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this code written by Forest Hale, on 2004-10-17, and placed into public domain
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this implements Quadratic BSpline surfaces as seen in Quake3 by id Software
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a small rant on misuse of the name 'bezier': many people seem to think that
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bezier is a generic term for splines, but it is not, it is a term for a
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specific type of bspline (4 control points, cubic bspline), bsplines are the
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generalization of the bezier spline to support dimensions other than cubic.
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example equations for 1-5 control point bsplines being sampled as t=0...1
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1: flat (0th dimension)
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2: linear (1st dimension)
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o = a * (1 - t) + b * t
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3: quadratic bspline (2nd dimension)
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o = a * (1 - t) * (1 - t) + 2 * b * (1 - t) * t + c * t * t
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4: cubic (bezier) bspline (3rd dimension)
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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
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5: quartic bspline (4th dimension)
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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
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arbitrary dimension bspline
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double factorial(int n)
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double bsplinesample(int dimensions, double t, double *param)
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for (i = 0;i < dimensions + 1;i++)
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o += param[i] * factorial(dimensions)/(factorial(i)*factorial(dimensions-i)) * pow(t, i) * pow(1 - t, dimensions - i);
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// Calculate number of resulting vertex rows/columns by given patch size and tesselation factor
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// tess=0 means that we reduce detalization of base 3x3 patches by removing middle row and column of vertices
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// "DimForTess" is "DIMension FOR TESSelation factor"
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// 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)
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int Q3PatchDimForTess(int size, int tess)
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return (size - 1) * tess + 1;
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return (size - 1) / 2 + 1;
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return 0; // Maybe warn about wrong tess here?
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// to expand a 5x5 patch to 21x21 vertices (4x4 tesselation), one might use this call:
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// Q3PatchSubdivideFloat(3, sizeof(float[3]), outvertices, 5, 5, sizeof(float[3]), patchvertices, 4, 4);
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void Q3PatchTesselateFloat(int numcomponents, int outputstride, float *outputvertices, int patchwidth, int patchheight, int inputstride, float *patchvertices, int tesselationwidth, int tesselationheight)
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int k, l, x, y, component, outputwidth = Q3PatchDimForTess(patchwidth, tesselationwidth);
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float px, py, *v, a, b, c, *cp[3][3], temp[3][64];
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int xmax = max(1, 2*tesselationwidth);
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int ymax = max(1, 2*tesselationheight);
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// iterate over the individual 3x3 quadratic spline surfaces one at a time
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// expanding them to fill the output array (with some overlap to ensure
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// the edges are filled)
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for (k = 0;k < patchheight-1;k += 2)
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for (l = 0;l < patchwidth-1;l += 2)
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// set up control point pointers for quicker lookup later
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cp[y][x] = (float *)((unsigned char *)patchvertices + ((k+y)*patchwidth+(l+x)) * inputstride);
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for (y = 0;y <= ymax;y++)
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// calculate control points for this row by collapsing the 3
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// rows of control points to one row using py
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py = (float)y / (float)ymax;
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// calculate quadratic spline weights for py
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a = ((1.0f - py) * (1.0f - py));
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b = ((1.0f - py) * (2.0f * py));
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for (component = 0;component < numcomponents;component++)
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temp[0][component] = cp[0][0][component] * a + cp[1][0][component] * b + cp[2][0][component] * c;
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temp[1][component] = cp[0][1][component] * a + cp[1][1][component] * b + cp[2][1][component] * c;
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temp[2][component] = cp[0][2][component] * a + cp[1][2][component] * b + cp[2][2][component] * c;
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// fetch a pointer to the beginning of the output vertex row
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v = (float *)((unsigned char *)outputvertices + ((k * ymax / 2 + y) * outputwidth + l * xmax / 2) * outputstride);
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// for each column of the row...
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for (x = 0;x <= xmax;x++)
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// calculate point based on the row control points
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px = (float)x / (float)xmax;
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// calculate quadratic spline weights for px
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// (could be precalculated)
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a = ((1.0f - px) * (1.0f - px));
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b = ((1.0f - px) * (2.0f * px));
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for (component = 0;component < numcomponents;component++)
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v[component] = temp[0][component] * a + temp[1][component] * b + temp[2][component] * c;
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// advance to next output vertex using outputstride
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// (the next vertex may not be directly following this
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// one, as this may be part of a larger structure)
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v = (float *)((unsigned char *)v + outputstride);
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// enable this if you want results printed out
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printf("vertices[%i][%i] =\n{\n", (patchheight-1)*tesselationheight+1, (patchwidth-1)*tesselationwidth+1);
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for (y = 0;y < (patchheight-1)*tesselationheight+1;y++)
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for (x = 0;x < (patchwidth-1)*tesselationwidth+1;x++)
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for (component = 0;component < numcomponents;component++)
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printf("%f ", outputvertices[(y*((patchwidth-1)*tesselationwidth+1)+x)*numcomponents+component]);
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static int Q3PatchTesselation(float largestsquared3xcurvearea, float tolerance)
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// f is actually a squared 2x curve area... so the formula had to be adjusted to give roughly the same subdivisions
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f = pow(largestsquared3xcurvearea / 64.0f, 0.25f) / tolerance;
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//if(f < 0.25) // VERY flat patches
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if(f < 0.0001) // TOTALLY flat patches
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return (int) floor(log(f) / log(2.0f)) + 1;
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// this is always at least 2
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// maps [0.25..0.5[ to -1 (actually, 1 is returned)
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// maps [0.5..1[ to 0 (actually, 1 is returned)
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float Squared3xCurveArea(const float *a, const float *control, const float *b, int components)
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// mimicing the old behaviour with the new code...
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float quartercurvearea = 0;
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for (c = 0;c < components;c++)
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deviation = control[c] * 0.5f - a[c] * 0.25f - b[c] * 0.25f;
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quartercurvearea += deviation*deviation;
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// But as the new code now works on the squared 2x curve area, let's scale the value
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return quartercurvearea * quartercurvearea * 64.0;
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// ideally, we'd like the area between the spline a->control->b and the line a->b.
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// but as this is hard to calculate, let's calculate an upper bound of it:
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// the area of the triangle a->control->b->a.
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// one can prove that the area of a quadratic spline = 2/3 * the area of
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// the triangle of its control points!
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// to do it, first prove it for the spline through (0,0), (1,1), (2,0)
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// (which is a parabola) and then note that moving the control point
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// left/right is just shearing and keeps the area of both the spline and
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// the triangle invariant.
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// why are we going for the spline area anyway?
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// the area between the spline and the line a->b is a measure of the
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// error of approximation of the spline by the line.
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// also, on circle-like or parabola-like curves, you easily get that the
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// double amount of line approximation segments reduces the error to its quarter
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// (also, easy to prove for splines by doing it for one specific one, and using
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// affine transforms to get all other splines)
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// let's calculate the area! but we have to avoid the cross product, as
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// components is not necessarily 3
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// the area of a triangle spanned by vectors a and b is
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// 0.5 * |a| |b| sin gamma
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// 0.5 * sqrt(|a|^2 |b|^2 - (a.b)^2)
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float aa = 0, bb = 0, ab = 0;
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for (c = 0;c < components;c++)
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float xa = a[c] - control[c];
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float xb = b[c] - control[c];
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// area is 0.5 * sqrt(aa*bb - ab*ab)
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// 2x TRIANGLE area is sqrt(aa*bb - ab*ab)
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// 3x CURVE area is sqrt(aa*bb - ab*ab)
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return aa * bb - ab * ab;
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// returns how much tesselation of each segment is needed to remain under tolerance
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int Q3PatchTesselationOnX(int patchwidth, int patchheight, int components, const float *in, float tolerance)
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float squared3xcurvearea, largestsquared3xcurvearea;
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largestsquared3xcurvearea = 0;
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for (y = 0;y < patchheight;y++)
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for (x = 0;x < patchwidth-1;x += 2)
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patch = in + ((y * patchwidth) + x) * components;
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squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[components], &patch[2*components], components);
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if (largestsquared3xcurvearea < squared3xcurvearea)
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largestsquared3xcurvearea = squared3xcurvearea;
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return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);
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// returns how much tesselation of each segment is needed to remain under tolerance
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int Q3PatchTesselationOnY(int patchwidth, int patchheight, int components, const float *in, float tolerance)
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float squared3xcurvearea, largestsquared3xcurvearea;
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largestsquared3xcurvearea = 0;
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for (y = 0;y < patchheight-1;y += 2)
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for (x = 0;x < patchwidth;x++)
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patch = in + ((y * patchwidth) + x) * components;
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squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[patchwidth*components], &patch[2*patchwidth*components], components);
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if (largestsquared3xcurvearea < squared3xcurvearea)
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largestsquared3xcurvearea = squared3xcurvearea;
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return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);
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// Find an equal vertex in array. Check only vertices with odd X and Y
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static int FindEqualOddVertexInArray(int numcomponents, float *vertex, float *vertices, int width, int height)
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for (y=0; y<height; y+=2)
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for (x=0; x<width; x+=2)
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qboolean found = true;
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for (j=0; j<numcomponents; j++)
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if (fabs(*(vertex+j) - *(vertices+j)) > 0.05)
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// div0: this is notably smaller than the smallest radiant grid
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// but large enough so we don't need to get scared of roundoff
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vertices += numcomponents*2;
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vertices += numcomponents*(width-1);
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#define SIDE_INVALID -1
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static int GetSide(int p1, int p2, int width, int height, int *pointdist)
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int x1 = p1 % width, y1 = p1 / width;
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int x2 = p2 % width, y2 = p2 / width;
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if (p1 < 0 || p2 < 0)
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*pointdist = abs(y2 - y1);
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*pointdist = abs(x2 - x1);
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// Increase tesselation of one of two touching patches to make a seamless connection between them
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// Returns 0 in case if patches were not modified, otherwise 1
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int Q3PatchAdjustTesselation(int numcomponents, patchinfo_t *patch1, float *patchvertices1, patchinfo_t *patch2, float *patchvertices2)
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// what we are doing here is:
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// we take for each corner of one patch
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// and check if the other patch contains that corner
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// once we have a pair of such matches
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struct {int id1,id2;} commonverts[8];
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int i, j, k, side1, side2, *tess1, *tess2;
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qboolean modified = false;
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// Potential paired vertices (corners of the first patch)
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commonverts[0].id1 = 0;
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commonverts[1].id1 = patch1->xsize-1;
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commonverts[2].id1 = patch1->xsize*(patch1->ysize-1);
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commonverts[3].id1 = patch1->xsize*patch1->ysize-1;
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commonverts[i].id2 = FindEqualOddVertexInArray(numcomponents, patchvertices1+numcomponents*commonverts[i].id1, patchvertices2, patch2->xsize, patch2->ysize);
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// Corners of the second patch
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commonverts[4].id2 = 0;
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commonverts[5].id2 = patch2->xsize-1;
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commonverts[6].id2 = patch2->xsize*(patch2->ysize-1);
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commonverts[7].id2 = patch2->xsize*patch2->ysize-1;
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commonverts[i].id1 = FindEqualOddVertexInArray(numcomponents, patchvertices2+numcomponents*commonverts[i].id2, patchvertices1, patch1->xsize, patch1->ysize);
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side1 = GetSide(commonverts[i].id1,commonverts[j].id1,patch1->xsize,patch1->ysize,&dist1);
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side2 = GetSide(commonverts[i].id2,commonverts[j].id2,patch2->xsize,patch2->ysize,&dist2);
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if (side1 == SIDE_INVALID || side2 == SIDE_INVALID)
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// no patch welding if the resolutions mismatch
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// Update every lod level
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for (k=0;k<PATCH_LODS_NUM;++k)
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tess1 = side1 == SIDE_X ? &patch1->lods[k].xtess : &patch1->lods[k].ytess;
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tess2 = side2 == SIDE_X ? &patch2->lods[k].xtess : &patch2->lods[k].ytess;
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if (*tess1 != *tess2)
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// calculates elements for a grid of vertices
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// (such as those produced by Q3PatchTesselate)
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// (note: width and height are the actual vertex size, this produces
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// (width-1)*(height-1)*2 triangles, 3 elements each)
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void Q3PatchTriangleElements(int *elements, int width, int height, int firstvertex)
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int x, y, row0, row1;
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for (y = 0;y < height - 1;y++)
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// swap the triangle order in odd rows as optimization for collision stride
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row0 = firstvertex + (y + 0) * width + width - 2;
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row1 = firstvertex + (y + 1) * width + width - 2;
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for (x = 0;x < width - 1;x++)
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*elements++ = row1 + 1;
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*elements++ = row0 + 1;
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*elements++ = row0 + 1;
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row0 = firstvertex + (y + 0) * width;
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row1 = firstvertex + (y + 1) * width;
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for (x = 0;x < width - 1;x++)
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*elements++ = row0 + 1;
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*elements++ = row1 + 1;
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*elements++ = row0 + 1;
added
removed
curves.c
