3 this code written by Forest Hale, on 2004-10-17, and placed into public domain
4 this implements Quadratic BSpline surfaces as seen in Quake3 by id Software
6 a small rant on misuse of the name 'bezier': many people seem to think that
7 bezier is a generic term for splines, but it is not, it is a term for a
8 specific type of bspline (4 control points, cubic bspline), bsplines are the
9 generalization of the bezier spline to support dimensions other than cubic.
11 example equations for 1-5 control point bsplines being sampled as t=0...1
12 1: flat (0th dimension)
14 2: linear (1st dimension)
15 o = a * (1 - t) + b * t
16 3: quadratic bspline (2nd dimension)
17 o = a * (1 - t) * (1 - t) + 2 * b * (1 - t) * t + c * t * t
18 4: cubic (bezier) bspline (3rd dimension)
19 o = a * (1 - t) * (1 - t) * (1 - t) + 3 * b * (1 - t) * (1 - t) * t + 3 * c * (1 - t) * t * t + d * t * t * t
20 5: quartic bspline (4th dimension)
21 o = a * (1 - t) * (1 - t) * (1 - t) * (1 - t) + 4 * b * (1 - t) * (1 - t) * (1 - t) * t + 6 * c * (1 - t) * (1 - t) * t * t + 4 * d * (1 - t) * t * t * t + e * t * t * t * t
23 arbitrary dimension bspline
24 double factorial(int n)
33 double bsplinesample(int dimensions, double t, double *param)
36 for (i = 0;i < dimensions + 1;i++)
37 o += param[i] * factorial(dimensions)/(factorial(i)*factorial(dimensions-i)) * pow(t, i) * pow(1 - t, dimensions - i);
48 // Calculate number of resulting vertex rows/columns by given patch size and tesselation factor
49 // tess=0 means that we reduce detalization of base 3x3 patches by removing middle row and column of vertices
50 // "DimForTess" is "DIMension FOR TESSelation factor"
51 // NB: tess=0 actually means that tess must be 0.5, but obviously it can't because it is of int type. (so "a*tess"-like code is replaced by "a/2" if tess=0)
52 int Q3PatchDimForTess(int size, int tess)
55 return (size - 1) * tess + 1;
57 return (size - 1) / 2 + 1;
59 return 0; // Maybe warn about wrong tess here?
63 // to expand a 5x5 patch to 21x21 vertices (4x4 tesselation), one might use this call:
64 // Q3PatchSubdivideFloat(3, sizeof(float[3]), outvertices, 5, 5, sizeof(float[3]), patchvertices, 4, 4);
65 void Q3PatchTesselateFloat(int numcomponents, int outputstride, float *outputvertices, int patchwidth, int patchheight, int inputstride, float *patchvertices, int tesselationwidth, int tesselationheight)
67 int k, l, x, y, component, outputwidth = Q3PatchDimForTess(patchwidth, tesselationwidth);
68 float px, py, *v, a, b, c, *cp[3][3], temp[3][64];
69 int xmax = max(1, 2*tesselationwidth);
70 int ymax = max(1, 2*tesselationheight);
72 // iterate over the individual 3x3 quadratic spline surfaces one at a time
73 // expanding them to fill the output array (with some overlap to ensure
74 // the edges are filled)
75 for (k = 0;k < patchheight-1;k += 2)
77 for (l = 0;l < patchwidth-1;l += 2)
79 // set up control point pointers for quicker lookup later
82 cp[y][x] = (float *)((unsigned char *)patchvertices + ((k+y)*patchwidth+(l+x)) * inputstride);
84 for (y = 0;y <= ymax;y++)
86 // calculate control points for this row by collapsing the 3
87 // rows of control points to one row using py
88 py = (float)y / (float)ymax;
89 // calculate quadratic spline weights for py
90 a = ((1.0f - py) * (1.0f - py));
91 b = ((1.0f - py) * (2.0f * py));
93 for (component = 0;component < numcomponents;component++)
95 temp[0][component] = cp[0][0][component] * a + cp[1][0][component] * b + cp[2][0][component] * c;
96 temp[1][component] = cp[0][1][component] * a + cp[1][1][component] * b + cp[2][1][component] * c;
97 temp[2][component] = cp[0][2][component] * a + cp[1][2][component] * b + cp[2][2][component] * c;
99 // fetch a pointer to the beginning of the output vertex row
100 v = (float *)((unsigned char *)outputvertices + ((k * ymax / 2 + y) * outputwidth + l * xmax / 2) * outputstride);
101 // for each column of the row...
102 for (x = 0;x <= xmax;x++)
104 // calculate point based on the row control points
105 px = (float)x / (float)xmax;
106 // calculate quadratic spline weights for px
107 // (could be precalculated)
108 a = ((1.0f - px) * (1.0f - px));
109 b = ((1.0f - px) * (2.0f * px));
111 for (component = 0;component < numcomponents;component++)
112 v[component] = temp[0][component] * a + temp[1][component] * b + temp[2][component] * c;
113 // advance to next output vertex using outputstride
114 // (the next vertex may not be directly following this
115 // one, as this may be part of a larger structure)
116 v = (float *)((unsigned char *)v + outputstride);
122 // enable this if you want results printed out
123 printf("vertices[%i][%i] =\n{\n", (patchheight-1)*tesselationheight+1, (patchwidth-1)*tesselationwidth+1);
124 for (y = 0;y < (patchheight-1)*tesselationheight+1;y++)
126 for (x = 0;x < (patchwidth-1)*tesselationwidth+1;x++)
129 for (component = 0;component < numcomponents;component++)
130 printf("%f ", outputvertices[(y*((patchwidth-1)*tesselationwidth+1)+x)*numcomponents+component]);
139 static int Q3PatchTesselation(float largestsquared3xcurvearea, float tolerance)
142 // f is actually a squared 2x curve area... so the formula had to be adjusted to give roughly the same subdivisions
143 f = pow(largestsquared3xcurvearea / 64.0f, 0.25f) / tolerance;
144 //if(f < 0.25) // VERY flat patches
145 if(f < 0.0001) // TOTALLY flat patches
150 return (int) floor(log(f) / log(2.0f)) + 1;
151 // this is always at least 2
152 // maps [0.25..0.5[ to -1 (actually, 1 is returned)
153 // maps [0.5..1[ to 0 (actually, 1 is returned)
159 float Squared3xCurveArea(const float *a, const float *control, const float *b, int components)
162 // mimicing the old behaviour with the new code...
165 float quartercurvearea = 0;
167 for (c = 0;c < components;c++)
169 deviation = control[c] * 0.5f - a[c] * 0.25f - b[c] * 0.25f;
170 quartercurvearea += deviation*deviation;
173 // But as the new code now works on the squared 2x curve area, let's scale the value
174 return quartercurvearea * quartercurvearea * 64.0;
177 // ideally, we'd like the area between the spline a->control->b and the line a->b.
178 // but as this is hard to calculate, let's calculate an upper bound of it:
179 // the area of the triangle a->control->b->a.
181 // one can prove that the area of a quadratic spline = 2/3 * the area of
182 // the triangle of its control points!
183 // to do it, first prove it for the spline through (0,0), (1,1), (2,0)
184 // (which is a parabola) and then note that moving the control point
185 // left/right is just shearing and keeps the area of both the spline and
186 // the triangle invariant.
188 // why are we going for the spline area anyway?
191 // the area between the spline and the line a->b is a measure of the
192 // error of approximation of the spline by the line.
194 // also, on circle-like or parabola-like curves, you easily get that the
195 // double amount of line approximation segments reduces the error to its quarter
196 // (also, easy to prove for splines by doing it for one specific one, and using
197 // affine transforms to get all other splines)
201 // let's calculate the area! but we have to avoid the cross product, as
202 // components is not necessarily 3
204 // the area of a triangle spanned by vectors a and b is
206 // 0.5 * |a| |b| sin gamma
214 // 0.5 * sqrt(|a|^2 |b|^2 - (a.b)^2)
216 float aa = 0, bb = 0, ab = 0;
217 for (c = 0;c < components;c++)
219 float xa = a[c] - control[c];
220 float xb = b[c] - control[c];
225 // area is 0.5 * sqrt(aa*bb - ab*ab)
226 // 2x TRIANGLE area is sqrt(aa*bb - ab*ab)
227 // 3x CURVE area is sqrt(aa*bb - ab*ab)
228 return aa * bb - ab * ab;
232 // returns how much tesselation of each segment is needed to remain under tolerance
233 int Q3PatchTesselationOnX(int patchwidth, int patchheight, int components, const float *in, float tolerance)
237 float squared3xcurvearea, largestsquared3xcurvearea;
238 largestsquared3xcurvearea = 0;
239 for (y = 0;y < patchheight;y++)
241 for (x = 0;x < patchwidth-1;x += 2)
243 patch = in + ((y * patchwidth) + x) * components;
244 squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[components], &patch[2*components], components);
245 if (largestsquared3xcurvearea < squared3xcurvearea)
246 largestsquared3xcurvearea = squared3xcurvearea;
249 return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);
252 // returns how much tesselation of each segment is needed to remain under tolerance
253 int Q3PatchTesselationOnY(int patchwidth, int patchheight, int components, const float *in, float tolerance)
257 float squared3xcurvearea, largestsquared3xcurvearea;
258 largestsquared3xcurvearea = 0;
259 for (y = 0;y < patchheight-1;y += 2)
261 for (x = 0;x < patchwidth;x++)
263 patch = in + ((y * patchwidth) + x) * components;
264 squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[patchwidth*components], &patch[2*patchwidth*components], components);
265 if (largestsquared3xcurvearea < squared3xcurvearea)
266 largestsquared3xcurvearea = squared3xcurvearea;
269 return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);
272 // Find an equal vertex in array. Check only vertices with odd X and Y
273 static int FindEqualOddVertexInArray(int numcomponents, float *vertex, float *vertices, int width, int height)
276 for (y=0; y<height; y+=2)
278 for (x=0; x<width; x+=2)
280 qboolean found = true;
281 for (j=0; j<numcomponents; j++)
282 if (fabs(*(vertex+j) - *(vertices+j)) > 0.05)
283 // div0: this is notably smaller than the smallest radiant grid
284 // but large enough so we don't need to get scared of roundoff
292 vertices += numcomponents*2;
294 vertices += numcomponents*(width-1);
299 #define SIDE_INVALID -1
303 static int GetSide(int p1, int p2, int width, int height, int *pointdist)
305 int x1 = p1 % width, y1 = p1 / width;
306 int x2 = p2 % width, y2 = p2 / width;
307 if (p1 < 0 || p2 < 0)
313 *pointdist = abs(y2 - y1);
321 *pointdist = abs(x2 - x1);
328 // Increase tesselation of one of two touching patches to make a seamless connection between them
329 // Returns 0 in case if patches were not modified, otherwise 1
330 int Q3PatchAdjustTesselation(int numcomponents, patchinfo_t *patch1, float *patchvertices1, patchinfo_t *patch2, float *patchvertices2)
332 // what we are doing here is:
333 // we take for each corner of one patch
334 // and check if the other patch contains that corner
335 // once we have a pair of such matches
337 struct {int id1,id2;} commonverts[8];
338 int i, j, k, side1, side2, *tess1, *tess2;
340 qboolean modified = false;
342 // Potential paired vertices (corners of the first patch)
343 commonverts[0].id1 = 0;
344 commonverts[1].id1 = patch1->xsize-1;
345 commonverts[2].id1 = patch1->xsize*(patch1->ysize-1);
346 commonverts[3].id1 = patch1->xsize*patch1->ysize-1;
348 commonverts[i].id2 = FindEqualOddVertexInArray(numcomponents, patchvertices1+numcomponents*commonverts[i].id1, patchvertices2, patch2->xsize, patch2->ysize);
350 // Corners of the second patch
351 commonverts[4].id2 = 0;
352 commonverts[5].id2 = patch2->xsize-1;
353 commonverts[6].id2 = patch2->xsize*(patch2->ysize-1);
354 commonverts[7].id2 = patch2->xsize*patch2->ysize-1;
356 commonverts[i].id1 = FindEqualOddVertexInArray(numcomponents, patchvertices2+numcomponents*commonverts[i].id2, patchvertices1, patch1->xsize, patch1->ysize);
361 side1 = GetSide(commonverts[i].id1,commonverts[j].id1,patch1->xsize,patch1->ysize,&dist1);
362 side2 = GetSide(commonverts[i].id2,commonverts[j].id2,patch2->xsize,patch2->ysize,&dist2);
364 if (side1 == SIDE_INVALID || side2 == SIDE_INVALID)
369 // no patch welding if the resolutions mismatch
373 // Update every lod level
374 for (k=0;k<PATCH_LODS_NUM;++k)
376 tess1 = side1 == SIDE_X ? &patch1->lods[k].xtess : &patch1->lods[k].ytess;
377 tess2 = side2 == SIDE_X ? &patch2->lods[k].xtess : &patch2->lods[k].ytess;
378 if (*tess1 != *tess2)
396 // calculates elements for a grid of vertices
397 // (such as those produced by Q3PatchTesselate)
398 // (note: width and height are the actual vertex size, this produces
399 // (width-1)*(height-1)*2 triangles, 3 elements each)
400 void Q3PatchTriangleElements(int *elements, int width, int height, int firstvertex)
402 int x, y, row0, row1;
403 for (y = 0;y < height - 1;y++)
407 // swap the triangle order in odd rows as optimization for collision stride
408 row0 = firstvertex + (y + 0) * width + width - 2;
409 row1 = firstvertex + (y + 1) * width + width - 2;
410 for (x = 0;x < width - 1;x++)
413 *elements++ = row1 + 1;
414 *elements++ = row0 + 1;
417 *elements++ = row0 + 1;
424 row0 = firstvertex + (y + 0) * width;
425 row1 = firstvertex + (y + 1) * width;
426 for (x = 0;x < width - 1;x++)
430 *elements++ = row0 + 1;
432 *elements++ = row1 + 1;
433 *elements++ = row0 + 1;