curves.c

   1
   2 /*
   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
   5
   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.
  10
  11 example equations for 1-5 control point bsplines being sampled as t=0...1
  12 1: flat (0th dimension)
  13 o = a
  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
  22
  23 arbitrary dimension bspline
  24 double factorial(int n)
  25 {
  26         int i;
  27         double f;
  28         f = 1;
  29         for (i = 1;i < n;i++)
  30                 f = f * i;
  31         return f;
  32 }
  33 double bsplinesample(int dimensions, double t, double *param)
  34 {
  35         double o = 0;
  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);
  38 }
  39 */
  40
  41 #include <math.h>
  42 #include "curves.h"
  43
  44 // usage:
  45 // to expand a 5x5 patch to 21x21 vertices (4x4 tesselation), one might use this call:
  46 // Q3PatchSubdivideFloat(3, sizeof(float[3]), outvertices, 5, 5, sizeof(float[3]), patchvertices, 4, 4);
  47 void Q3PatchTesselateFloat(int numcomponents, int outputstride, float *outputvertices, int patchwidth, int patchheight, int inputstride, float *patchvertices, int tesselationwidth, int tesselationheight)
  48 {
  49         int k, l, x, y, component, outputwidth = (patchwidth-1)*tesselationwidth+1;
  50         float px, py, *v, a, b, c, *cp[3][3], temp[3][64];
  51         // iterate over the individual 3x3 quadratic spline surfaces one at a time
  52         // expanding them to fill the output array (with some overlap to ensure
  53         // the edges are filled)
  54         for (k = 0;k < patchheight-1;k += 2)
  55         {
  56                 for (l = 0;l < patchwidth-1;l += 2)
  57                 {
  58                         // set up control point pointers for quicker lookup later
  59                         for (y = 0;y < 3;y++)
  60                                 for (x = 0;x < 3;x++)
  61                                         cp[y][x] = (float *)((unsigned char *)patchvertices + ((k+y)*patchwidth+(l+x)) * inputstride);
  62                         // for each row...
  63                         for (y = 0;y <= tesselationheight*2;y++)
  64                         {
  65                                 // calculate control points for this row by collapsing the 3
  66                                 // rows of control points to one row using py
  67                                 py = (float)y / (float)(tesselationheight*2);
  68                                 // calculate quadratic spline weights for py
  69                                 a = ((1.0f - py) * (1.0f - py));
  70                                 b = ((1.0f - py) * (2.0f * py));
  71                                 c = ((       py) * (       py));
  72                                 for (component = 0;component < numcomponents;component++)
  73                                 {
  74                                         temp[0][component] = cp[0][0][component] * a + cp[1][0][component] * b + cp[2][0][component] * c;
  75                                         temp[1][component] = cp[0][1][component] * a + cp[1][1][component] * b + cp[2][1][component] * c;
  76                                         temp[2][component] = cp[0][2][component] * a + cp[1][2][component] * b + cp[2][2][component] * c;
  77                                 }
  78                                 // fetch a pointer to the beginning of the output vertex row
  79                                 v = (float *)((unsigned char *)outputvertices + ((k * tesselationheight + y) * outputwidth + l * tesselationwidth) * outputstride);
  80                                 // for each column of the row...
  81                                 for (x = 0;x <= (tesselationwidth*2);x++)
  82                                 {
  83                                         // calculate point based on the row control points
  84                                         px = (float)x / (float)(tesselationwidth*2);
  85                                         // calculate quadratic spline weights for px
  86                                         // (could be precalculated)
  87                                         a = ((1.0f - px) * (1.0f - px));
  88                                         b = ((1.0f - px) * (2.0f * px));
  89                                         c = ((       px) * (       px));
  90                                         for (component = 0;component < numcomponents;component++)
  91                                                 v[component] = temp[0][component] * a + temp[1][component] * b + temp[2][component] * c;
  92                                         // advance to next output vertex using outputstride
  93                                         // (the next vertex may not be directly following this
  94                                         // one, as this may be part of a larger structure)
  95                                         v = (float *)((unsigned char *)v + outputstride);
  96                                 }
  97                         }
  98                 }
  99         }
 100 #if 0
 101         // enable this if you want results printed out
 102         printf("vertices[%i][%i] =\n{\n", (patchheight-1)*tesselationheight+1, (patchwidth-1)*tesselationwidth+1);
 103         for (y = 0;y < (patchheight-1)*tesselationheight+1;y++)
 104         {
 105                 for (x = 0;x < (patchwidth-1)*tesselationwidth+1;x++)
 106                 {
 107                         printf("(");
 108                         for (component = 0;component < numcomponents;component++)
 109                                 printf("%f ", outputvertices[(y*((patchwidth-1)*tesselationwidth+1)+x)*numcomponents+component]);
 110                         printf(") ");
 111                 }
 112                 printf("\n");
 113         }
 114         printf("}\n");
 115 #endif
 116 }
 117
 118 // returns how much tesselation of each segment is needed to remain under tolerance
 119 int Q3PatchTesselationOnX(int patchwidth, int patchheight, int components, const float *in, float tolerance)
 120 {
 121         int c, x, y;
 122         const float *patch;
 123         float deviation, squareddeviation, bestsquareddeviation;
 124         bestsquareddeviation = 0;
 125         for (y = 0;y < patchheight;y++)
 126         {
 127                 for (x = 0;x < patchwidth-1;x += 2)
 128                 {
 129                         squareddeviation = 0;
 130                         for (c = 0, patch = in + ((y * patchwidth) + x) * components;c < components;c++, patch++)
 131                         {
 132                                 deviation = patch[components] * 0.5f - patch[0] * 0.25f - patch[2*components] * 0.25f;
 133                                 squareddeviation += deviation*deviation;
 134                         }
 135                         if (bestsquareddeviation < squareddeviation)
 136                                 bestsquareddeviation = squareddeviation;
 137                 }
 138         }
 139         return (int)floor(log(sqrt(bestsquareddeviation) / tolerance) / log(2)) + 1;
 140 }
 141
 142 // returns how much tesselation of each segment is needed to remain under tolerance
 143 int Q3PatchTesselationOnY(int patchwidth, int patchheight, int components, const float *in, float tolerance)
 144 {
 145         int c, x, y;
 146         const float *patch;
 147         float deviation, squareddeviation, bestsquareddeviation;
 148         bestsquareddeviation = 0;
 149         for (y = 0;y < patchheight-1;y += 2)
 150         {
 151                 for (x = 0;x < patchwidth;x++)
 152                 {
 153                         squareddeviation = 0;
 154                         for (c = 0, patch = in + ((y * patchwidth) + x) * components;c < components;c++, patch++)
 155                         {
 156                                 deviation = patch[patchwidth*components] * 0.5f - patch[0] * 0.25f - patch[2*patchwidth*components] * 0.25f;
 157                                 squareddeviation += deviation*deviation;
 158                         }
 159                         if (bestsquareddeviation < squareddeviation)
 160                                 bestsquareddeviation = squareddeviation;
 161                 }
 162         }
 163         return (int)floor(log(sqrt(bestsquareddeviation) / tolerance) / log(2)) + 1;
 164 }
 165
 166 // calculates elements for a grid of vertices
 167 // (such as those produced by Q3PatchTesselate)
 168 // (note: width and height are the actual vertex size, this produces
 169 //  (width-1)*(height-1)*2 triangles, 3 elements each)
 170 void Q3PatchTriangleElements(int *elements, int width, int height, int firstvertex)
 171 {
 172         int x, y, row0, row1;
 173         for (y = 0;y < height - 1;y++)
 174         {
 175                 row0 = firstvertex + (y + 0) * width;
 176                 row1 = firstvertex + (y + 1) * width;
 177                 for (x = 0;x < width - 1;x++)
 178                 {
 179                         *elements++ = row0;
 180                         *elements++ = row1;
 181                         *elements++ = row0 + 1;
 182                         *elements++ = row1;
 183                         *elements++ = row1 + 1;
 184                         *elements++ = row0 + 1;
 185                         row0++;
 186                         row1++;
 187                 }
 188         }
 189 }
 190