */
9728 /* triangulation by inserting a point at its midpoint, */
9729 /* and recursively forcing in the two half-segments if */
9732 /* Generates a sequence of edges connecting `endpoint1' to `endpoint2'. */
9733 /* `newmark' is the boundary marker of the segment, assigned to each new */
9734 /* splitting point and shell edge. */
9736 /* Note that conformingedge() does not always maintain the conforming */
9737 /* Delaunay property. Once inserted, segments are locked into place; */
9738 /* points inserted later (to force other segments in) may render these */
9739 /* fixed segments non-Delaunay. The conforming Delaunay property will be */
9740 /* restored by enforcequality() by splitting encroached segments. */
9742 /*****************************************************************************/
9749 void conformingedge( endpoint1, endpoint2, newmark )
9754 struct triedge searchtri1, searchtri2;
9755 struct edge brokenshelle;
9757 point midpoint1, midpoint2;
9758 enum insertsiteresult success;
9759 int result1, result2;
9761 shelle sptr; /* Temporary variable used by tspivot(). */
9763 if ( verbose > 2 ) {
9764 printf( "Forcing segment into triangulation by recursive splitting:\n" );
9765 printf( " (%.12g, %.12g) (%.12g, %.12g)\n", endpoint1[0], endpoint1[1],
9766 endpoint2[0], endpoint2[1] );
9768 /* Create a new point to insert in the middle of the segment. */
9769 newpoint = (point) poolalloc( &points );
9770 /* Interpolate coordinates and attributes. */
9771 for ( i = 0; i < 2 + nextras; i++ ) {
9772 newpoint[i] = 0.5 * ( endpoint1[i] + endpoint2[i] );
9774 setpointmark( newpoint, newmark );
9775 /* Find a boundary triangle to search from. */
9776 searchtri1.tri = (triangle *) NULL;
9777 /* Attempt to insert the new point. */
9778 success = insertsite( newpoint, &searchtri1, (struct edge *) NULL, 0, 0 );
9779 if ( success == DUPLICATEPOINT ) {
9780 if ( verbose > 2 ) {
9781 printf( " Segment intersects existing point (%.12g, %.12g).\n",
9782 newpoint[0], newpoint[1] );
9784 /* Use the point that's already there. */
9785 pointdealloc( newpoint );
9786 org( searchtri1, newpoint );
9789 if ( success == VIOLATINGPOINT ) {
9790 if ( verbose > 2 ) {
9791 printf( " Two segments intersect at (%.12g, %.12g).\n",
9792 newpoint[0], newpoint[1] );
9794 /* By fluke, we've landed right on another segment. Split it. */
9795 tspivot( searchtri1, brokenshelle );
9796 success = insertsite( newpoint, &searchtri1, &brokenshelle, 0, 0 );
9797 if ( success != SUCCESSFULPOINT ) {
9798 printf( "Internal error in conformingedge():\n" );
9799 printf( " Failure to split a segment.\n" );
9803 /* The point has been inserted successfully. */
9804 if ( steinerleft > 0 ) {
9808 triedgecopy( searchtri1, searchtri2 );
9809 result1 = scoutsegment( &searchtri1, endpoint1, newmark );
9810 result2 = scoutsegment( &searchtri2, endpoint2, newmark );
9812 /* The origin of searchtri1 may have changed if a collision with an */
9813 /* intervening vertex on the segment occurred. */
9814 org( searchtri1, midpoint1 );
9815 conformingedge( midpoint1, endpoint1, newmark );
9818 /* The origin of searchtri2 may have changed if a collision with an */
9819 /* intervening vertex on the segment occurred. */
9820 org( searchtri2, midpoint2 );
9821 conformingedge( midpoint2, endpoint2, newmark );
9825 #endif /* not CDT_ONLY */
9826 #endif /* not REDUCED */
9828 /*****************************************************************************/
9830 /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */
9831 /* recursively from an existing point. Pay special */
9832 /* attention to stacking inverted triangles. */
9834 /* This is a support routine for inserting segments into a constrained */
9835 /* Delaunay triangulation. */
9837 /* The origin of fixuptri is treated as if it has just been inserted, and */
9838 /* the local Delaunay condition needs to be enforced. It is only enforced */
9839 /* in one sector, however, that being the angular range defined by */
9842 /* This routine also needs to make decisions regarding the "stacking" of */
9843 /* triangles. (Read the description of constrainededge() below before */
9844 /* reading on here, so you understand the algorithm.) If the position of */
9845 /* the new point (the origin of fixuptri) indicates that the vertex before */
9846 /* it on the polygon is a reflex vertex, then "stack" the triangle by */
9847 /* doing nothing. (fixuptri is an inverted triangle, which is how stacked */
9848 /* triangles are identified.) */
9850 /* Otherwise, check whether the vertex before that was a reflex vertex. */
9851 /* If so, perform an edge flip, thereby eliminating an inverted triangle */
9852 /* (popping it off the stack). The edge flip may result in the creation */
9853 /* of a new inverted triangle, depending on whether or not the new vertex */
9854 /* is visible to the vertex three edges behind on the polygon. */
9856 /* If neither of the two vertices behind the new vertex are reflex */
9857 /* vertices, fixuptri and fartri, the triangle opposite it, are not */
9858 /* inverted; hence, ensure that the edge between them is locally Delaunay. */
9860 /* `leftside' indicates whether or not fixuptri is to the left of the */
9861 /* segment being inserted. (Imagine that the segment is pointing up from */
9862 /* endpoint1 to endpoint2.) */
9864 /*****************************************************************************/
9866 void delaunayfixup( fixuptri, leftside )
9867 struct triedge *fixuptri;
9870 struct triedge neartri;
9871 struct triedge fartri;
9872 struct edge faredge;
9873 point nearpoint, leftpoint, rightpoint, farpoint;
9874 triangle ptr; /* Temporary variable used by sym(). */
9875 shelle sptr; /* Temporary variable used by tspivot(). */
9877 lnext( *fixuptri, neartri );
9878 sym( neartri, fartri );
9879 /* Check if the edge opposite the origin of fixuptri can be flipped. */
9880 if ( fartri.tri == dummytri ) {
9883 tspivot( neartri, faredge );
9884 if ( faredge.sh != dummysh ) {
9887 /* Find all the relevant vertices. */
9888 apex( neartri, nearpoint );
9889 org( neartri, leftpoint );
9890 dest( neartri, rightpoint );
9891 apex( fartri, farpoint );
9892 /* Check whether the previous polygon vertex is a reflex vertex. */
9894 if ( counterclockwise( nearpoint, leftpoint, farpoint ) <= 0.0 ) {
9895 /* leftpoint is a reflex vertex too. Nothing can */
9896 /* be done until a convex section is found. */
9901 if ( counterclockwise( farpoint, rightpoint, nearpoint ) <= 0.0 ) {
9902 /* rightpoint is a reflex vertex too. Nothing can */
9903 /* be done until a convex section is found. */
9907 if ( counterclockwise( rightpoint, leftpoint, farpoint ) > 0.0 ) {
9908 /* fartri is not an inverted triangle, and farpoint is not a reflex */
9909 /* vertex. As there are no reflex vertices, fixuptri isn't an */
9910 /* inverted triangle, either. Hence, test the edge between the */
9911 /* triangles to ensure it is locally Delaunay. */
9912 if ( incircle( leftpoint, farpoint, rightpoint, nearpoint ) <= 0.0 ) {
9915 /* Not locally Delaunay; go on to an edge flip. */
9916 } /* else fartri is inverted; remove it from the stack by flipping. */
9918 lprevself( *fixuptri ); /* Restore the origin of fixuptri after the flip. */
9919 /* Recursively process the two triangles that result from the flip. */
9920 delaunayfixup( fixuptri, leftside );
9921 delaunayfixup( &fartri, leftside );
9924 /*****************************************************************************/
9926 /* constrainededge() Force a segment into a constrained Delaunay */
9927 /* triangulation by deleting the triangles it */
9928 /* intersects, and triangulating the polygons that */
9929 /* form on each side of it. */
9931 /* Generates a single edge connecting `endpoint1' to `endpoint2'. The */
9932 /* triangle `starttri' has `endpoint1' as its origin. `newmark' is the */
9933 /* boundary marker of the segment. */
9935 /* To insert a segment, every triangle whose interior intersects the */
9936 /* segment is deleted. The union of these deleted triangles is a polygon */
9937 /* (which is not necessarily monotone, but is close enough), which is */
9938 /* divided into two polygons by the new segment. This routine's task is */
9939 /* to generate the Delaunay triangulation of these two polygons. */
9941 /* You might think of this routine's behavior as a two-step process. The */
9942 /* first step is to walk from endpoint1 to endpoint2, flipping each edge */
9943 /* encountered. This step creates a fan of edges connected to endpoint1, */
9944 /* including the desired edge to endpoint2. The second step enforces the */
9945 /* Delaunay condition on each side of the segment in an incremental manner: */
9946 /* proceeding along the polygon from endpoint1 to endpoint2 (this is done */
9947 /* independently on each side of the segment), each vertex is "enforced" */
9948 /* as if it had just been inserted, but affecting only the previous */
9949 /* vertices. The result is the same as if the vertices had been inserted */
9950 /* in the order they appear on the polygon, so the result is Delaunay. */
9952 /* In truth, constrainededge() interleaves these two steps. The procedure */
9953 /* walks from endpoint1 to endpoint2, and each time an edge is encountered */
9954 /* and flipped, the newly exposed vertex (at the far end of the flipped */
9955 /* edge) is "enforced" upon the previously flipped edges, usually affecting */
9956 /* only one side of the polygon (depending upon which side of the segment */
9957 /* the vertex falls on). */
9959 /* The algorithm is complicated by the need to handle polygons that are not */
9960 /* convex. Although the polygon is not necessarily monotone, it can be */
9961 /* triangulated in a manner similar to the stack-based algorithms for */
9962 /* monotone polygons. For each reflex vertex (local concavity) of the */
9963 /* polygon, there will be an inverted triangle formed by one of the edge */
9964 /* flips. (An inverted triangle is one with negative area - that is, its */
9965 /* vertices are arranged in clockwise order - and is best thought of as a */
9966 /* wrinkle in the fabric of the mesh.) Each inverted triangle can be */
9967 /* thought of as a reflex vertex pushed on the stack, waiting to be fixed */
9970 /* A reflex vertex is popped from the stack when a vertex is inserted that */
9971 /* is visible to the reflex vertex. (However, if the vertex behind the */
9972 /* reflex vertex is not visible to the reflex vertex, a new inverted */
9973 /* triangle will take its place on the stack.) These details are handled */
9974 /* by the delaunayfixup() routine above. */
9976 /*****************************************************************************/
9978 void constrainededge( starttri, endpoint2, newmark )
9979 struct triedge *starttri;
9983 struct triedge fixuptri, fixuptri2;
9984 struct edge fixupedge;
9990 triangle ptr; /* Temporary variable used by sym() and oprev(). */
9991 shelle sptr; /* Temporary variable used by tspivot(). */
9993 org( *starttri, endpoint1 );
9994 lnext( *starttri, fixuptri );
9996 /* `collision' indicates whether we have found a point directly */
9997 /* between endpoint1 and endpoint2. */
10001 org( fixuptri, farpoint );
10002 /* `farpoint' is the extreme point of the polygon we are "digging" */
10003 /* to get from endpoint1 to endpoint2. */
10004 if (( farpoint[0] == endpoint2[0] ) && ( farpoint[1] == endpoint2[1] )) {
10005 oprev( fixuptri, fixuptri2 );
10006 /* Enforce the Delaunay condition around endpoint2. */
10007 delaunayfixup( &fixuptri, 0 );
10008 delaunayfixup( &fixuptri2, 1 );
10012 /* Check whether farpoint is to the left or right of the segment */
10013 /* being inserted, to decide which edge of fixuptri to dig */
10014 /* through next. */
10015 area = counterclockwise( endpoint1, endpoint2, farpoint );
10016 if ( area == 0.0 ) {
10017 /* We've collided with a point between endpoint1 and endpoint2. */
10019 oprev( fixuptri, fixuptri2 );
10020 /* Enforce the Delaunay condition around farpoint. */
10021 delaunayfixup( &fixuptri, 0 );
10022 delaunayfixup( &fixuptri2, 1 );
10026 if ( area > 0.0 ) { /* farpoint is to the left of the segment. */
10027 oprev( fixuptri, fixuptri2 );
10028 /* Enforce the Delaunay condition around farpoint, on the */
10029 /* left side of the segment only. */
10030 delaunayfixup( &fixuptri2, 1 );
10031 /* Flip the edge that crosses the segment. After the edge is */
10032 /* flipped, one of its endpoints is the fan vertex, and the */
10033 /* destination of fixuptri is the fan vertex. */
10034 lprevself( fixuptri );
10036 else { /* farpoint is to the right of the segment. */
10037 delaunayfixup( &fixuptri, 0 );
10038 /* Flip the edge that crosses the segment. After the edge is */
10039 /* flipped, one of its endpoints is the fan vertex, and the */
10040 /* destination of fixuptri is the fan vertex. */
10041 oprevself( fixuptri );
10043 /* Check for two intersecting segments. */
10044 tspivot( fixuptri, fixupedge );
10045 if ( fixupedge.sh == dummysh ) {
10046 flip( &fixuptri ); /* May create an inverted triangle on the left. */
10049 /* We've collided with a segment between endpoint1 and endpoint2. */
10051 /* Insert a point at the intersection. */
10052 segmentintersection( &fixuptri, &fixupedge, endpoint2 );
10058 /* Insert a shell edge to make the segment permanent. */
10059 insertshelle( &fixuptri, newmark );
10060 /* If there was a collision with an interceding vertex, install another */
10061 /* segment connecting that vertex with endpoint2. */
10063 /* Insert the remainder of the segment. */
10064 if ( !scoutsegment( &fixuptri, endpoint2, newmark )) {
10065 constrainededge( &fixuptri, endpoint2, newmark );
10070 /*****************************************************************************/
10072 /* insertsegment() Insert a PSLG segment into a triangulation. */
10074 /*****************************************************************************/
10076 void insertsegment( endpoint1, endpoint2, newmark )
10081 struct triedge searchtri1, searchtri2;
10082 triangle encodedtri;
10084 triangle ptr; /* Temporary variable used by sym(). */
10086 if ( verbose > 1 ) {
10087 printf( " Connecting (%.12g, %.12g) to (%.12g, %.12g).\n",
10088 endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1] );
10091 /* Find a triangle whose origin is the segment's first endpoint. */
10092 checkpoint = (point) NULL;
10093 encodedtri = point2tri( endpoint1 );
10094 if ( encodedtri != (triangle) NULL ) {
10095 decode( encodedtri, searchtri1 );
10096 org( searchtri1, checkpoint );
10098 if ( checkpoint != endpoint1 ) {
10099 /* Find a boundary triangle to search from. */
10100 searchtri1.tri = dummytri;
10101 searchtri1.orient = 0;
10102 symself( searchtri1 );
10103 /* Search for the segment's first endpoint by point location. */
10104 if ( locate( endpoint1, &searchtri1 ) != ONVERTEX ) {
10106 "Internal error in insertsegment(): Unable to locate PSLG point\n" );
10107 printf( " (%.12g, %.12g) in triangulation.\n",
10108 endpoint1[0], endpoint1[1] );
10112 /* Remember this triangle to improve subsequent point location. */
10113 triedgecopy( searchtri1, recenttri );
10114 /* Scout the beginnings of a path from the first endpoint */
10115 /* toward the second. */
10116 if ( scoutsegment( &searchtri1, endpoint2, newmark )) {
10117 /* The segment was easily inserted. */
10120 /* The first endpoint may have changed if a collision with an intervening */
10121 /* vertex on the segment occurred. */
10122 org( searchtri1, endpoint1 );
10124 /* Find a triangle whose origin is the segment's second endpoint. */
10125 checkpoint = (point) NULL;
10126 encodedtri = point2tri( endpoint2 );
10127 if ( encodedtri != (triangle) NULL ) {
10128 decode( encodedtri, searchtri2 );
10129 org( searchtri2, checkpoint );
10131 if ( checkpoint != endpoint2 ) {
10132 /* Find a boundary triangle to search from. */
10133 searchtri2.tri = dummytri;
10134 searchtri2.orient = 0;
10135 symself( searchtri2 );
10136 /* Search for the segment's second endpoint by point location. */
10137 if ( locate( endpoint2, &searchtri2 ) != ONVERTEX ) {
10139 "Internal error in insertsegment(): Unable to locate PSLG point\n" );
10140 printf( " (%.12g, %.12g) in triangulation.\n",
10141 endpoint2[0], endpoint2[1] );
10145 /* Remember this triangle to improve subsequent point location. */
10146 triedgecopy( searchtri2, recenttri );
10147 /* Scout the beginnings of a path from the second endpoint */
10148 /* toward the first. */
10149 if ( scoutsegment( &searchtri2, endpoint1, newmark )) {
10150 /* The segment was easily inserted. */
10153 /* The second endpoint may have changed if a collision with an intervening */
10154 /* vertex on the segment occurred. */
10155 org( searchtri2, endpoint2 );
10162 /* Insert vertices to force the segment into the triangulation. */
10163 conformingedge( endpoint1, endpoint2, newmark );
10166 #endif /* not CDT_ONLY */
10167 #endif /* not REDUCED */
10168 /* Insert the segment directly into the triangulation. */
10169 constrainededge( &searchtri1, endpoint2, newmark );
10175 #endif /* not CDT_ONLY */
10176 #endif /* not REDUCED */
10179 /*****************************************************************************/
10181 /* markhull() Cover the convex hull of a triangulation with shell edges. */
10183 /*****************************************************************************/
10186 struct triedge hulltri;
10187 struct triedge nexttri;
10188 struct triedge starttri;
10189 triangle ptr; /* Temporary variable used by sym() and oprev(). */
10191 /* Find a triangle handle on the hull. */
10192 hulltri.tri = dummytri;
10193 hulltri.orient = 0;
10194 symself( hulltri );
10195 /* Remember where we started so we know when to stop. */
10196 triedgecopy( hulltri, starttri );
10197 /* Go once counterclockwise around the convex hull. */
10199 /* Create a shell edge if there isn't already one here. */
10200 insertshelle( &hulltri, 1 );
10201 /* To find the next hull edge, go clockwise around the next vertex. */
10202 lnextself( hulltri );
10203 oprev( hulltri, nexttri );
10204 while ( nexttri.tri != dummytri ) {
10205 triedgecopy( nexttri, hulltri );
10206 oprev( hulltri, nexttri );
10208 } while ( !triedgeequal( hulltri, starttri ));
10211 /*****************************************************************************/
10213 /* formskeleton() Create the shell edges of a triangulation, including */
10214 /* PSLG edges and edges on the convex hull. */
10216 /* The PSLG edges are read from a .poly file. The return value is the */
10217 /* number of segments in the file. */
10219 /*****************************************************************************/
10224 int formskeleton( segmentlist, segmentmarkerlist, numberofsegments )
10226 int *segmentmarkerlist;
10227 int numberofsegments;
10229 #else /* not TRILIBRARY */
10231 int formskeleton( polyfile, polyfilename )
10233 char *polyfilename;
10235 #endif /* not TRILIBRARY */
10240 char polyfilename[6];
10242 #else /* not TRILIBRARY */
10243 char inputline[INPUTLINESIZE];
10245 #endif /* not TRILIBRARY */
10246 point endpoint1, endpoint2;
10248 int segmentmarkers;
10255 printf( "Inserting segments into Delaunay triangulation.\n" );
10259 strcpy( polyfilename, "input" );
10260 segments = numberofsegments;
10261 segmentmarkers = segmentmarkerlist != (int *) NULL;
10263 #else /* not TRILIBRARY */
10264 /* Read the segments from a .poly file. */
10265 /* Read number of segments and number of boundary markers. */
10266 stringptr = readline( inputline, polyfile, polyfilename );
10267 segments = (int) strtol( stringptr, &stringptr, 0 );
10268 stringptr = findfield( stringptr );
10269 if ( *stringptr == '\0' ) {
10270 segmentmarkers = 0;
10273 segmentmarkers = (int) strtol( stringptr, &stringptr, 0 );
10275 #endif /* not TRILIBRARY */
10276 /* If segments are to be inserted, compute a mapping */
10277 /* from points to triangles. */
10278 if ( segments > 0 ) {
10280 printf( " Inserting PSLG segments.\n" );
10286 /* Read and insert the segments. */
10287 for ( i = 1; i <= segments; i++ ) {
10290 end1 = segmentlist[index++];
10291 end2 = segmentlist[index++];
10292 if ( segmentmarkers ) {
10293 boundmarker = segmentmarkerlist[i - 1];
10295 #else /* not TRILIBRARY */
10296 stringptr = readline( inputline, polyfile, inpolyfilename );
10297 stringptr = findfield( stringptr );
10298 if ( *stringptr == '\0' ) {
10299 printf( "Error: Segment %d has no endpoints in %s.\n", i,
10304 end1 = (int) strtol( stringptr, &stringptr, 0 );
10306 stringptr = findfield( stringptr );
10307 if ( *stringptr == '\0' ) {
10308 printf( "Error: Segment %d is missing its second endpoint in %s.\n", i,
10313 end2 = (int) strtol( stringptr, &stringptr, 0 );
10315 if ( segmentmarkers ) {
10316 stringptr = findfield( stringptr );
10317 if ( *stringptr == '\0' ) {
10321 boundmarker = (int) strtol( stringptr, &stringptr, 0 );
10324 #endif /* not TRILIBRARY */
10325 if (( end1 < firstnumber ) || ( end1 >= firstnumber + inpoints )) {
10327 printf( "Warning: Invalid first endpoint of segment %d in %s.\n", i,
10331 else if (( end2 < firstnumber ) || ( end2 >= firstnumber + inpoints )) {
10333 printf( "Warning: Invalid second endpoint of segment %d in %s.\n", i,
10338 endpoint1 = getpoint( end1 );
10339 endpoint2 = getpoint( end2 );
10340 if (( endpoint1[0] == endpoint2[0] ) && ( endpoint1[1] == endpoint2[1] )) {
10342 printf( "Warning: Endpoints of segment %d are coincident in %s.\n",
10347 insertsegment( endpoint1, endpoint2, boundmarker );
10355 if ( convex || !poly ) {
10356 /* Enclose the convex hull with shell edges. */
10358 printf( " Enclosing convex hull with segments.\n" );
10367 /********* Segment (shell edge) insertion ends here *********/
10369 /********* Carving out holes and concavities begins here *********/
10373 /*****************************************************************************/
10375 /* infecthull() Virally infect all of the triangles of the convex hull */
10376 /* that are not protected by shell edges. Where there are */
10377 /* shell edges, set boundary markers as appropriate. */
10379 /*****************************************************************************/
10382 struct triedge hulltri;
10383 struct triedge nexttri;
10384 struct triedge starttri;
10385 struct edge hulledge;
10386 triangle **deadtri;
10388 triangle ptr; /* Temporary variable used by sym(). */
10389 shelle sptr; /* Temporary variable used by tspivot(). */
10392 printf( " Marking concavities (external triangles) for elimination.\n" );
10394 /* Find a triangle handle on the hull. */
10395 hulltri.tri = dummytri;
10396 hulltri.orient = 0;
10397 symself( hulltri );
10398 /* Remember where we started so we know when to stop. */
10399 triedgecopy( hulltri, starttri );
10400 /* Go once counterclockwise around the convex hull. */
10402 /* Ignore triangles that are already infected. */
10403 if ( !infected( hulltri )) {
10404 /* Is the triangle protected by a shell edge? */
10405 tspivot( hulltri, hulledge );
10406 if ( hulledge.sh == dummysh ) {
10407 /* The triangle is not protected; infect it. */
10409 deadtri = (triangle **) poolalloc( &viri );
10410 *deadtri = hulltri.tri;
10413 /* The triangle is protected; set boundary markers if appropriate. */
10414 if ( mark( hulledge ) == 0 ) {
10415 setmark( hulledge, 1 );
10416 org( hulltri, horg );
10417 dest( hulltri, hdest );
10418 if ( pointmark( horg ) == 0 ) {
10419 setpointmark( horg, 1 );
10421 if ( pointmark( hdest ) == 0 ) {
10422 setpointmark( hdest, 1 );
10427 /* To find the next hull edge, go clockwise around the next vertex. */
10428 lnextself( hulltri );
10429 oprev( hulltri, nexttri );
10430 while ( nexttri.tri != dummytri ) {
10431 triedgecopy( nexttri, hulltri );
10432 oprev( hulltri, nexttri );
10434 } while ( !triedgeequal( hulltri, starttri ));
10437 /*****************************************************************************/
10439 /* plague() Spread the virus from all infected triangles to any neighbors */
10440 /* not protected by shell edges. Delete all infected triangles. */
10442 /* This is the procedure that actually creates holes and concavities. */
10444 /* This procedure operates in two phases. The first phase identifies all */
10445 /* the triangles that will die, and marks them as infected. They are */
10446 /* marked to ensure that each triangle is added to the virus pool only */
10447 /* once, so the procedure will terminate. */
10449 /* The second phase actually eliminates the infected triangles. It also */
10450 /* eliminates orphaned points. */
10452 /*****************************************************************************/
10455 struct triedge testtri;
10456 struct triedge neighbor;
10457 triangle **virusloop;
10458 triangle **deadtri;
10459 struct edge neighborshelle;
10462 point deadorg, deaddest, deadapex;
10464 triangle ptr; /* Temporary variable used by sym() and onext(). */
10465 shelle sptr; /* Temporary variable used by tspivot(). */
10468 printf( " Marking neighbors of marked triangles.\n" );
10470 /* Loop through all the infected triangles, spreading the virus to */
10471 /* their neighbors, then to their neighbors' neighbors. */
10472 traversalinit( &viri );
10473 virusloop = (triangle **) traverse( &viri );
10474 while ( virusloop != (triangle **) NULL ) {
10475 testtri.tri = *virusloop;
10476 /* A triangle is marked as infected by messing with one of its shell */
10477 /* edges, setting it to an illegal value. Hence, we have to */
10478 /* temporarily uninfect this triangle so that we can examine its */
10479 /* adjacent shell edges. */
10480 uninfect( testtri );
10481 if ( verbose > 2 ) {
10482 /* Assign the triangle an orientation for convenience in */
10483 /* checking its points. */
10484 testtri.orient = 0;
10485 org( testtri, deadorg );
10486 dest( testtri, deaddest );
10487 apex( testtri, deadapex );
10488 printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10489 deadorg[0], deadorg[1], deaddest[0], deaddest[1],
10490 deadapex[0], deadapex[1] );
10492 /* Check each of the triangle's three neighbors. */
10493 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10494 /* Find the neighbor. */
10495 sym( testtri, neighbor );
10496 /* Check for a shell between the triangle and its neighbor. */
10497 tspivot( testtri, neighborshelle );
10498 /* Check if the neighbor is nonexistent or already infected. */
10499 if (( neighbor.tri == dummytri ) || infected( neighbor )) {
10500 if ( neighborshelle.sh != dummysh ) {
10501 /* There is a shell edge separating the triangle from its */
10502 /* neighbor, but both triangles are dying, so the shell */
10503 /* edge dies too. */
10504 shelledealloc( neighborshelle.sh );
10505 if ( neighbor.tri != dummytri ) {
10506 /* Make sure the shell edge doesn't get deallocated again */
10507 /* later when the infected neighbor is visited. */
10508 uninfect( neighbor );
10509 tsdissolve( neighbor );
10510 infect( neighbor );
10514 else { /* The neighbor exists and is not infected. */
10515 if ( neighborshelle.sh == dummysh ) {
10516 /* There is no shell edge protecting the neighbor, so */
10517 /* the neighbor becomes infected. */
10518 if ( verbose > 2 ) {
10519 org( neighbor, deadorg );
10520 dest( neighbor, deaddest );
10521 apex( neighbor, deadapex );
10523 " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10524 deadorg[0], deadorg[1], deaddest[0], deaddest[1],
10525 deadapex[0], deadapex[1] );
10527 infect( neighbor );
10528 /* Ensure that the neighbor's neighbors will be infected. */
10529 deadtri = (triangle **) poolalloc( &viri );
10530 *deadtri = neighbor.tri;
10532 else { /* The neighbor is protected by a shell edge. */
10533 /* Remove this triangle from the shell edge. */
10534 stdissolve( neighborshelle );
10535 /* The shell edge becomes a boundary. Set markers accordingly. */
10536 if ( mark( neighborshelle ) == 0 ) {
10537 setmark( neighborshelle, 1 );
10539 org( neighbor, norg );
10540 dest( neighbor, ndest );
10541 if ( pointmark( norg ) == 0 ) {
10542 setpointmark( norg, 1 );
10544 if ( pointmark( ndest ) == 0 ) {
10545 setpointmark( ndest, 1 );
10550 /* Remark the triangle as infected, so it doesn't get added to the */
10551 /* virus pool again. */
10553 virusloop = (triangle **) traverse( &viri );
10557 printf( " Deleting marked triangles.\n" );
10559 traversalinit( &viri );
10560 virusloop = (triangle **) traverse( &viri );
10561 while ( virusloop != (triangle **) NULL ) {
10562 testtri.tri = *virusloop;
10564 /* Check each of the three corners of the triangle for elimination. */
10565 /* This is done by walking around each point, checking if it is */
10566 /* still connected to at least one live triangle. */
10567 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10568 org( testtri, testpoint );
10569 /* Check if the point has already been tested. */
10570 if ( testpoint != (point) NULL ) {
10572 /* Mark the corner of the triangle as having been tested. */
10573 setorg( testtri, NULL );
10574 /* Walk counterclockwise about the point. */
10575 onext( testtri, neighbor );
10576 /* Stop upon reaching a boundary or the starting triangle. */
10577 while (( neighbor.tri != dummytri )
10578 && ( !triedgeequal( neighbor, testtri ))) {
10579 if ( infected( neighbor )) {
10580 /* Mark the corner of this triangle as having been tested. */
10581 setorg( neighbor, NULL );
10584 /* A live triangle. The point survives. */
10587 /* Walk counterclockwise about the point. */
10588 onextself( neighbor );
10590 /* If we reached a boundary, we must walk clockwise as well. */
10591 if ( neighbor.tri == dummytri ) {
10592 /* Walk clockwise about the point. */
10593 oprev( testtri, neighbor );
10594 /* Stop upon reaching a boundary. */
10595 while ( neighbor.tri != dummytri ) {
10596 if ( infected( neighbor )) {
10597 /* Mark the corner of this triangle as having been tested. */
10598 setorg( neighbor, NULL );
10601 /* A live triangle. The point survives. */
10604 /* Walk clockwise about the point. */
10605 oprevself( neighbor );
10609 if ( verbose > 1 ) {
10610 printf( " Deleting point (%.12g, %.12g)\n",
10611 testpoint[0], testpoint[1] );
10613 pointdealloc( testpoint );
10618 /* Record changes in the number of boundary edges, and disconnect */
10619 /* dead triangles from their neighbors. */
10620 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10621 sym( testtri, neighbor );
10622 if ( neighbor.tri == dummytri ) {
10623 /* There is no neighboring triangle on this edge, so this edge */
10624 /* is a boundary edge. This triangle is being deleted, so this */
10625 /* boundary edge is deleted. */
10629 /* Disconnect the triangle from its neighbor. */
10630 dissolve( neighbor );
10631 /* There is a neighboring triangle on this edge, so this edge */
10632 /* becomes a boundary edge when this triangle is deleted. */
10636 /* Return the dead triangle to the pool of triangles. */
10637 triangledealloc( testtri.tri );
10638 virusloop = (triangle **) traverse( &viri );
10640 /* Empty the virus pool. */
10641 poolrestart( &viri );
10644 /*****************************************************************************/
10646 /* regionplague() Spread regional attributes and/or area constraints */
10647 /* (from a .poly file) throughout the mesh. */
10649 /* This procedure operates in two phases. The first phase spreads an */
10650 /* attribute and/or an area constraint through a (segment-bounded) region. */
10651 /* The triangles are marked to ensure that each triangle is added to the */
10652 /* virus pool only once, so the procedure will terminate. */
10654 /* The second phase uninfects all infected triangles, returning them to */
10657 /*****************************************************************************/
10659 void regionplague( attribute, area )
10663 struct triedge testtri;
10664 struct triedge neighbor;
10665 triangle **virusloop;
10666 triangle **regiontri;
10667 struct edge neighborshelle;
10668 point regionorg, regiondest, regionapex;
10669 triangle ptr; /* Temporary variable used by sym() and onext(). */
10670 shelle sptr; /* Temporary variable used by tspivot(). */
10672 if ( verbose > 1 ) {
10673 printf( " Marking neighbors of marked triangles.\n" );
10675 /* Loop through all the infected triangles, spreading the attribute */
10676 /* and/or area constraint to their neighbors, then to their neighbors' */
10678 traversalinit( &viri );
10679 virusloop = (triangle **) traverse( &viri );
10680 while ( virusloop != (triangle **) NULL ) {
10681 testtri.tri = *virusloop;
10682 /* A triangle is marked as infected by messing with one of its shell */
10683 /* edges, setting it to an illegal value. Hence, we have to */
10684 /* temporarily uninfect this triangle so that we can examine its */
10685 /* adjacent shell edges. */
10686 uninfect( testtri );
10687 if ( regionattrib ) {
10688 /* Set an attribute. */
10689 setelemattribute( testtri, eextras, attribute );
10692 /* Set an area constraint. */
10693 setareabound( testtri, area );
10695 if ( verbose > 2 ) {
10696 /* Assign the triangle an orientation for convenience in */
10697 /* checking its points. */
10698 testtri.orient = 0;
10699 org( testtri, regionorg );
10700 dest( testtri, regiondest );
10701 apex( testtri, regionapex );
10702 printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10703 regionorg[0], regionorg[1], regiondest[0], regiondest[1],
10704 regionapex[0], regionapex[1] );
10706 /* Check each of the triangle's three neighbors. */
10707 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10708 /* Find the neighbor. */
10709 sym( testtri, neighbor );
10710 /* Check for a shell between the triangle and its neighbor. */
10711 tspivot( testtri, neighborshelle );
10712 /* Make sure the neighbor exists, is not already infected, and */
10713 /* isn't protected by a shell edge. */
10714 if (( neighbor.tri != dummytri ) && !infected( neighbor )
10715 && ( neighborshelle.sh == dummysh )) {
10716 if ( verbose > 2 ) {
10717 org( neighbor, regionorg );
10718 dest( neighbor, regiondest );
10719 apex( neighbor, regionapex );
10720 printf( " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10721 regionorg[0], regionorg[1], regiondest[0], regiondest[1],
10722 regionapex[0], regionapex[1] );
10724 /* Infect the neighbor. */
10725 infect( neighbor );
10726 /* Ensure that the neighbor's neighbors will be infected. */
10727 regiontri = (triangle **) poolalloc( &viri );
10728 *regiontri = neighbor.tri;
10731 /* Remark the triangle as infected, so it doesn't get added to the */
10732 /* virus pool again. */
10734 virusloop = (triangle **) traverse( &viri );
10737 /* Uninfect all triangles. */
10738 if ( verbose > 1 ) {
10739 printf( " Unmarking marked triangles.\n" );
10741 traversalinit( &viri );
10742 virusloop = (triangle **) traverse( &viri );
10743 while ( virusloop != (triangle **) NULL ) {
10744 testtri.tri = *virusloop;
10745 uninfect( testtri );
10746 virusloop = (triangle **) traverse( &viri );
10748 /* Empty the virus pool. */
10749 poolrestart( &viri );
10752 /*****************************************************************************/
10754 /* carveholes() Find the holes and infect them. Find the area */
10755 /* constraints and infect them. Infect the convex hull. */
10756 /* Spread the infection and kill triangles. Spread the */
10757 /* area constraints. */
10759 /* This routine mainly calls other routines to carry out all these */
10762 /*****************************************************************************/
10764 void carveholes( holelist, holes, regionlist, regions )
10770 struct triedge searchtri;
10771 struct triedge triangleloop;
10772 struct triedge *regiontris;
10773 triangle **holetri;
10774 triangle **regiontri;
10775 point searchorg, searchdest;
10776 enum locateresult intersect;
10778 triangle ptr; /* Temporary variable used by sym(). */
10780 if ( !( quiet || ( noholes && convex ))) {
10781 printf( "Removing unwanted triangles.\n" );
10782 if ( verbose && ( holes > 0 )) {
10783 printf( " Marking holes for elimination.\n" );
10787 if ( regions > 0 ) {
10788 /* Allocate storage for the triangles in which region points fall. */
10789 regiontris = (struct triedge *) malloc( regions * sizeof( struct triedge ));
10790 if ( regiontris == (struct triedge *) NULL ) {
10791 printf( "Error: Out of memory.\n" );
10796 if ((( holes > 0 ) && !noholes ) || !convex || ( regions > 0 )) {
10797 /* Initialize a pool of viri to be used for holes, concavities, */
10798 /* regional attributes, and/or regional area constraints. */
10799 poolinit( &viri, sizeof( triangle * ), VIRUSPERBLOCK, POINTER, 0 );
10803 /* Mark as infected any unprotected triangles on the boundary. */
10804 /* This is one way by which concavities are created. */
10808 if (( holes > 0 ) && !noholes ) {
10809 /* Infect each triangle in which a hole lies. */
10810 for ( i = 0; i < 2 * holes; i += 2 ) {
10811 /* Ignore holes that aren't within the bounds of the mesh. */
10812 if (( holelist[i] >= xmin ) && ( holelist[i] <= xmax )
10813 && ( holelist[i + 1] >= ymin ) && ( holelist[i + 1] <= ymax )) {
10814 /* Start searching from some triangle on the outer boundary. */
10815 searchtri.tri = dummytri;
10816 searchtri.orient = 0;
10817 symself( searchtri );
10818 /* Ensure that the hole is to the left of this boundary edge; */
10819 /* otherwise, locate() will falsely report that the hole */
10820 /* falls within the starting triangle. */
10821 org( searchtri, searchorg );
10822 dest( searchtri, searchdest );
10823 if ( counterclockwise( searchorg, searchdest, &holelist[i] ) > 0.0 ) {
10824 /* Find a triangle that contains the hole. */
10825 intersect = locate( &holelist[i], &searchtri );
10826 if (( intersect != OUTSIDE ) && ( !infected( searchtri ))) {
10827 /* Infect the triangle. This is done by marking the triangle */
10828 /* as infect and including the triangle in the virus pool. */
10829 infect( searchtri );
10830 holetri = (triangle **) poolalloc( &viri );
10831 *holetri = searchtri.tri;
10838 /* Now, we have to find all the regions BEFORE we carve the holes, because */
10839 /* locate() won't work when the triangulation is no longer convex. */
10840 /* (Incidentally, this is the reason why regional attributes and area */
10841 /* constraints can't be used when refining a preexisting mesh, which */
10842 /* might not be convex; they can only be used with a freshly */
10843 /* triangulated PSLG.) */
10844 if ( regions > 0 ) {
10845 /* Find the starting triangle for each region. */
10846 for ( i = 0; i < regions; i++ ) {
10847 regiontris[i].tri = dummytri;
10848 /* Ignore region points that aren't within the bounds of the mesh. */
10849 if (( regionlist[4 * i] >= xmin ) && ( regionlist[4 * i] <= xmax ) &&
10850 ( regionlist[4 * i + 1] >= ymin ) && ( regionlist[4 * i + 1] <= ymax )) {
10851 /* Start searching from some triangle on the outer boundary. */
10852 searchtri.tri = dummytri;
10853 searchtri.orient = 0;
10854 symself( searchtri );
10855 /* Ensure that the region point is to the left of this boundary */
10856 /* edge; otherwise, locate() will falsely report that the */
10857 /* region point falls within the starting triangle. */
10858 org( searchtri, searchorg );
10859 dest( searchtri, searchdest );
10860 if ( counterclockwise( searchorg, searchdest, ®ionlist[4 * i] ) >
10862 /* Find a triangle that contains the region point. */
10863 intersect = locate( ®ionlist[4 * i], &searchtri );
10864 if (( intersect != OUTSIDE ) && ( !infected( searchtri ))) {
10865 /* Record the triangle for processing after the */
10866 /* holes have been carved. */
10867 triedgecopy( searchtri, regiontris[i] );
10874 if ( viri.items > 0 ) {
10875 /* Carve the holes and concavities. */
10878 /* The virus pool should be empty now. */
10880 if ( regions > 0 ) {
10882 if ( regionattrib ) {
10884 printf( "Spreading regional attributes and area constraints.\n" );
10887 printf( "Spreading regional attributes.\n" );
10891 printf( "Spreading regional area constraints.\n" );
10894 if ( regionattrib && !refine ) {
10895 /* Assign every triangle a regional attribute of zero. */
10896 traversalinit( &triangles );
10897 triangleloop.orient = 0;
10898 triangleloop.tri = triangletraverse();
10899 while ( triangleloop.tri != (triangle *) NULL ) {
10900 setelemattribute( triangleloop, eextras, 0.0 );
10901 triangleloop.tri = triangletraverse();
10904 for ( i = 0; i < regions; i++ ) {
10905 if ( regiontris[i].tri != dummytri ) {
10906 /* Make sure the triangle under consideration still exists. */
10907 /* It may have been eaten by the virus. */
10908 if ( regiontris[i].tri[3] != (triangle) NULL ) {
10909 /* Put one triangle in the virus pool. */
10910 infect( regiontris[i] );
10911 regiontri = (triangle **) poolalloc( &viri );
10912 *regiontri = regiontris[i].tri;
10913 /* Apply one region's attribute and/or area constraint. */
10914 regionplague( regionlist[4 * i + 2], regionlist[4 * i + 3] );
10915 /* The virus pool should be empty now. */
10919 if ( regionattrib && !refine ) {
10920 /* Note the fact that each triangle has an additional attribute. */
10925 /* Free up memory. */
10926 if ((( holes > 0 ) && !noholes ) || !convex || ( regions > 0 )) {
10927 pooldeinit( &viri );
10929 if ( regions > 0 ) {
10930 free( regiontris );
10936 /********* Carving out holes and concavities ends here *********/
10938 /********* Mesh quality maintenance begins here *********/
10942 /*****************************************************************************/
10944 /* tallyencs() Traverse the entire list of shell edges, check each edge */
10945 /* to see if it is encroached. If so, add it to the list. */
10947 /*****************************************************************************/
10953 struct edge edgeloop;
10956 traversalinit( &shelles );
10957 edgeloop.shorient = 0;
10958 edgeloop.sh = shelletraverse();
10959 while ( edgeloop.sh != (shelle *) NULL ) {
10960 /* If the segment is encroached, add it to the list. */
10961 dummy = checkedge4encroach( &edgeloop );
10962 edgeloop.sh = shelletraverse();
10966 #endif /* not CDT_ONLY */
10968 /*****************************************************************************/
10970 /* precisionerror() Print an error message for precision problems. */
10972 /*****************************************************************************/
10977 void precisionerror(){
10978 printf( "Try increasing the area criterion and/or reducing the minimum\n" );
10979 printf( " allowable angle so that tiny triangles are not created.\n" );
10982 printf( "Alternatively, try recompiling me with double precision\n" );
10983 printf( " arithmetic (by removing \"#define SINGLE\" from the\n" );
10984 printf( " source file or \"-DSINGLE\" from the makefile).\n" );
10985 #endif /* SINGLE */
10988 #endif /* not CDT_ONLY */
10990 /*****************************************************************************/
10992 /* repairencs() Find and repair all the encroached segments. */
10994 /* Encroached segments are repaired by splitting them by inserting a point */
10995 /* at or near their centers. */
10997 /* `flaws' is a flag that specifies whether one should take note of new */
10998 /* encroached segments and bad triangles that result from inserting points */
10999 /* to repair existing encroached segments. */
11001 /* When a segment is split, the two resulting subsegments are always */
11002 /* tested to see if they are encroached upon, regardless of the value */
11005 /*****************************************************************************/
11010 void repairencs( flaws )
11013 struct triedge enctri;
11014 struct triedge testtri;
11015 struct edge *encloop;
11016 struct edge testsh;
11019 enum insertsiteresult success;
11020 REAL segmentlength, nearestpoweroftwo;
11022 int acuteorg, acutedest;
11025 triangle ptr; /* Temporary variable used by stpivot(). */
11026 shelle sptr; /* Temporary variable used by snext(). */
11028 while (( badsegments.items > 0 ) && ( steinerleft != 0 )) {
11029 traversalinit( &badsegments );
11030 encloop = badsegmenttraverse();
11031 while (( encloop != (struct edge *) NULL ) && ( steinerleft != 0 )) {
11032 /* To decide where to split a segment, we need to know if the */
11033 /* segment shares an endpoint with an adjacent segment. */
11034 /* The concern is that, if we simply split every encroached */
11035 /* segment in its center, two adjacent segments with a small */
11036 /* angle between them might lead to an infinite loop; each */
11037 /* point added to split one segment will encroach upon the */
11038 /* other segment, which must then be split with a point that */
11039 /* will encroach upon the first segment, and so on forever. */
11040 /* To avoid this, imagine a set of concentric circles, whose */
11041 /* radii are powers of two, about each segment endpoint. */
11042 /* These concentric circles determine where the segment is */
11043 /* split. (If both endpoints are shared with adjacent */
11044 /* segments, split the segment in the middle, and apply the */
11045 /* concentric shells for later splittings.) */
11047 /* Is the origin shared with another segment? */
11048 stpivot( *encloop, enctri );
11049 lnext( enctri, testtri );
11050 tspivot( testtri, testsh );
11051 acuteorg = testsh.sh != dummysh;
11052 /* Is the destination shared with another segment? */
11053 lnextself( testtri );
11054 tspivot( testtri, testsh );
11055 acutedest = testsh.sh != dummysh;
11056 /* Now, check the other side of the segment, if there's a triangle */
11058 sym( enctri, testtri );
11059 if ( testtri.tri != dummytri ) {
11060 /* Is the destination shared with another segment? */
11061 lnextself( testtri );
11062 tspivot( testtri, testsh );
11063 acutedest = acutedest || ( testsh.sh != dummysh );
11064 /* Is the origin shared with another segment? */
11065 lnextself( testtri );
11066 tspivot( testtri, testsh );
11067 acuteorg = acuteorg || ( testsh.sh != dummysh );
11070 sorg( *encloop, eorg );
11071 sdest( *encloop, edest );
11072 /* Use the concentric circles if exactly one endpoint is shared */
11073 /* with another adjacent segment. */
11074 if ( acuteorg ^ acutedest ) {
11075 segmentlength = sqrt(( edest[0] - eorg[0] ) * ( edest[0] - eorg[0] )
11076 + ( edest[1] - eorg[1] ) * ( edest[1] - eorg[1] ));
11077 /* Find the power of two nearest the segment's length. */
11078 nearestpoweroftwo = 1.0;
11079 while ( segmentlength > SQUAREROOTTWO * nearestpoweroftwo ) {
11080 nearestpoweroftwo *= 2.0;
11082 while ( segmentlength < ( 0.5 * SQUAREROOTTWO ) * nearestpoweroftwo ) {
11083 nearestpoweroftwo *= 0.5;
11085 /* Where do we split the segment? */
11086 split = 0.5 * nearestpoweroftwo / segmentlength;
11088 split = 1.0 - split;
11092 /* If we're not worried about adjacent segments, split */
11093 /* this segment in the middle. */
11097 /* Create the new point. */
11098 newpoint = (point) poolalloc( &points );
11099 /* Interpolate its coordinate and attributes. */
11100 for ( i = 0; i < 2 + nextras; i++ ) {
11101 newpoint[i] = ( 1.0 - split ) * eorg[i] + split * edest[i];
11103 setpointmark( newpoint, mark( *encloop ));
11104 if ( verbose > 1 ) {
11106 " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n",
11107 eorg[0], eorg[1], edest[0], edest[1], newpoint[0], newpoint[1] );
11109 /* Check whether the new point lies on an endpoint. */
11110 if ((( newpoint[0] == eorg[0] ) && ( newpoint[1] == eorg[1] ))
11111 || (( newpoint[0] == edest[0] ) && ( newpoint[1] == edest[1] ))) {
11112 printf( "Error: Ran out of precision at (%.12g, %.12g).\n",
11113 newpoint[0], newpoint[1] );
11114 printf( "I attempted to split a segment to a smaller size than can\n" );
11115 printf( " be accommodated by the finite precision of floating point\n"
11117 printf( " arithmetic.\n" );
11121 /* Insert the splitting point. This should always succeed. */
11122 success = insertsite( newpoint, &enctri, encloop, flaws, flaws );
11123 if (( success != SUCCESSFULPOINT ) && ( success != ENCROACHINGPOINT )) {
11124 printf( "Internal error in repairencs():\n" );
11125 printf( " Failure to split a segment.\n" );
11128 if ( steinerleft > 0 ) {
11131 /* Check the two new subsegments to see if they're encroached. */
11132 dummy = checkedge4encroach( encloop );
11133 snextself( *encloop );
11134 dummy = checkedge4encroach( encloop );
11136 badsegmentdealloc( encloop );
11137 encloop = badsegmenttraverse();
11142 #endif /* not CDT_ONLY */
11144 /*****************************************************************************/
11146 /* tallyfaces() Test every triangle in the mesh for quality measures. */
11148 /*****************************************************************************/
11154 struct triedge triangleloop;
11157 printf( " Making a list of bad triangles.\n" );
11159 traversalinit( &triangles );
11160 triangleloop.orient = 0;
11161 triangleloop.tri = triangletraverse();
11162 while ( triangleloop.tri != (triangle *) NULL ) {
11163 /* If the triangle is bad, enqueue it. */
11164 testtriangle( &triangleloop );
11165 triangleloop.tri = triangletraverse();
11169 #endif /* not CDT_ONLY */
11171 /*****************************************************************************/
11173 /* findcircumcenter() Find the circumcenter of a triangle. */
11175 /* The result is returned both in terms of x-y coordinates and xi-eta */
11176 /* coordinates. The xi-eta coordinate system is defined in terms of the */
11177 /* triangle: the origin of the triangle is the origin of the coordinate */
11178 /* system; the destination of the triangle is one unit along the xi axis; */
11179 /* and the apex of the triangle is one unit along the eta axis. */
11181 /* The return value indicates which edge of the triangle is shortest. */
11183 /*****************************************************************************/
11185 enum circumcenterresult findcircumcenter( torg, tdest, tapex, circumcenter,
11190 point circumcenter;
11194 REAL xdo, ydo, xao, yao, xad, yad;
11195 REAL dodist, aodist, addist;
11199 circumcentercount++;
11201 /* Compute the circumcenter of the triangle. */
11202 xdo = tdest[0] - torg[0];
11203 ydo = tdest[1] - torg[1];
11204 xao = tapex[0] - torg[0];
11205 yao = tapex[1] - torg[1];
11206 dodist = xdo * xdo + ydo * ydo;
11207 aodist = xao * xao + yao * yao;
11209 denominator = (REAL)( 0.5 / ( xdo * yao - xao * ydo ));
11212 /* Use the counterclockwise() routine to ensure a positive (and */
11213 /* reasonably accurate) result, avoiding any possibility of */
11214 /* division by zero. */
11215 denominator = (REAL)( 0.5 / counterclockwise( tdest, tapex, torg ));
11216 /* Don't count the above as an orientation test. */
11217 counterclockcount--;
11219 circumcenter[0] = torg[0] - ( ydo * aodist - yao * dodist ) * denominator;
11220 circumcenter[1] = torg[1] + ( xdo * aodist - xao * dodist ) * denominator;
11222 /* To interpolate point attributes for the new point inserted at */
11223 /* the circumcenter, define a coordinate system with a xi-axis, */
11224 /* directed from the triangle's origin to its destination, and */
11225 /* an eta-axis, directed from its origin to its apex. */
11226 /* Calculate the xi and eta coordinates of the circumcenter. */
11227 dx = circumcenter[0] - torg[0];
11228 dy = circumcenter[1] - torg[1];
11229 *xi = (REAL)(( dx * yao - xao * dy ) * ( 2.0 * denominator ));
11230 *eta = (REAL)(( xdo * dy - dx * ydo ) * ( 2.0 * denominator ));
11232 xad = tapex[0] - tdest[0];
11233 yad = tapex[1] - tdest[1];
11234 addist = xad * xad + yad * yad;
11235 if (( addist < dodist ) && ( addist < aodist )) {
11236 return OPPOSITEORG;
11238 else if ( dodist < aodist ) {
11239 return OPPOSITEAPEX;
11242 return OPPOSITEDEST;
11246 /*****************************************************************************/
11248 /* splittriangle() Inserts a point at the circumcenter of a triangle. */
11249 /* Deletes the newly inserted point if it encroaches upon */
11252 /*****************************************************************************/
11257 void splittriangle( badtri )
11258 struct badface *badtri;
11260 point borg, bdest, bapex;
11263 enum insertsiteresult success;
11264 enum circumcenterresult shortedge;
11268 org( badtri->badfacetri, borg );
11269 dest( badtri->badfacetri, bdest );
11270 apex( badtri->badfacetri, bapex );
11271 /* Make sure that this triangle is still the same triangle it was */
11272 /* when it was tested and determined to be of bad quality. */
11273 /* Subsequent transformations may have made it a different triangle. */
11274 if (( borg == badtri->faceorg ) && ( bdest == badtri->facedest ) &&
11275 ( bapex == badtri->faceapex )) {
11276 if ( verbose > 1 ) {
11277 printf( " Splitting this triangle at its circumcenter:\n" );
11278 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0],
11279 borg[1], bdest[0], bdest[1], bapex[0], bapex[1] );
11282 /* Create a new point at the triangle's circumcenter. */
11283 newpoint = (point) poolalloc( &points );
11284 shortedge = findcircumcenter( borg, bdest, bapex, newpoint, &xi, &eta );
11285 /* Check whether the new point lies on a triangle vertex. */
11286 if ((( newpoint[0] == borg[0] ) && ( newpoint[1] == borg[1] ))
11287 || (( newpoint[0] == bdest[0] ) && ( newpoint[1] == bdest[1] ))
11288 || (( newpoint[0] == bapex[0] ) && ( newpoint[1] == bapex[1] ))) {
11290 printf( "Warning: New point (%.12g, %.12g) falls on existing vertex.\n"
11291 , newpoint[0], newpoint[1] );
11294 pointdealloc( newpoint );
11297 for ( i = 2; i < 2 + nextras; i++ ) {
11298 /* Interpolate the point attributes at the circumcenter. */
11299 newpoint[i] = borg[i] + xi * ( bdest[i] - borg[i] )
11300 + eta * ( bapex[i] - borg[i] );
11302 /* The new point must be in the interior, and have a marker of zero. */
11303 setpointmark( newpoint, 0 );
11304 /* Ensure that the handle `badtri->badfacetri' represents the shortest */
11305 /* edge of the triangle. This ensures that the circumcenter must */
11306 /* fall to the left of this edge, so point location will work. */
11307 if ( shortedge == OPPOSITEORG ) {
11308 lnextself( badtri->badfacetri );
11310 else if ( shortedge == OPPOSITEDEST ) {
11311 lprevself( badtri->badfacetri );
11313 /* Insert the circumcenter, searching from the edge of the triangle, */
11314 /* and maintain the Delaunay property of the triangulation. */
11315 success = insertsite( newpoint, &( badtri->badfacetri ),
11316 (struct edge *) NULL, 1, 1 );
11317 if ( success == SUCCESSFULPOINT ) {
11318 if ( steinerleft > 0 ) {
11322 else if ( success == ENCROACHINGPOINT ) {
11323 /* If the newly inserted point encroaches upon a segment, delete it. */
11324 deletesite( &( badtri->badfacetri ));
11326 else if ( success == VIOLATINGPOINT ) {
11327 /* Failed to insert the new point, but some segment was */
11328 /* marked as being encroached. */
11329 pointdealloc( newpoint );
11331 else { /* success == DUPLICATEPOINT */
11332 /* Failed to insert the new point because a vertex is already there. */
11335 "Warning: New point (%.12g, %.12g) falls on existing vertex.\n"
11336 , newpoint[0], newpoint[1] );
11339 pointdealloc( newpoint );
11344 printf( " The new point is at the circumcenter of triangle\n" );
11345 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
11346 borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1] );
11348 printf( "This probably means that I am trying to refine triangles\n" );
11349 printf( " to a smaller size than can be accommodated by the finite\n" );
11350 printf( " precision of floating point arithmetic. (You can be\n" );
11351 printf( " sure of this if I fail to terminate.)\n" );
11355 /* Return the bad triangle to the pool. */
11356 pooldealloc( &badtriangles, (VOID *) badtri );
11359 #endif /* not CDT_ONLY */
11361 /*****************************************************************************/
11363 /* enforcequality() Remove all the encroached edges and bad triangles */
11364 /* from the triangulation. */
11366 /*****************************************************************************/
11371 void enforcequality(){
11375 printf( "Adding Steiner points to enforce quality.\n" );
11377 /* Initialize the pool of encroached segments. */
11378 poolinit( &badsegments, sizeof( struct edge ), BADSEGMENTPERBLOCK, POINTER, 0 );
11380 printf( " Looking for encroached segments.\n" );
11382 /* Test all segments to see if they're encroached. */
11384 if ( verbose && ( badsegments.items > 0 )) {
11385 printf( " Splitting encroached segments.\n" );
11387 /* Note that steinerleft == -1 if an unlimited number */
11388 /* of Steiner points is allowed. */
11389 while (( badsegments.items > 0 ) && ( steinerleft != 0 )) {
11390 /* Fix the segments without noting newly encroached segments or */
11391 /* bad triangles. The reason we don't want to note newly */
11392 /* encroached segments is because some encroached segments are */
11393 /* likely to be noted multiple times, and would then be blindly */
11394 /* split multiple times. I should fix that some time. */
11396 /* Now, find all the segments that became encroached while adding */
11397 /* points to split encroached segments. */
11400 /* At this point, if we haven't run out of Steiner points, the */
11401 /* triangulation should be (conforming) Delaunay. */
11403 /* Next, we worry about enforcing triangle quality. */
11404 if (( minangle > 0.0 ) || vararea || fixedarea ) {
11405 /* Initialize the pool of bad triangles. */
11406 poolinit( &badtriangles, sizeof( struct badface ), BADTRIPERBLOCK, POINTER,
11408 /* Initialize the queues of bad triangles. */
11409 for ( i = 0; i < 64; i++ ) {
11410 queuefront[i] = (struct badface *) NULL;
11411 queuetail[i] = &queuefront[i];
11413 /* Test all triangles to see if they're bad. */
11416 printf( " Splitting bad triangles.\n" );
11418 while (( badtriangles.items > 0 ) && ( steinerleft != 0 )) {
11419 /* Fix one bad triangle by inserting a point at its circumcenter. */
11420 splittriangle( dequeuebadtri());
11421 /* Fix any encroached segments that may have resulted. Record */
11422 /* any new bad triangles or encroached segments that result. */
11423 if ( badsegments.items > 0 ) {
11428 /* At this point, if we haven't run out of Steiner points, the */
11429 /* triangulation should be (conforming) Delaunay and have no */
11430 /* low-quality triangles. */
11432 /* Might we have run out of Steiner points too soon? */
11433 if ( !quiet && ( badsegments.items > 0 ) && ( steinerleft == 0 )) {
11434 printf( "\nWarning: I ran out of Steiner points, but the mesh has\n" );
11435 if ( badsegments.items == 1 ) {
11436 printf( " an encroached segment, and therefore might not be truly\n" );
11439 printf( " %ld encroached segments, and therefore might not be truly\n",
11440 badsegments.items );
11442 printf( " Delaunay. If the Delaunay property is important to you,\n" );
11443 printf( " try increasing the number of Steiner points (controlled by\n" );
11444 printf( " the -S switch) slightly and try again.\n\n" );
11448 #endif /* not CDT_ONLY */
11452 /********* Mesh quality maintenance ends here *********/
11454 /*****************************************************************************/
11456 /* highorder() Create extra nodes for quadratic subparametric elements. */
11458 /*****************************************************************************/
11461 struct triedge triangleloop, trisym;
11462 struct edge checkmark;
11466 triangle ptr; /* Temporary variable used by sym(). */
11467 shelle sptr; /* Temporary variable used by tspivot(). */
11470 printf( "Adding vertices for second-order triangles.\n" );
11472 /* The following line ensures that dead items in the pool of nodes */
11473 /* cannot be allocated for the extra nodes associated with high */
11474 /* order elements. This ensures that the primary nodes (at the */
11475 /* corners of elements) will occur earlier in the output files, and */
11476 /* have lower indices, than the extra nodes. */
11477 points.deaditemstack = (VOID *) NULL;
11479 traversalinit( &triangles );
11480 triangleloop.tri = triangletraverse();
11481 /* To loop over the set of edges, loop over all triangles, and look at */
11482 /* the three edges of each triangle. If there isn't another triangle */
11483 /* adjacent to the edge, operate on the edge. If there is another */
11484 /* adjacent triangle, operate on the edge only if the current triangle */
11485 /* has a smaller pointer than its neighbor. This way, each edge is */
11486 /* considered only once. */
11487 while ( triangleloop.tri != (triangle *) NULL ) {
11488 for ( triangleloop.orient = 0; triangleloop.orient < 3;
11489 triangleloop.orient++ ) {
11490 sym( triangleloop, trisym );
11491 if (( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri )) {
11492 org( triangleloop, torg );
11493 dest( triangleloop, tdest );
11494 /* Create a new node in the middle of the edge. Interpolate */
11495 /* its attributes. */
11496 newpoint = (point) poolalloc( &points );
11497 for ( i = 0; i < 2 + nextras; i++ ) {
11498 newpoint[i] = (REAL)( 0.5 * ( torg[i] + tdest[i] ));
11500 /* Set the new node's marker to zero or one, depending on */
11501 /* whether it lies on a boundary. */
11502 setpointmark( newpoint, trisym.tri == dummytri );
11503 if ( useshelles ) {
11504 tspivot( triangleloop, checkmark );
11505 /* If this edge is a segment, transfer the marker to the new node. */
11506 if ( checkmark.sh != dummysh ) {
11507 setpointmark( newpoint, mark( checkmark ));
11510 if ( verbose > 1 ) {
11511 printf( " Creating (%.12g, %.12g).\n", newpoint[0], newpoint[1] );
11513 /* Record the new node in the (one or two) adjacent elements. */
11514 triangleloop.tri[highorderindex + triangleloop.orient] =
11515 (triangle) newpoint;
11516 if ( trisym.tri != dummytri ) {
11517 trisym.tri[highorderindex + trisym.orient] = (triangle) newpoint;
11521 triangleloop.tri = triangletraverse();
11525 /********* File I/O routines begin here *********/
11529 /*****************************************************************************/
11531 /* readline() Read a nonempty line from a file. */
11533 /* A line is considered "nonempty" if it contains something that looks like */
11536 /*****************************************************************************/
11541 char *readline( string, infile, infilename )
11548 /* Search for something that looks like a number. */
11550 result = fgets( string, INPUTLINESIZE, infile );
11551 if ( result == (char *) NULL ) {
11552 printf( " Error: Unexpected end of file in %s.\n", infilename );
11555 /* Skip anything that doesn't look like a number, a comment, */
11556 /* or the end of a line. */
11557 while (( *result != '\0' ) && ( *result != '#' )
11558 && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' )
11559 && (( *result < '0' ) || ( *result > '9' ))) {
11562 /* If it's a comment or end of line, read another line and try again. */
11563 } while (( *result == '#' ) || ( *result == '\0' ));
11567 #endif /* not TRILIBRARY */
11569 /*****************************************************************************/
11571 /* findfield() Find the next field of a string. */
11573 /* Jumps past the current field by searching for whitespace, then jumps */
11574 /* past the whitespace to find the next field. */
11576 /*****************************************************************************/
11581 char *findfield( string )
11587 /* Skip the current field. Stop upon reaching whitespace. */
11588 while (( *result != '\0' ) && ( *result != '#' )
11589 && ( *result != ' ' ) && ( *result != '\t' )) {
11592 /* Now skip the whitespace and anything else that doesn't look like a */
11593 /* number, a comment, or the end of a line. */
11594 while (( *result != '\0' ) && ( *result != '#' )
11595 && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' )
11596 && (( *result < '0' ) || ( *result > '9' ))) {
11599 /* Check for a comment (prefixed with `#'). */
11600 if ( *result == '#' ) {
11606 #endif /* not TRILIBRARY */
11608 /*****************************************************************************/
11610 /* readnodes() Read the points from a file, which may be a .node or .poly */
11613 /*****************************************************************************/
11618 void readnodes( nodefilename, polyfilename, polyfile )
11619 char *nodefilename;
11620 char *polyfilename;
11625 char inputline[INPUTLINESIZE];
11635 /* Read the points from a .poly file. */
11637 printf( "Opening %s.\n", polyfilename );
11639 *polyfile = fopen( polyfilename, "r" );
11640 if ( *polyfile == (FILE *) NULL ) {
11641 printf( " Error: Cannot access file %s.\n", polyfilename );
11644 /* Read number of points, number of dimensions, number of point */
11645 /* attributes, and number of boundary markers. */
11646 stringptr = readline( inputline, *polyfile, polyfilename );
11647 inpoints = (int) strtol( stringptr, &stringptr, 0 );
11648 stringptr = findfield( stringptr );
11649 if ( *stringptr == '\0' ) {
11653 mesh_dim = (int) strtol( stringptr, &stringptr, 0 );
11655 stringptr = findfield( stringptr );
11656 if ( *stringptr == '\0' ) {
11660 nextras = (int) strtol( stringptr, &stringptr, 0 );
11662 stringptr = findfield( stringptr );
11663 if ( *stringptr == '\0' ) {
11667 nodemarkers = (int) strtol( stringptr, &stringptr, 0 );
11669 if ( inpoints > 0 ) {
11670 infile = *polyfile;
11671 infilename = polyfilename;
11675 /* If the .poly file claims there are zero points, that means that */
11676 /* the points should be read from a separate .node file. */
11678 infilename = innodefilename;
11683 infilename = innodefilename;
11684 *polyfile = (FILE *) NULL;
11687 if ( readnodefile ) {
11688 /* Read the points from a .node file. */
11690 printf( "Opening %s.\n", innodefilename );
11692 infile = fopen( innodefilename, "r" );
11693 if ( infile == (FILE *) NULL ) {
11694 printf( " Error: Cannot access file %s.\n", innodefilename );
11697 /* Read number of points, number of dimensions, number of point */
11698 /* attributes, and number of boundary markers. */
11699 stringptr = readline( inputline, infile, innodefilename );
11700 inpoints = (int) strtol( stringptr, &stringptr, 0 );
11701 stringptr = findfield( stringptr );
11702 if ( *stringptr == '\0' ) {
11706 mesh_dim = (int) strtol( stringptr, &stringptr, 0 );
11708 stringptr = findfield( stringptr );
11709 if ( *stringptr == '\0' ) {
11713 nextras = (int) strtol( stringptr, &stringptr, 0 );
11715 stringptr = findfield( stringptr );
11716 if ( *stringptr == '\0' ) {
11720 nodemarkers = (int) strtol( stringptr, &stringptr, 0 );
11724 if ( inpoints < 3 ) {
11725 printf( "Error: Input must have at least three input points.\n" );
11728 if ( mesh_dim != 2 ) {
11729 printf( "Error: Triangle only works with two-dimensional meshes.\n" );
11733 initializepointpool();
11735 /* Read the points. */
11736 for ( i = 0; i < inpoints; i++ ) {
11737 pointloop = (point) poolalloc( &points );
11738 stringptr = readline( inputline, infile, infilename );
11740 firstnode = (int) strtol( stringptr, &stringptr, 0 );
11741 if (( firstnode == 0 ) || ( firstnode == 1 )) {
11742 firstnumber = firstnode;
11745 stringptr = findfield( stringptr );
11746 if ( *stringptr == '\0' ) {
11747 printf( "Error: Point %d has no x coordinate.\n", firstnumber + i );
11750 x = (REAL) strtod( stringptr, &stringptr );
11751 stringptr = findfield( stringptr );
11752 if ( *stringptr == '\0' ) {
11753 printf( "Error: Point %d has no y coordinate.\n", firstnumber + i );
11756 y = (REAL) strtod( stringptr, &stringptr );
11759 /* Read the point attributes. */
11760 for ( j = 2; j < 2 + nextras; j++ ) {
11761 stringptr = findfield( stringptr );
11762 if ( *stringptr == '\0' ) {
11763 pointloop[j] = 0.0;
11766 pointloop[j] = (REAL) strtod( stringptr, &stringptr );
11769 if ( nodemarkers ) {
11770 /* Read a point marker. */
11771 stringptr = findfield( stringptr );
11772 if ( *stringptr == '\0' ) {
11773 setpointmark( pointloop, 0 );
11776 currentmarker = (int) strtol( stringptr, &stringptr, 0 );
11777 setpointmark( pointloop, currentmarker );
11781 /* If no markers are specified in the file, they default to zero. */
11782 setpointmark( pointloop, 0 );
11784 /* Determine the smallest and largest x and y coordinates. */
11790 xmin = ( x < xmin ) ? x : xmin;
11791 xmax = ( x > xmax ) ? x : xmax;
11792 ymin = ( y < ymin ) ? y : ymin;
11793 ymax = ( y > ymax ) ? y : ymax;
11796 if ( readnodefile ) {
11800 /* Nonexistent x value used as a flag to mark circle events in sweepline */
11801 /* Delaunay algorithm. */
11802 xminextreme = 10 * xmin - 9 * xmax;
11805 #endif /* not TRILIBRARY */
11807 /*****************************************************************************/
11809 /* transfernodes() Read the points from memory. */
11811 /*****************************************************************************/
11816 void transfernodes( pointlist, pointattriblist, pointmarkerlist, numberofpoints,
11817 numberofpointattribs )
11819 REAL *pointattriblist;
11820 int *pointmarkerlist;
11821 int numberofpoints;
11822 int numberofpointattribs;
11830 inpoints = numberofpoints;
11832 nextras = numberofpointattribs;
11834 if ( inpoints < 3 ) {
11835 printf( "Error: Input must have at least three input points.\n" );
11839 initializepointpool();
11841 /* Read the points. */
11844 for ( i = 0; i < inpoints; i++ ) {
11845 pointloop = (point) poolalloc( &points );
11846 /* Read the point coordinates. */
11847 x = pointloop[0] = pointlist[coordindex++];
11848 y = pointloop[1] = pointlist[coordindex++];
11849 /* Read the point attributes. */
11850 for ( j = 0; j < numberofpointattribs; j++ ) {
11851 pointloop[2 + j] = pointattriblist[attribindex++];
11853 if ( pointmarkerlist != (int *) NULL ) {
11854 /* Read a point marker. */
11855 setpointmark( pointloop, pointmarkerlist[i] );
11858 /* If no markers are specified, they default to zero. */
11859 setpointmark( pointloop, 0 );
11863 /* Determine the smallest and largest x and y coordinates. */
11869 xmin = ( x < xmin ) ? x : xmin;
11870 xmax = ( x > xmax ) ? x : xmax;
11871 ymin = ( y < ymin ) ? y : ymin;
11872 ymax = ( y > ymax ) ? y : ymax;
11876 /* Nonexistent x value used as a flag to mark circle events in sweepline */
11877 /* Delaunay algorithm. */
11878 xminextreme = 10 * xmin - 9 * xmax;
11881 #endif /* TRILIBRARY */
11883 /*****************************************************************************/
11885 /* readholes() Read the holes, and possibly regional attributes and area */
11886 /* constraints, from a .poly file. */
11888 /*****************************************************************************/
11893 void readholes( polyfile, polyfilename, hlist, holes, rlist, regions )
11895 char *polyfilename;
11903 char inputline[INPUTLINESIZE];
11908 /* Read the holes. */
11909 stringptr = readline( inputline, polyfile, polyfilename );
11910 *holes = (int) strtol( stringptr, &stringptr, 0 );
11911 if ( *holes > 0 ) {
11912 holelist = (REAL *) malloc( 2 * *holes * sizeof( REAL ));
11914 if ( holelist == (REAL *) NULL ) {
11915 printf( "Error: Out of memory.\n" );
11918 for ( i = 0; i < 2 * *holes; i += 2 ) {
11919 stringptr = readline( inputline, polyfile, polyfilename );
11920 stringptr = findfield( stringptr );
11921 if ( *stringptr == '\0' ) {
11922 printf( "Error: Hole %d has no x coordinate.\n",
11923 firstnumber + ( i >> 1 ));
11927 holelist[i] = (REAL) strtod( stringptr, &stringptr );
11929 stringptr = findfield( stringptr );
11930 if ( *stringptr == '\0' ) {
11931 printf( "Error: Hole %d has no y coordinate.\n",
11932 firstnumber + ( i >> 1 ));
11936 holelist[i + 1] = (REAL) strtod( stringptr, &stringptr );
11941 *hlist = (REAL *) NULL;
11946 if (( regionattrib || vararea ) && !refine ) {
11947 /* Read the area constraints. */
11948 stringptr = readline( inputline, polyfile, polyfilename );
11949 *regions = (int) strtol( stringptr, &stringptr, 0 );
11950 if ( *regions > 0 ) {
11951 regionlist = (REAL *) malloc( 4 * *regions * sizeof( REAL ));
11952 *rlist = regionlist;
11953 if ( regionlist == (REAL *) NULL ) {
11954 printf( "Error: Out of memory.\n" );
11958 for ( i = 0; i < *regions; i++ ) {
11959 stringptr = readline( inputline, polyfile, polyfilename );
11960 stringptr = findfield( stringptr );
11961 if ( *stringptr == '\0' ) {
11962 printf( "Error: Region %d has no x coordinate.\n",
11967 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11969 stringptr = findfield( stringptr );
11970 if ( *stringptr == '\0' ) {
11971 printf( "Error: Region %d has no y coordinate.\n",
11976 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11978 stringptr = findfield( stringptr );
11979 if ( *stringptr == '\0' ) {
11981 "Error: Region %d has no region attribute or area constraint.\n",
11986 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11988 stringptr = findfield( stringptr );
11989 if ( *stringptr == '\0' ) {
11990 regionlist[index] = regionlist[index - 1];
11993 regionlist[index] = (REAL) strtod( stringptr, &stringptr );
12000 /* Set `*regions' to zero to avoid an accidental free() later. */
12002 *rlist = (REAL *) NULL;
12004 #endif /* not CDT_ONLY */
12006 fclose( polyfile );
12009 #endif /* not TRILIBRARY */
12011 /*****************************************************************************/
12013 /* finishfile() Write the command line to the output file so the user */
12014 /* can remember how the file was generated. Close the file. */
12016 /*****************************************************************************/
12021 void finishfile( outfile, argc, argv )
12028 fprintf( outfile, "# Generated by" );
12029 for ( i = 0; i < argc; i++ ) {
12030 fprintf( outfile, " " );
12031 fputs( argv[i], outfile );
12033 fprintf( outfile, "\n" );
12037 #endif /* not TRILIBRARY */
12039 /*****************************************************************************/
12041 /* writenodes() Number the points and write them to a .node file. */
12043 /* To save memory, the point numbers are written over the shell markers */
12044 /* after the points are written to a file. */
12046 /*****************************************************************************/
12051 void writenodes( pointlist, pointattriblist, pointmarkerlist )
12053 REAL **pointattriblist;
12054 int **pointmarkerlist;
12056 #else /* not TRILIBRARY */
12058 void writenodes( nodefilename, argc, argv )
12059 char *nodefilename;
12063 #endif /* not TRILIBRARY */
12073 #else /* not TRILIBRARY */
12075 #endif /* not TRILIBRARY */
12083 printf( "Writing points.\n" );
12085 /* Allocate memory for output points if necessary. */
12086 if ( *pointlist == (REAL *) NULL ) {
12087 *pointlist = (REAL *) malloc( points.items * 2 * sizeof( REAL ));
12088 if ( *pointlist == (REAL *) NULL ) {
12089 printf( "Error: Out of memory.\n" );
12093 /* Allocate memory for output point attributes if necessary. */
12094 if (( nextras > 0 ) && ( *pointattriblist == (REAL *) NULL )) {
12095 *pointattriblist = (REAL *) malloc( points.items * nextras * sizeof( REAL ));
12096 if ( *pointattriblist == (REAL *) NULL ) {
12097 printf( "Error: Out of memory.\n" );
12101 /* Allocate memory for output point markers if necessary. */
12102 if ( !nobound && ( *pointmarkerlist == (int *) NULL )) {
12103 *pointmarkerlist = (int *) malloc( points.items * sizeof( int ));
12104 if ( *pointmarkerlist == (int *) NULL ) {
12105 printf( "Error: Out of memory.\n" );
12109 plist = *pointlist;
12110 palist = *pointattriblist;
12111 pmlist = *pointmarkerlist;
12114 #else /* not TRILIBRARY */
12116 printf( "Writing %s.\n", nodefilename );
12118 outfile = fopen( nodefilename, "w" );
12119 if ( outfile == (FILE *) NULL ) {
12120 printf( " Error: Cannot create file %s.\n", nodefilename );
12123 /* Number of points, number of dimensions, number of point attributes, */
12124 /* and number of boundary markers (zero or one). */
12125 fprintf( outfile, "%ld %d %d %d\n", points.items, mesh_dim, nextras,
12127 #endif /* not TRILIBRARY */
12129 traversalinit( &points );
12130 pointloop = pointtraverse();
12131 pointnumber = firstnumber;
12132 while ( pointloop != (point) NULL ) {
12135 /* X and y coordinates. */
12136 plist[coordindex++] = pointloop[0];
12137 plist[coordindex++] = pointloop[1];
12138 /* Point attributes. */
12139 for ( i = 0; i < nextras; i++ ) {
12140 palist[attribindex++] = pointloop[2 + i];
12143 /* Copy the boundary marker. */
12144 pmlist[pointnumber - firstnumber] = pointmark( pointloop );
12146 #else /* not TRILIBRARY */
12147 /* Point number, x and y coordinates. */
12148 fprintf( outfile, "%4d %.17g %.17g", pointnumber, pointloop[0],
12150 for ( i = 0; i < nextras; i++ ) {
12151 /* Write an attribute. */
12152 fprintf( outfile, " %.17g", pointloop[i + 2] );
12155 fprintf( outfile, "\n" );
12158 /* Write the boundary marker. */
12159 fprintf( outfile, " %d\n", pointmark( pointloop ));
12161 #endif /* not TRILIBRARY */
12163 setpointmark( pointloop, pointnumber );
12164 pointloop = pointtraverse();
12170 finishfile( outfile, argc, argv );
12171 #endif /* not TRILIBRARY */
12174 /*****************************************************************************/
12176 /* numbernodes() Number the points. */
12178 /* Each point is assigned a marker equal to its number. */
12180 /* Used when writenodes() is not called because no .node file is written. */
12182 /*****************************************************************************/
12184 void numbernodes(){
12188 traversalinit( &points );
12189 pointloop = pointtraverse();
12190 pointnumber = firstnumber;
12191 while ( pointloop != (point) NULL ) {
12192 setpointmark( pointloop, pointnumber );
12193 pointloop = pointtraverse();
12198 /*****************************************************************************/
12200 /* writeelements() Write the triangles to an .ele file. */
12202 /*****************************************************************************/
12207 void writeelements( trianglelist, triangleattriblist )
12208 int **trianglelist;
12209 REAL **triangleattriblist;
12211 #else /* not TRILIBRARY */
12213 void writeelements( elefilename, argc, argv )
12218 #endif /* not TRILIBRARY */
12227 #else /* not TRILIBRARY */
12229 #endif /* not TRILIBRARY */
12230 struct triedge triangleloop;
12232 point mid1, mid2, mid3;
12239 printf( "Writing triangles.\n" );
12241 /* Allocate memory for output triangles if necessary. */
12242 if ( *trianglelist == (int *) NULL ) {
12243 *trianglelist = (int *) malloc( triangles.items *
12244 (( order + 1 ) * ( order + 2 ) / 2 ) * sizeof( int ));
12245 if ( *trianglelist == (int *) NULL ) {
12246 printf( "Error: Out of memory.\n" );
12250 /* Allocate memory for output triangle attributes if necessary. */
12251 if (( eextras > 0 ) && ( *triangleattriblist == (REAL *) NULL )) {
12252 *triangleattriblist = (REAL *) malloc( triangles.items * eextras *
12254 if ( *triangleattriblist == (REAL *) NULL ) {
12255 printf( "Error: Out of memory.\n" );
12259 tlist = *trianglelist;
12260 talist = *triangleattriblist;
12263 #else /* not TRILIBRARY */
12265 printf( "Writing %s.\n", elefilename );
12267 outfile = fopen( elefilename, "w" );
12268 if ( outfile == (FILE *) NULL ) {
12269 printf( " Error: Cannot create file %s.\n", elefilename );
12272 /* Number of triangles, points per triangle, attributes per triangle. */
12273 fprintf( outfile, "%ld %d %d\n", triangles.items,
12274 ( order + 1 ) * ( order + 2 ) / 2, eextras );
12275 #endif /* not TRILIBRARY */
12277 traversalinit( &triangles );
12278 triangleloop.tri = triangletraverse();
12279 triangleloop.orient = 0;
12280 elementnumber = firstnumber;
12281 while ( triangleloop.tri != (triangle *) NULL ) {
12282 org( triangleloop, p1 );
12283 dest( triangleloop, p2 );
12284 apex( triangleloop, p3 );
12285 if ( order == 1 ) {
12288 tlist[pointindex++] = pointmark( p1 );
12289 tlist[pointindex++] = pointmark( p2 );
12290 tlist[pointindex++] = pointmark( p3 );
12291 #else /* not TRILIBRARY */
12292 /* Triangle number, indices for three points. */
12293 fprintf( outfile, "%4d %4d %4d %4d", elementnumber,
12294 pointmark( p1 ), pointmark( p2 ), pointmark( p3 ));
12295 #endif /* not TRILIBRARY */
12298 mid1 = (point) triangleloop.tri[highorderindex + 1];
12299 mid2 = (point) triangleloop.tri[highorderindex + 2];
12300 mid3 = (point) triangleloop.tri[highorderindex];
12303 tlist[pointindex++] = pointmark( p1 );
12304 tlist[pointindex++] = pointmark( p2 );
12305 tlist[pointindex++] = pointmark( p3 );
12306 tlist[pointindex++] = pointmark( mid1 );
12307 tlist[pointindex++] = pointmark( mid2 );
12308 tlist[pointindex++] = pointmark( mid3 );
12309 #else /* not TRILIBRARY */
12310 /* Triangle number, indices for six points. */
12311 fprintf( outfile, "%4d %4d %4d %4d %4d %4d %4d", elementnumber,
12312 pointmark( p1 ), pointmark( p2 ), pointmark( p3 ), pointmark( mid1 ),
12313 pointmark( mid2 ), pointmark( mid3 ));
12314 #endif /* not TRILIBRARY */
12319 for ( i = 0; i < eextras; i++ ) {
12320 talist[attribindex++] = elemattribute( triangleloop, i );
12322 #else /* not TRILIBRARY */
12323 for ( i = 0; i < eextras; i++ ) {
12324 fprintf( outfile, " %.17g", elemattribute( triangleloop, i ));
12326 fprintf( outfile, "\n" );
12327 #endif /* not TRILIBRARY */
12329 triangleloop.tri = triangletraverse();
12335 finishfile( outfile, argc, argv );
12336 #endif /* not TRILIBRARY */
12339 /*****************************************************************************/
12341 /* writepoly() Write the segments and holes to a .poly file. */
12343 /*****************************************************************************/
12348 void writepoly( segmentlist, segmentmarkerlist )
12350 int **segmentmarkerlist;
12352 #else /* not TRILIBRARY */
12354 void writepoly( polyfilename, holelist, holes, regionlist, regions, argc, argv )
12355 char *polyfilename;
12363 #endif /* not TRILIBRARY */
12371 #else /* not TRILIBRARY */
12374 #endif /* not TRILIBRARY */
12375 struct edge shelleloop;
12376 point endpoint1, endpoint2;
12382 printf( "Writing segments.\n" );
12384 /* Allocate memory for output segments if necessary. */
12385 if ( *segmentlist == (int *) NULL ) {
12386 *segmentlist = (int *) malloc( shelles.items * 2 * sizeof( int ));
12387 if ( *segmentlist == (int *) NULL ) {
12388 printf( "Error: Out of memory.\n" );
12392 /* Allocate memory for output segment markers if necessary. */
12393 if ( !nobound && ( *segmentmarkerlist == (int *) NULL )) {
12394 *segmentmarkerlist = (int *) malloc( shelles.items * sizeof( int ));
12395 if ( *segmentmarkerlist == (int *) NULL ) {
12396 printf( "Error: Out of memory.\n" );
12400 slist = *segmentlist;
12401 smlist = *segmentmarkerlist;
12403 #else /* not TRILIBRARY */
12405 printf( "Writing %s.\n", polyfilename );
12407 outfile = fopen( polyfilename, "w" );
12408 if ( outfile == (FILE *) NULL ) {
12409 printf( " Error: Cannot create file %s.\n", polyfilename );
12412 /* The zero indicates that the points are in a separate .node file. */
12413 /* Followed by number of dimensions, number of point attributes, */
12414 /* and number of boundary markers (zero or one). */
12415 fprintf( outfile, "%d %d %d %d\n", 0, mesh_dim, nextras, 1 - nobound );
12416 /* Number of segments, number of boundary markers (zero or one). */
12417 fprintf( outfile, "%ld %d\n", shelles.items, 1 - nobound );
12418 #endif /* not TRILIBRARY */
12420 traversalinit( &shelles );
12421 shelleloop.sh = shelletraverse();
12422 shelleloop.shorient = 0;
12423 shellenumber = firstnumber;
12424 while ( shelleloop.sh != (shelle *) NULL ) {
12425 sorg( shelleloop, endpoint1 );
12426 sdest( shelleloop, endpoint2 );
12429 /* Copy indices of the segment's two endpoints. */
12430 slist[index++] = pointmark( endpoint1 );
12431 slist[index++] = pointmark( endpoint2 );
12433 /* Copy the boundary marker. */
12434 smlist[shellenumber - firstnumber] = mark( shelleloop );
12436 #else /* not TRILIBRARY */
12437 /* Segment number, indices of its two endpoints, and possibly a marker. */
12439 fprintf( outfile, "%4d %4d %4d\n", shellenumber,
12440 pointmark( endpoint1 ), pointmark( endpoint2 ));
12443 fprintf( outfile, "%4d %4d %4d %4d\n", shellenumber,
12444 pointmark( endpoint1 ), pointmark( endpoint2 ), mark( shelleloop ));
12446 #endif /* not TRILIBRARY */
12448 shelleloop.sh = shelletraverse();
12456 fprintf( outfile, "%d\n", holes );
12458 for ( i = 0; i < holes; i++ ) {
12459 /* Hole number, x and y coordinates. */
12460 fprintf( outfile, "%4d %.17g %.17g\n", firstnumber + i,
12461 holelist[2 * i], holelist[2 * i + 1] );
12464 if ( regions > 0 ) {
12465 fprintf( outfile, "%d\n", regions );
12466 for ( i = 0; i < regions; i++ ) {
12467 /* Region number, x and y coordinates, attribute, maximum area. */
12468 fprintf( outfile, "%4d %.17g %.17g %.17g %.17g\n", firstnumber + i,
12469 regionlist[4 * i], regionlist[4 * i + 1],
12470 regionlist[4 * i + 2], regionlist[4 * i + 3] );
12473 #endif /* not CDT_ONLY */
12475 finishfile( outfile, argc, argv );
12476 #endif /* not TRILIBRARY */
12479 /*****************************************************************************/
12481 /* writeedges() Write the edges to a .edge file. */
12483 /*****************************************************************************/
12488 void writeedges( edgelist, edgemarkerlist )
12490 int **edgemarkerlist;
12492 #else /* not TRILIBRARY */
12494 void writeedges( edgefilename, argc, argv )
12495 char *edgefilename;
12499 #endif /* not TRILIBRARY */
12507 #else /* not TRILIBRARY */
12509 #endif /* not TRILIBRARY */
12510 struct triedge triangleloop, trisym;
12511 struct edge checkmark;
12514 triangle ptr; /* Temporary variable used by sym(). */
12515 shelle sptr; /* Temporary variable used by tspivot(). */
12520 printf( "Writing edges.\n" );
12522 /* Allocate memory for edges if necessary. */
12523 if ( *edgelist == (int *) NULL ) {
12524 *edgelist = (int *) malloc( edges * 2 * sizeof( int ));
12525 if ( *edgelist == (int *) NULL ) {
12526 printf( "Error: Out of memory.\n" );
12530 /* Allocate memory for edge markers if necessary. */
12531 if ( !nobound && ( *edgemarkerlist == (int *) NULL )) {
12532 *edgemarkerlist = (int *) malloc( edges * sizeof( int ));
12533 if ( *edgemarkerlist == (int *) NULL ) {
12534 printf( "Error: Out of memory.\n" );
12539 emlist = *edgemarkerlist;
12541 #else /* not TRILIBRARY */
12543 printf( "Writing %s.\n", edgefilename );
12545 outfile = fopen( edgefilename, "w" );
12546 if ( outfile == (FILE *) NULL ) {
12547 printf( " Error: Cannot create file %s.\n", edgefilename );
12550 /* Number of edges, number of boundary markers (zero or one). */
12551 fprintf( outfile, "%ld %d\n", edges, 1 - nobound );
12552 #endif /* not TRILIBRARY */
12554 traversalinit( &triangles );
12555 triangleloop.tri = triangletraverse();
12556 edgenumber = firstnumber;
12557 /* To loop over the set of edges, loop over all triangles, and look at */
12558 /* the three edges of each triangle. If there isn't another triangle */
12559 /* adjacent to the edge, operate on the edge. If there is another */
12560 /* adjacent triangle, operate on the edge only if the current triangle */
12561 /* has a smaller pointer than its neighbor. This way, each edge is */
12562 /* considered only once. */
12563 while ( triangleloop.tri != (triangle *) NULL ) {
12564 for ( triangleloop.orient = 0; triangleloop.orient < 3;
12565 triangleloop.orient++ ) {
12566 sym( triangleloop, trisym );
12567 if (( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri )) {
12568 org( triangleloop, p1 );
12569 dest( triangleloop, p2 );
12572 elist[index++] = pointmark( p1 );
12573 elist[index++] = pointmark( p2 );
12574 #endif /* TRILIBRARY */
12578 /* Edge number, indices of two endpoints. */
12579 fprintf( outfile, "%4d %d %d\n", edgenumber,
12580 pointmark( p1 ), pointmark( p2 ));
12581 #endif /* not TRILIBRARY */
12584 /* Edge number, indices of two endpoints, and a boundary marker. */
12585 /* If there's no shell edge, the boundary marker is zero. */
12586 if ( useshelles ) {
12587 tspivot( triangleloop, checkmark );
12588 if ( checkmark.sh == dummysh ) {
12591 emlist[edgenumber - firstnumber] = 0;
12592 #else /* not TRILIBRARY */
12593 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12594 pointmark( p1 ), pointmark( p2 ), 0 );
12595 #endif /* not TRILIBRARY */
12600 emlist[edgenumber - firstnumber] = mark( checkmark );
12601 #else /* not TRILIBRARY */
12602 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12603 pointmark( p1 ), pointmark( p2 ), mark( checkmark ));
12604 #endif /* not TRILIBRARY */
12610 emlist[edgenumber - firstnumber] = trisym.tri == dummytri;
12611 #else /* not TRILIBRARY */
12612 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12613 pointmark( p1 ), pointmark( p2 ), trisym.tri == dummytri );
12614 #endif /* not TRILIBRARY */
12620 triangleloop.tri = triangletraverse();
12625 finishfile( outfile, argc, argv );
12626 #endif /* not TRILIBRARY */
12629 /*****************************************************************************/
12631 /* writevoronoi() Write the Voronoi diagram to a .v.node and .v.edge */
12634 /* The Voronoi diagram is the geometric dual of the Delaunay triangulation. */
12635 /* Hence, the Voronoi vertices are listed by traversing the Delaunay */
12636 /* triangles, and the Voronoi edges are listed by traversing the Delaunay */
12639 /* WARNING: In order to assign numbers to the Voronoi vertices, this */
12640 /* procedure messes up the shell edges or the extra nodes of every */
12641 /* element. Hence, you should call this procedure last. */
12643 /*****************************************************************************/
12648 void writevoronoi( vpointlist, vpointattriblist, vpointmarkerlist, vedgelist,
12649 vedgemarkerlist, vnormlist )
12650 REAL * *vpointlist;
12651 REAL **vpointattriblist;
12652 int **vpointmarkerlist;
12654 int **vedgemarkerlist;
12657 #else /* not TRILIBRARY */
12659 void writevoronoi( vnodefilename, vedgefilename, argc, argv )
12660 char *vnodefilename;
12661 char *vedgefilename;
12665 #endif /* not TRILIBRARY */
12676 #else /* not TRILIBRARY */
12678 #endif /* not TRILIBRARY */
12679 struct triedge triangleloop, trisym;
12680 point torg, tdest, tapex;
12681 REAL circumcenter[2];
12683 int vnodenumber, vedgenumber;
12686 triangle ptr; /* Temporary variable used by sym(). */
12691 printf( "Writing Voronoi vertices.\n" );
12693 /* Allocate memory for Voronoi vertices if necessary. */
12694 if ( *vpointlist == (REAL *) NULL ) {
12695 *vpointlist = (REAL *) malloc( triangles.items * 2 * sizeof( REAL ));
12696 if ( *vpointlist == (REAL *) NULL ) {
12697 printf( "Error: Out of memory.\n" );
12701 /* Allocate memory for Voronoi vertex attributes if necessary. */
12702 if ( *vpointattriblist == (REAL *) NULL ) {
12703 *vpointattriblist = (REAL *) malloc( triangles.items * nextras *
12705 if ( *vpointattriblist == (REAL *) NULL ) {
12706 printf( "Error: Out of memory.\n" );
12710 *vpointmarkerlist = (int *) NULL;
12711 plist = *vpointlist;
12712 palist = *vpointattriblist;
12715 #else /* not TRILIBRARY */
12717 printf( "Writing %s.\n", vnodefilename );
12719 outfile = fopen( vnodefilename, "w" );
12720 if ( outfile == (FILE *) NULL ) {
12721 printf( " Error: Cannot create file %s.\n", vnodefilename );
12724 /* Number of triangles, two dimensions, number of point attributes, */
12725 /* zero markers. */
12726 fprintf( outfile, "%ld %d %d %d\n", triangles.items, 2, nextras, 0 );
12727 #endif /* not TRILIBRARY */
12729 traversalinit( &triangles );
12730 triangleloop.tri = triangletraverse();
12731 triangleloop.orient = 0;
12732 vnodenumber = firstnumber;
12733 while ( triangleloop.tri != (triangle *) NULL ) {
12734 org( triangleloop, torg );
12735 dest( triangleloop, tdest );
12736 apex( triangleloop, tapex );
12737 findcircumcenter( torg, tdest, tapex, circumcenter, &xi, &eta );
12740 /* X and y coordinates. */
12741 plist[coordindex++] = circumcenter[0];
12742 plist[coordindex++] = circumcenter[1];
12743 for ( i = 2; i < 2 + nextras; i++ ) {
12744 /* Interpolate the point attributes at the circumcenter. */
12745 palist[attribindex++] = torg[i] + xi * ( tdest[i] - torg[i] )
12746 + eta * ( tapex[i] - torg[i] );
12748 #else /* not TRILIBRARY */
12749 /* Voronoi vertex number, x and y coordinates. */
12750 fprintf( outfile, "%4d %.17g %.17g", vnodenumber, circumcenter[0],
12752 for ( i = 2; i < 2 + nextras; i++ ) {
12753 /* Interpolate the point attributes at the circumcenter. */
12754 fprintf( outfile, " %.17g", torg[i] + xi * ( tdest[i] - torg[i] )
12755 + eta * ( tapex[i] - torg[i] ));
12757 fprintf( outfile, "\n" );
12758 #endif /* not TRILIBRARY */
12760 *(int *) ( triangleloop.tri + 6 ) = vnodenumber;
12761 triangleloop.tri = triangletraverse();
12767 finishfile( outfile, argc, argv );
12768 #endif /* not TRILIBRARY */
12773 printf( "Writing Voronoi edges.\n" );
12775 /* Allocate memory for output Voronoi edges if necessary. */
12776 if ( *vedgelist == (int *) NULL ) {
12777 *vedgelist = (int *) malloc( edges * 2 * sizeof( int ));
12778 if ( *vedgelist == (int *) NULL ) {
12779 printf( "Error: Out of memory.\n" );
12783 *vedgemarkerlist = (int *) NULL;
12784 /* Allocate memory for output Voronoi norms if necessary. */
12785 if ( *vnormlist == (REAL *) NULL ) {
12786 *vnormlist = (REAL *) malloc( edges * 2 * sizeof( REAL ));
12787 if ( *vnormlist == (REAL *) NULL ) {
12788 printf( "Error: Out of memory.\n" );
12792 elist = *vedgelist;
12793 normlist = *vnormlist;
12795 #else /* not TRILIBRARY */
12797 printf( "Writing %s.\n", vedgefilename );
12799 outfile = fopen( vedgefilename, "w" );
12800 if ( outfile == (FILE *) NULL ) {
12801 printf( " Error: Cannot create file %s.\n", vedgefilename );
12804 /* Number of edges, zero boundary markers. */
12805 fprintf( outfile, "%ld %d\n", edges, 0 );
12806 #endif /* not TRILIBRARY */
12808 traversalinit( &triangles );
12809 triangleloop.tri = triangletraverse();
12810 vedgenumber = firstnumber;
12811 /* To loop over the set of edges, loop over all triangles, and look at */
12812 /* the three edges of each triangle. If there isn't another triangle */
12813 /* adjacent to the edge, operate on the edge. If there is another */
12814 /* adjacent triangle, operate on the edge only if the current triangle */
12815 /* has a smaller pointer than its neighbor. This way, each edge is */
12816 /* considered only once. */
12817 while ( triangleloop.tri != (triangle *) NULL ) {
12818 for ( triangleloop.orient = 0; triangleloop.orient < 3;
12819 triangleloop.orient++ ) {
12820 sym( triangleloop, trisym );
12821 if (( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri )) {
12822 /* Find the number of this triangle (and Voronoi vertex). */
12823 p1 = *(int *) ( triangleloop.tri + 6 );
12824 if ( trisym.tri == dummytri ) {
12825 org( triangleloop, torg );
12826 dest( triangleloop, tdest );
12829 /* Copy an infinite ray. Index of one endpoint, and -1. */
12830 elist[coordindex] = p1;
12831 normlist[coordindex++] = tdest[1] - torg[1];
12832 elist[coordindex] = -1;
12833 normlist[coordindex++] = torg[0] - tdest[0];
12834 #else /* not TRILIBRARY */
12835 /* Write an infinite ray. Edge number, index of one endpoint, -1, */
12836 /* and x and y coordinates of a vector representing the */
12837 /* direction of the ray. */
12838 fprintf( outfile, "%4d %d %d %.17g %.17g\n", vedgenumber,
12839 p1, -1, tdest[1] - torg[1], torg[0] - tdest[0] );
12840 #endif /* not TRILIBRARY */
12843 /* Find the number of the adjacent triangle (and Voronoi vertex). */
12844 p2 = *(int *) ( trisym.tri + 6 );
12845 /* Finite edge. Write indices of two endpoints. */
12848 elist[coordindex] = p1;
12849 normlist[coordindex++] = 0.0;
12850 elist[coordindex] = p2;
12851 normlist[coordindex++] = 0.0;
12852 #else /* not TRILIBRARY */
12853 fprintf( outfile, "%4d %d %d\n", vedgenumber, p1, p2 );
12854 #endif /* not TRILIBRARY */
12859 triangleloop.tri = triangletraverse();
12864 finishfile( outfile, argc, argv );
12865 #endif /* not TRILIBRARY */
12871 void writeneighbors( neighborlist )
12872 int **neighborlist;
12874 #else /* not TRILIBRARY */
12876 void writeneighbors( neighborfilename, argc, argv )
12877 char *neighborfilename;
12881 #endif /* not TRILIBRARY */
12888 #else /* not TRILIBRARY */
12890 #endif /* not TRILIBRARY */
12891 struct triedge triangleloop, trisym;
12893 int neighbor1, neighbor2, neighbor3;
12894 triangle ptr; /* Temporary variable used by sym(). */
12899 printf( "Writing neighbors.\n" );
12901 /* Allocate memory for neighbors if necessary. */
12902 if ( *neighborlist == (int *) NULL ) {
12903 *neighborlist = (int *) malloc( triangles.items * 3 * sizeof( int ));
12904 if ( *neighborlist == (int *) NULL ) {
12905 printf( "Error: Out of memory.\n" );
12909 nlist = *neighborlist;
12911 #else /* not TRILIBRARY */
12913 printf( "Writing %s.\n", neighborfilename );
12915 outfile = fopen( neighborfilename, "w" );
12916 if ( outfile == (FILE *) NULL ) {
12917 printf( " Error: Cannot create file %s.\n", neighborfilename );
12920 /* Number of triangles, three edges per triangle. */
12921 fprintf( outfile, "%ld %d\n", triangles.items, 3 );
12922 #endif /* not TRILIBRARY */
12924 traversalinit( &triangles );
12925 triangleloop.tri = triangletraverse();
12926 triangleloop.orient = 0;
12927 elementnumber = firstnumber;
12928 while ( triangleloop.tri != (triangle *) NULL ) {
12929 *(int *) ( triangleloop.tri + 6 ) = elementnumber;
12930 triangleloop.tri = triangletraverse();
12933 *(int *) ( dummytri + 6 ) = -1;
12935 traversalinit( &triangles );
12936 triangleloop.tri = triangletraverse();
12937 elementnumber = firstnumber;
12938 while ( triangleloop.tri != (triangle *) NULL ) {
12939 triangleloop.orient = 1;
12940 sym( triangleloop, trisym );
12941 neighbor1 = *(int *) ( trisym.tri + 6 );
12942 triangleloop.orient = 2;
12943 sym( triangleloop, trisym );
12944 neighbor2 = *(int *) ( trisym.tri + 6 );
12945 triangleloop.orient = 0;
12946 sym( triangleloop, trisym );
12947 neighbor3 = *(int *) ( trisym.tri + 6 );
12950 nlist[index++] = neighbor1;
12951 nlist[index++] = neighbor2;
12952 nlist[index++] = neighbor3;
12953 #else /* not TRILIBRARY */
12954 /* Triangle number, neighboring triangle numbers. */
12955 fprintf( outfile, "%4d %d %d %d\n", elementnumber,
12956 neighbor1, neighbor2, neighbor3 );
12957 #endif /* not TRILIBRARY */
12959 triangleloop.tri = triangletraverse();
12965 finishfile( outfile, argc, argv );
12966 #endif /* TRILIBRARY */
12969 /*****************************************************************************/
12971 /* writeoff() Write the triangulation to an .off file. */
12973 /* OFF stands for the Object File Format, a format used by the Geometry */
12974 /* Center's Geomview package. */
12976 /*****************************************************************************/
12981 void writeoff( offfilename, argc, argv )
12987 struct triedge triangleloop;
12992 printf( "Writing %s.\n", offfilename );
12994 outfile = fopen( offfilename, "w" );
12995 if ( outfile == (FILE *) NULL ) {
12996 printf( " Error: Cannot create file %s.\n", offfilename );
12999 /* Number of points, triangles, and edges. */
13000 fprintf( outfile, "OFF\n%ld %ld %ld\n", points.items, triangles.items,
13003 /* Write the points. */
13004 traversalinit( &points );
13005 pointloop = pointtraverse();
13006 while ( pointloop != (point) NULL ) {
13007 /* The "0.0" is here because the OFF format uses 3D coordinates. */
13008 fprintf( outfile, " %.17g %.17g %.17g\n", pointloop[0],
13009 pointloop[1], 0.0 );
13010 pointloop = pointtraverse();
13013 /* Write the triangles. */
13014 traversalinit( &triangles );
13015 triangleloop.tri = triangletraverse();
13016 triangleloop.orient = 0;
13017 while ( triangleloop.tri != (triangle *) NULL ) {
13018 org( triangleloop, p1 );
13019 dest( triangleloop, p2 );
13020 apex( triangleloop, p3 );
13021 /* The "3" means a three-vertex polygon. */
13022 fprintf( outfile, " 3 %4d %4d %4d\n", pointmark( p1 ) - 1,
13023 pointmark( p2 ) - 1, pointmark( p3 ) - 1 );
13024 triangleloop.tri = triangletraverse();
13026 finishfile( outfile, argc, argv );
13029 #endif /* not TRILIBRARY */
13033 /********* File I/O routines end here *********/
13035 /*****************************************************************************/
13037 /* quality_statistics() Print statistics about the quality of the mesh. */
13039 /*****************************************************************************/
13041 void quality_statistics(){
13042 struct triedge triangleloop;
13044 REAL cossquaretable[8];
13045 REAL ratiotable[16];
13047 REAL edgelength[3];
13051 REAL shortest, longest;
13053 REAL smallestarea, biggestarea;
13054 REAL triminaltitude2;
13058 REAL smallestangle, biggestangle;
13059 REAL radconst, degconst;
13060 int angletable[18];
13061 int aspecttable[16];
13067 printf( "Mesh quality statistics:\n\n" );
13068 radconst = (REAL)( PI / 18.0 );
13069 degconst = (REAL)( 180.0 / PI );
13070 for ( i = 0; i < 8; i++ ) {
13071 cossquaretable[i] = (REAL)( cos( radconst * (REAL) ( i + 1 )));
13072 cossquaretable[i] = cossquaretable[i] * cossquaretable[i];
13074 for ( i = 0; i < 18; i++ ) {
13078 ratiotable[0] = 1.5; ratiotable[1] = 2.0;
13079 ratiotable[2] = 2.5; ratiotable[3] = 3.0;
13080 ratiotable[4] = 4.0; ratiotable[5] = 6.0;
13081 ratiotable[6] = 10.0; ratiotable[7] = 15.0;
13082 ratiotable[8] = 25.0; ratiotable[9] = 50.0;
13083 ratiotable[10] = 100.0; ratiotable[11] = 300.0;
13084 ratiotable[12] = 1000.0; ratiotable[13] = 10000.0;
13085 ratiotable[14] = 100000.0; ratiotable[15] = 0.0;
13086 for ( i = 0; i < 16; i++ ) {
13087 aspecttable[i] = 0;
13091 minaltitude = xmax - xmin + ymax - ymin;
13092 minaltitude = minaltitude * minaltitude;
13093 shortest = minaltitude;
13095 smallestarea = minaltitude;
13098 smallestangle = 0.0;
13099 biggestangle = 2.0;
13102 traversalinit( &triangles );
13103 triangleloop.tri = triangletraverse();
13104 triangleloop.orient = 0;
13105 while ( triangleloop.tri != (triangle *) NULL ) {
13106 org( triangleloop, p[0] );
13107 dest( triangleloop, p[1] );
13108 apex( triangleloop, p[2] );
13111 for ( i = 0; i < 3; i++ ) {
13114 dx[i] = p[j][0] - p[k][0];
13115 dy[i] = p[j][1] - p[k][1];
13116 edgelength[i] = dx[i] * dx[i] + dy[i] * dy[i];
13117 if ( edgelength[i] > trilongest2 ) {
13118 trilongest2 = edgelength[i];
13120 if ( edgelength[i] > longest ) {
13121 longest = edgelength[i];
13123 if ( edgelength[i] < shortest ) {
13124 shortest = edgelength[i];
13128 triarea = counterclockwise( p[0], p[1], p[2] );
13129 if ( triarea < smallestarea ) {
13130 smallestarea = triarea;
13132 if ( triarea > biggestarea ) {
13133 biggestarea = triarea;
13135 triminaltitude2 = triarea * triarea / trilongest2;
13136 if ( triminaltitude2 < minaltitude ) {
13137 minaltitude = triminaltitude2;
13139 triaspect2 = trilongest2 / triminaltitude2;
13140 if ( triaspect2 > worstaspect ) {
13141 worstaspect = triaspect2;
13144 while (( triaspect2 > ratiotable[aspectindex] * ratiotable[aspectindex] )
13145 && ( aspectindex < 15 )) {
13148 aspecttable[aspectindex]++;
13150 for ( i = 0; i < 3; i++ ) {
13153 dotproduct = dx[j] * dx[k] + dy[j] * dy[k];
13154 cossquare = dotproduct * dotproduct / ( edgelength[j] * edgelength[k] );
13156 for ( ii = 7; ii >= 0; ii-- ) {
13157 if ( cossquare > cossquaretable[ii] ) {
13161 if ( dotproduct <= 0.0 ) {
13162 angletable[tendegree]++;
13163 if ( cossquare > smallestangle ) {
13164 smallestangle = cossquare;
13166 if ( acutebiggest && ( cossquare < biggestangle )) {
13167 biggestangle = cossquare;
13171 angletable[17 - tendegree]++;
13172 if ( acutebiggest || ( cossquare > biggestangle )) {
13173 biggestangle = cossquare;
13178 triangleloop.tri = triangletraverse();
13181 shortest = (REAL)sqrt( shortest );
13182 longest = (REAL)sqrt( longest );
13183 minaltitude = (REAL)sqrt( minaltitude );
13184 worstaspect = (REAL)sqrt( worstaspect );
13185 smallestarea *= 2.0;
13186 biggestarea *= 2.0;
13187 if ( smallestangle >= 1.0 ) {
13188 smallestangle = 0.0;
13191 smallestangle = (REAL)( degconst * acos( sqrt( smallestangle )));
13193 if ( biggestangle >= 1.0 ) {
13194 biggestangle = 180.0;
13197 if ( acutebiggest ) {
13198 biggestangle = (REAL)( degconst * acos( sqrt( biggestangle )));
13201 biggestangle = (REAL)( 180.0 - degconst * acos( sqrt( biggestangle )));
13205 printf( " Smallest area: %16.5g | Largest area: %16.5g\n",
13206 smallestarea, biggestarea );
13207 printf( " Shortest edge: %16.5g | Longest edge: %16.5g\n",
13208 shortest, longest );
13209 printf( " Shortest altitude: %12.5g | Largest aspect ratio: %8.5g\n\n",
13210 minaltitude, worstaspect );
13211 printf( " Aspect ratio histogram:\n" );
13212 printf( " 1.1547 - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n",
13213 ratiotable[0], aspecttable[0], ratiotable[7], ratiotable[8],
13215 for ( i = 1; i < 7; i++ ) {
13216 printf( " %6.6g - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n",
13217 ratiotable[i - 1], ratiotable[i], aspecttable[i],
13218 ratiotable[i + 7], ratiotable[i + 8], aspecttable[i + 8] );
13220 printf( " %6.6g - %-6.6g : %8d | %6.6g - : %8d\n",
13221 ratiotable[6], ratiotable[7], aspecttable[7], ratiotable[14],
13224 " (Triangle aspect ratio is longest edge divided by shortest altitude)\n\n" );
13225 printf( " Smallest angle: %15.5g | Largest angle: %15.5g\n\n",
13226 smallestangle, biggestangle );
13227 printf( " Angle histogram:\n" );
13228 for ( i = 0; i < 9; i++ ) {
13229 printf( " %3d - %3d degrees: %8d | %3d - %3d degrees: %8d\n",
13230 i * 10, i * 10 + 10, angletable[i],
13231 i * 10 + 90, i * 10 + 100, angletable[i + 9] );
13236 /*****************************************************************************/
13238 /* statistics() Print all sorts of cool facts. */
13240 /*****************************************************************************/
13243 printf( "\nStatistics:\n\n" );
13244 printf( " Input points: %d\n", inpoints );
13246 printf( " Input triangles: %d\n", inelements );
13249 printf( " Input segments: %d\n", insegments );
13251 printf( " Input holes: %d\n", holes );
13255 printf( "\n Mesh points: %ld\n", points.items );
13256 printf( " Mesh triangles: %ld\n", triangles.items );
13257 printf( " Mesh edges: %ld\n", edges );
13258 if ( poly || refine ) {
13259 printf( " Mesh boundary edges: %ld\n", hullsize );
13260 printf( " Mesh segments: %ld\n\n", shelles.items );
13263 printf( " Mesh convex hull edges: %ld\n\n", hullsize );
13266 quality_statistics();
13267 printf( "Memory allocation statistics:\n\n" );
13268 printf( " Maximum number of points: %ld\n", points.maxitems );
13269 printf( " Maximum number of triangles: %ld\n", triangles.maxitems );
13270 if ( shelles.maxitems > 0 ) {
13271 printf( " Maximum number of segments: %ld\n", shelles.maxitems );
13273 if ( viri.maxitems > 0 ) {
13274 printf( " Maximum number of viri: %ld\n", viri.maxitems );
13276 if ( badsegments.maxitems > 0 ) {
13277 printf( " Maximum number of encroached segments: %ld\n",
13278 badsegments.maxitems );
13280 if ( badtriangles.maxitems > 0 ) {
13281 printf( " Maximum number of bad triangles: %ld\n",
13282 badtriangles.maxitems );
13284 if ( splaynodes.maxitems > 0 ) {
13285 printf( " Maximum number of splay tree nodes: %ld\n",
13286 splaynodes.maxitems );
13288 printf( " Approximate heap memory use (bytes): %ld\n\n",
13289 points.maxitems * points.itembytes
13290 + triangles.maxitems * triangles.itembytes
13291 + shelles.maxitems * shelles.itembytes
13292 + viri.maxitems * viri.itembytes
13293 + badsegments.maxitems * badsegments.itembytes
13294 + badtriangles.maxitems * badtriangles.itembytes
13295 + splaynodes.maxitems * splaynodes.itembytes );
13297 printf( "Algorithmic statistics:\n\n" );
13298 printf( " Number of incircle tests: %ld\n", incirclecount );
13299 printf( " Number of orientation tests: %ld\n", counterclockcount );
13300 if ( hyperbolacount > 0 ) {
13301 printf( " Number of right-of-hyperbola tests: %ld\n",
13304 if ( circumcentercount > 0 ) {
13305 printf( " Number of circumcenter computations: %ld\n",
13306 circumcentercount );
13308 if ( circletopcount > 0 ) {
13309 printf( " Number of circle top computations: %ld\n",
13316 /*****************************************************************************/
13318 /* main() or triangulate() Gosh, do everything. */
13320 /* The sequence is roughly as follows. Many of these steps can be skipped, */
13321 /* depending on the command line switches. */
13323 /* - Initialize constants and parse the command line. */
13324 /* - Read the points from a file and either */
13325 /* - triangulate them (no -r), or */
13326 /* - read an old mesh from files and reconstruct it (-r). */
13327 /* - Insert the PSLG segments (-p), and possibly segments on the convex */
13329 /* - Read the holes (-p), regional attributes (-pA), and regional area */
13330 /* constraints (-pa). Carve the holes and concavities, and spread the */
13331 /* regional attributes and area constraints. */
13332 /* - Enforce the constraints on minimum angle (-q) and maximum area (-a). */
13333 /* Also enforce the conforming Delaunay property (-q and -a). */
13334 /* - Compute the number of edges in the resulting mesh. */
13335 /* - Promote the mesh's linear triangles to higher order elements (-o). */
13336 /* - Write the output files and print the statistics. */
13337 /* - Check the consistency and Delaunay property of the mesh (-C). */
13339 /*****************************************************************************/
13344 void triangulate( triswitches, in, out, vorout )
13346 struct triangulateio *in;
13347 struct triangulateio *out;
13348 struct triangulateio *vorout;
13350 #else /* not TRILIBRARY */
13352 int main( argc, argv )
13356 #endif /* not TRILIBRARY */
13359 REAL *holearray; /* Array of holes. */
13360 REAL *regionarray; /* Array of regional attributes and area constraints. */
13364 #endif /* not TRILIBRARY */
13367 /* Variables for timing the performance of Triangle. The types are */
13368 /* defined in sys/time.h. */
13369 struct timeval tv0, tv1, tv2, tv3, tv4, tv5, tv6;
13370 struct timezone tz;
13371 #endif /* NO_TIMER */
13375 gettimeofday( &tv0, &tz );
13376 #endif /* NO_TIMER */
13381 parsecommandline( 1, &triswitches );
13382 #else /* not TRILIBRARY */
13383 parsecommandline( argc, argv );
13384 #endif /* not TRILIBRARY */
13388 transfernodes( in->pointlist, in->pointattributelist, in->pointmarkerlist,
13389 in->numberofpoints, in->numberofpointattributes );
13390 #else /* not TRILIBRARY */
13391 readnodes( innodefilename, inpolyfilename, &polyfile );
13392 #endif /* not TRILIBRARY */
13397 gettimeofday( &tv1, &tz );
13399 #endif /* NO_TIMER */
13403 hullsize = delaunay(); /* Triangulate the points. */
13404 #else /* not CDT_ONLY */
13406 /* Read and reconstruct a mesh. */
13409 hullsize = reconstruct( in->trianglelist, in->triangleattributelist,
13410 in->trianglearealist, in->numberoftriangles,
13411 in->numberofcorners, in->numberoftriangleattributes,
13412 in->segmentlist, in->segmentmarkerlist,
13413 in->numberofsegments );
13414 #else /* not TRILIBRARY */
13415 hullsize = reconstruct( inelefilename, areafilename, inpolyfilename,
13417 #endif /* not TRILIBRARY */
13420 hullsize = delaunay(); /* Triangulate the points. */
13422 #endif /* not CDT_ONLY */
13427 gettimeofday( &tv2, &tz );
13429 printf( "Mesh reconstruction" );
13432 printf( "Delaunay" );
13434 printf( " milliseconds: %ld\n", 1000l * ( tv2.tv_sec - tv1.tv_sec )
13435 + ( tv2.tv_usec - tv1.tv_usec ) / 1000l );
13437 #endif /* NO_TIMER */
13439 /* Ensure that no point can be mistaken for a triangular bounding */
13440 /* box point in insertsite(). */
13441 infpoint1 = (point) NULL;
13442 infpoint2 = (point) NULL;
13443 infpoint3 = (point) NULL;
13445 if ( useshelles ) {
13446 checksegments = 1; /* Segments will be introduced next. */
13448 /* Insert PSLG segments and/or convex hull segments. */
13451 insegments = formskeleton( in->segmentlist, in->segmentmarkerlist,
13452 in->numberofsegments );
13453 #else /* not TRILIBRARY */
13454 insegments = formskeleton( polyfile, inpolyfilename );
13455 #endif /* not TRILIBRARY */
13462 gettimeofday( &tv3, &tz );
13463 if ( useshelles && !refine ) {
13464 printf( "Segment milliseconds: %ld\n",
13465 1000l * ( tv3.tv_sec - tv2.tv_sec )
13466 + ( tv3.tv_usec - tv2.tv_usec ) / 1000l );
13469 #endif /* NO_TIMER */
13474 holearray = in->holelist;
13475 holes = in->numberofholes;
13476 regionarray = in->regionlist;
13477 regions = in->numberofregions;
13478 #else /* not TRILIBRARY */
13479 readholes( polyfile, inpolyfilename, &holearray, &holes,
13480 ®ionarray, ®ions );
13481 #endif /* not TRILIBRARY */
13483 /* Carve out holes and concavities. */
13484 carveholes( holearray, holes, regionarray, regions );
13488 /* Without a PSLG, there can be no holes or regional attributes */
13489 /* or area constraints. The following are set to zero to avoid */
13490 /* an accidental free() later. */
13498 gettimeofday( &tv4, &tz );
13499 if ( poly && !refine ) {
13500 printf( "Hole milliseconds: %ld\n", 1000l * ( tv4.tv_sec - tv3.tv_sec )
13501 + ( tv4.tv_usec - tv3.tv_usec ) / 1000l );
13504 #endif /* NO_TIMER */
13509 enforcequality(); /* Enforce angle and area constraints. */
13511 #endif /* not CDT_ONLY */
13516 gettimeofday( &tv5, &tz );
13520 printf( "Quality milliseconds: %ld\n",
13521 1000l * ( tv5.tv_sec - tv4.tv_sec )
13522 + ( tv5.tv_usec - tv4.tv_usec ) / 1000l );
13524 #endif /* not CDT_ONLY */
13526 #endif /* NO_TIMER */
13528 /* Compute the number of edges. */
13529 edges = ( 3l * triangles.items + hullsize ) / 2l;
13532 highorder(); /* Promote elements to higher polynomial order. */
13540 out->numberofpoints = points.items;
13541 out->numberofpointattributes = nextras;
13542 out->numberoftriangles = triangles.items;
13543 out->numberofcorners = ( order + 1 ) * ( order + 2 ) / 2;
13544 out->numberoftriangleattributes = eextras;
13545 out->numberofedges = edges;
13546 if ( useshelles ) {
13547 out->numberofsegments = shelles.items;
13550 out->numberofsegments = hullsize;
13552 if ( vorout != (struct triangulateio *) NULL ) {
13553 vorout->numberofpoints = triangles.items;
13554 vorout->numberofpointattributes = nextras;
13555 vorout->numberofedges = edges;
13557 #endif /* TRILIBRARY */
13558 /* If not using iteration numbers, don't write a .node file if one was */
13559 /* read, because the original one would be overwritten! */
13560 if ( nonodewritten || ( noiterationnum && readnodefile )) {
13564 printf( "NOT writing points.\n" );
13565 #else /* not TRILIBRARY */
13566 printf( "NOT writing a .node file.\n" );
13567 #endif /* not TRILIBRARY */
13569 numbernodes(); /* We must remember to number the points. */
13574 writenodes( &out->pointlist, &out->pointattributelist,
13575 &out->pointmarkerlist );
13576 #else /* not TRILIBRARY */
13577 writenodes( outnodefilename, argc, argv ); /* Numbers the points too. */
13578 #endif /* TRILIBRARY */
13580 if ( noelewritten ) {
13584 printf( "NOT writing triangles.\n" );
13585 #else /* not TRILIBRARY */
13586 printf( "NOT writing an .ele file.\n" );
13587 #endif /* not TRILIBRARY */
13593 writeelements( &out->trianglelist, &out->triangleattributelist );
13594 #else /* not TRILIBRARY */
13595 writeelements( outelefilename, argc, argv );
13596 #endif /* not TRILIBRARY */
13598 /* The -c switch (convex switch) causes a PSLG to be written */
13599 /* even if none was read. */
13600 if ( poly || convex ) {
13601 /* If not using iteration numbers, don't overwrite the .poly file. */
13602 if ( nopolywritten || noiterationnum ) {
13606 printf( "NOT writing segments.\n" );
13607 #else /* not TRILIBRARY */
13608 printf( "NOT writing a .poly file.\n" );
13609 #endif /* not TRILIBRARY */
13615 writepoly( &out->segmentlist, &out->segmentmarkerlist );
13616 out->numberofholes = holes;
13617 out->numberofregions = regions;
13619 out->holelist = in->holelist;
13620 out->regionlist = in->regionlist;
13623 out->holelist = (REAL *) NULL;
13624 out->regionlist = (REAL *) NULL;
13626 #else /* not TRILIBRARY */
13627 writepoly( outpolyfilename, holearray, holes, regionarray, regions,
13629 #endif /* not TRILIBRARY */
13636 if ( regions > 0 ) {
13637 free( regionarray );
13639 #endif /* not CDT_ONLY */
13644 writeoff( offfilename, argc, argv );
13646 #endif /* not TRILIBRARY */
13650 writeedges( &out->edgelist, &out->edgemarkerlist );
13651 #else /* not TRILIBRARY */
13652 writeedges( edgefilename, argc, argv );
13653 #endif /* not TRILIBRARY */
13658 writevoronoi( &vorout->pointlist, &vorout->pointattributelist,
13659 &vorout->pointmarkerlist, &vorout->edgelist,
13660 &vorout->edgemarkerlist, &vorout->normlist );
13661 #else /* not TRILIBRARY */
13662 writevoronoi( vnodefilename, vedgefilename, argc, argv );
13663 #endif /* not TRILIBRARY */
13668 writeneighbors( &out->neighborlist );
13669 #else /* not TRILIBRARY */
13670 writeneighbors( neighborfilename, argc, argv );
13671 #endif /* not TRILIBRARY */
13677 gettimeofday( &tv6, &tz );
13678 printf( "\nOutput milliseconds: %ld\n",
13679 1000l * ( tv6.tv_sec - tv5.tv_sec )
13680 + ( tv6.tv_usec - tv5.tv_usec ) / 1000l );
13681 printf( "Total running milliseconds: %ld\n",
13682 1000l * ( tv6.tv_sec - tv0.tv_sec )
13683 + ( tv6.tv_usec - tv0.tv_usec ) / 1000l );
13684 #endif /* NO_TIMER */
13695 #endif /* not REDUCED */
13701 #endif /* not TRILIBRARY */
projects / xonotic / netradiant.git / blob