/*

this code written by Ashley Rose Hale (LadyHavoc), on 2004-10-17, and placed into public domain

this implements Quadratic BSpline surfaces as seen in Quake3 by id Software

a small rant on misuse of the name 'bezier': many people seem to think that

bezier is a generic term for splines, but it is not, it is a term for a

specific type of bspline (4 control points, cubic bspline), bsplines are the

generalization of the bezier spline to support dimensions other than cubic.

example equations for 1-5 control point bsplines being sampled as t=0...1

1: flat (0th dimension)

o = a

2: linear (1st dimension)

o = a * (1 - t) + b * t

3: quadratic bspline (2nd dimension)

o = a * (1 - t) * (1 - t) + 2 * b * (1 - t) * t + c * t * t

4: cubic (bezier) bspline (3rd dimension)

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

5: quartic bspline (4th dimension)

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

arbitrary dimension bspline

double factorial(int n)

{

int i;

double f;

f = 1;

for (i = 1;i < n;i++)

f = f * i;

return f;

}

double bsplinesample(int dimensions, double t, double *param)

{

double o = 0;

for (i = 0;i < dimensions + 1;i++)

o += param[i] * factorial(dimensions)/(factorial(i)*factorial(dimensions-i)) * pow(t, i) * pow(1 - t, dimensions - i);

return o;

}

*/

#include "quakedef.h"

#include "mathlib.h"

#include <math.h>

#include "curves.h"

// Calculate number of resulting vertex rows/columns by given patch size and tesselation factor

// tess=0 means that we reduce detalization of base 3x3 patches by removing middle row and column of vertices

// "DimForTess" is "DIMension FOR TESSelation factor"

// 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)

int Q3PatchDimForTess(int size, int tess)

{

if (tess > 0)

return (size - 1) * tess + 1;

else if (tess == 0)

return (size - 1) / 2 + 1;

else

return 0; // Maybe warn about wrong tess here?

}

// usage:

// to expand a 5x5 patch to 21x21 vertices (4x4 tesselation), one might use this call:

// Q3PatchSubdivideFloat(3, sizeof(float[3]), outvertices, 5, 5, sizeof(float[3]), patchvertices, 4, 4);

void Q3PatchTesselateFloat(int numcomponents, int outputstride, float *outputvertices, int patchwidth, int patchheight, int inputstride, float *patchvertices, int tesselationwidth, int tesselationheight)

{

int k, l, x, y, component, outputwidth = Q3PatchDimForTess(patchwidth, tesselationwidth);

float px, py, *v, a, b, c, *cp[3][3], temp[3][64];

int xmax = max(1, 2*tesselationwidth);

int ymax = max(1, 2*tesselationheight);

// iterate over the individual 3x3 quadratic spline surfaces one at a time

// expanding them to fill the output array (with some overlap to ensure

// the edges are filled)

for (k = 0;k < patchheight-1;k += 2)

{

for (l = 0;l < patchwidth-1;l += 2)

{

// set up control point pointers for quicker lookup later

for (y = 0;y < 3;y++)

for (x = 0;x < 3;x++)

cp[y][x] = (float *)((unsigned char *)patchvertices + ((k+y)*patchwidth+(l+x)) * inputstride);

// for each row...

for (y = 0;y <= ymax;y++)

{

// calculate control points for this row by collapsing the 3

// rows of control points to one row using py

py = (float)y / (float)ymax;

// calculate quadratic spline weights for py

a = ((1.0f - py) * (1.0f - py));

b = ((1.0f - py) * (2.0f * py));

c = (( py) * ( py));

for (component = 0;component < numcomponents;component++)

{

temp[0][component] = cp[0][0][component] * a + cp[1][0][component] * b + cp[2][0][component] * c;

temp[1][component] = cp[0][1][component] * a + cp[1][1][component] * b + cp[2][1][component] * c;

temp[2][component] = cp[0][2][component] * a + cp[1][2][component] * b + cp[2][2][component] * c;

}

// fetch a pointer to the beginning of the output vertex row

v = (float *)((unsigned char *)outputvertices + ((k * ymax / 2 + y) * outputwidth + l * xmax / 2) * outputstride);

// for each column of the row...

for (x = 0;x <= xmax;x++)

{

// calculate point based on the row control points

px = (float)x / (float)xmax;

// calculate quadratic spline weights for px

// (could be precalculated)

a = ((1.0f - px) * (1.0f - px));

b = ((1.0f - px) * (2.0f * px));

c = (( px) * ( px));

for (component = 0;component < numcomponents;component++)

v[component] = temp[0][component] * a + temp[1][component] * b + temp[2][component] * c;

// advance to next output vertex using outputstride

// (the next vertex may not be directly following this

// one, as this may be part of a larger structure)

v = (float *)((unsigned char *)v + outputstride);

}

}

}

}

#if 0

// enable this if you want results printed out

printf("vertices[%i][%i] =\n{\n", (patchheight-1)*tesselationheight+1, (patchwidth-1)*tesselationwidth+1);

for (y = 0;y < (patchheight-1)*tesselationheight+1;y++)

{

for (x = 0;x < (patchwidth-1)*tesselationwidth+1;x++)

{

printf("(");

for (component = 0;component < numcomponents;component++)

printf("%f ", outputvertices[(y*((patchwidth-1)*tesselationwidth+1)+x)*numcomponents+component]);

printf(") ");

}

printf("\n");

}

printf("}\n");

#endif

}

static int Q3PatchTesselation(float largestsquared3xcurvearea, float tolerance)

{

float f;

// f is actually a squared 2x curve area... so the formula had to be adjusted to give roughly the same subdivisions

f = pow(largestsquared3xcurvearea / 64.0f, 0.25f) / tolerance;

//if(f < 0.25) // VERY flat patches

if(f < 0.0001) // TOTALLY flat patches

return 0;

else if(f < 2)

return 1;

else

return (int) floor(log(f) / log(2.0f)) + 1;

// this is always at least 2

// maps [0.25..0.5[ to -1 (actually, 1 is returned)

// maps [0.5..1[ to 0 (actually, 1 is returned)

// maps [1..2[ to 1

// maps [2..4[ to 2

// maps [4..8[ to 4

}

static float Squared3xCurveArea(const float *a, const float *control, const float *b, int components)

{

#if 0

// mimicing the old behaviour with the new code...

float deviation;

float quartercurvearea = 0;

int c;

for (c = 0;c < components;c++)

{

deviation = control[c] * 0.5f - a[c] * 0.25f - b[c] * 0.25f;

quartercurvearea += deviation*deviation;

}

// But as the new code now works on the squared 2x curve area, let's scale the value

return quartercurvearea * quartercurvearea * 64.0;

#else

// ideally, we'd like the area between the spline a->control->b and the line a->b.

// but as this is hard to calculate, let's calculate an upper bound of it:

// the area of the triangle a->control->b->a.

//

// one can prove that the area of a quadratic spline = 2/3 * the area of

// the triangle of its control points!

// to do it, first prove it for the spline through (0,0), (1,1), (2,0)

// (which is a parabola) and then note that moving the control point

// left/right is just shearing and keeps the area of both the spline and

// the triangle invariant.

//

// why are we going for the spline area anyway?

// we know that:

//

// the area between the spline and the line a->b is a measure of the

// error of approximation of the spline by the line.

//

// also, on circle-like or parabola-like curves, you easily get that the

// double amount of line approximation segments reduces the error to its quarter

// (also, easy to prove for splines by doing it for one specific one, and using

// affine transforms to get all other splines)

//

// so...

//

// let's calculate the area! but we have to avoid the cross product, as

// components is not necessarily 3

//

// the area of a triangle spanned by vectors a and b is

//

// 0.5 * |a| |b| sin gamma

//

// now, cos gamma is

//

// a.b / (|a| |b|)

//

// so the area is

//

// 0.5 * sqrt(|a|^2 |b|^2 - (a.b)^2)

int c;

float aa = 0, bb = 0, ab = 0;

for (c = 0;c < components;c++)

{

float xa = a[c] - control[c];

float xb = b[c] - control[c];

aa += xa * xa;

ab += xa * xb;

bb += xb * xb;

}

// area is 0.5 * sqrt(aa*bb - ab*ab)

// 2x TRIANGLE area is sqrt(aa*bb - ab*ab)

// 3x CURVE area is sqrt(aa*bb - ab*ab)

return aa * bb - ab * ab;

#endif

}

// returns how much tesselation of each segment is needed to remain under tolerance

int Q3PatchTesselationOnX(int patchwidth, int patchheight, int components, const float *in, float tolerance)

{

int x, y;

const float *patch;

float squared3xcurvearea, largestsquared3xcurvearea;

largestsquared3xcurvearea = 0;

for (y = 0;y < patchheight;y++)

{

for (x = 0;x < patchwidth-1;x += 2)

{

patch = in + ((y * patchwidth) + x) * components;

squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[components], &patch[2*components], components);

if (largestsquared3xcurvearea < squared3xcurvearea)

largestsquared3xcurvearea = squared3xcurvearea;

}

}

return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);

}

// returns how much tesselation of each segment is needed to remain under tolerance

int Q3PatchTesselationOnY(int patchwidth, int patchheight, int components, const float *in, float tolerance)

{

int x, y;

const float *patch;

float squared3xcurvearea, largestsquared3xcurvearea;

largestsquared3xcurvearea = 0;

for (y = 0;y < patchheight-1;y += 2)

{

for (x = 0;x < patchwidth;x++)

{

patch = in + ((y * patchwidth) + x) * components;

squared3xcurvearea = Squared3xCurveArea(&patch[0], &patch[patchwidth*components], &patch[2*patchwidth*components], components);

if (largestsquared3xcurvearea < squared3xcurvearea)

largestsquared3xcurvearea = squared3xcurvearea;

}

}

return Q3PatchTesselation(largestsquared3xcurvearea, tolerance);

}

// Find an equal vertex in array. Check only vertices with odd X and Y

static int FindEqualOddVertexInArray(int numcomponents, float *vertex, float *vertices, int width, int height)

{

int x, y, j;

for (y=0; y<height; y+=2)

{

for (x=0; x<width; x+=2)

{

qbool found = true;

for (j=0; j<numcomponents; j++)

if (fabs(*(vertex+j) - *(vertices+j)) > 0.05)

// div0: this is notably smaller than the smallest radiant grid

// but large enough so we don't need to get scared of roundoff

// errors

{

found = false;

break;

}

if(found)

return y*width+x;

vertices += numcomponents*2;

}

vertices += numcomponents*(width-1);

}

return -1;

}

#define SIDE_INVALID -1

#define SIDE_X 0

#define SIDE_Y 1

static int GetSide(int p1, int p2, int width, int height, int *pointdist)

{

int x1 = p1 % width, y1 = p1 / width;

int x2 = p2 % width, y2 = p2 / width;

if (p1 < 0 || p2 < 0)

return SIDE_INVALID;

if (x1 == x2)

{

if (y1 != y2)

{

*pointdist = abs(y2 - y1);

return SIDE_Y;

}

else

return SIDE_INVALID;

}

else if (y1 == y2)

{

*pointdist = abs(x2 - x1);

return SIDE_X;

}

else

return SIDE_INVALID;

}

// Increase tesselation of one of two touching patches to make a seamless connection between them

// Returns 0 in case if patches were not modified, otherwise 1

int Q3PatchAdjustTesselation(int numcomponents, patchinfo_t *patch1, float *patchvertices1, patchinfo_t *patch2, float *patchvertices2)

{

// what we are doing here is:

// we take for each corner of one patch

// and check if the other patch contains that corner

// once we have a pair of such matches

struct {int id1,id2;} commonverts[8];

int i, j, k, side1, side2, *tess1, *tess2;

int dist1 = 0, dist2 = 0;

qbool modified = false;

// Potential paired vertices (corners of the first patch)

commonverts[0].id1 = 0;

commonverts[1].id1 = patch1->xsize-1;

commonverts[2].id1 = patch1->xsize*(patch1->ysize-1);

commonverts[3].id1 = patch1->xsize*patch1->ysize-1;

for (i=0;i<4;++i)

commonverts[i].id2 = FindEqualOddVertexInArray(numcomponents, patchvertices1+numcomponents*commonverts[i].id1, patchvertices2, patch2->xsize, patch2->ysize);

// Corners of the second patch

commonverts[4].id2 = 0;

commonverts[5].id2 = patch2->xsize-1;

commonverts[6].id2 = patch2->xsize*(patch2->ysize-1);

commonverts[7].id2 = patch2->xsize*patch2->ysize-1;

for (i=4;i<8;++i)

commonverts[i].id1 = FindEqualOddVertexInArray(numcomponents, patchvertices2+numcomponents*commonverts[i].id2, patchvertices1, patch1->xsize, patch1->ysize);

for (i=0;i<8;++i)

for (j=i+1;j<8;++j)

{

side1 = GetSide(commonverts[i].id1,commonverts[j].id1,patch1->xsize,patch1->ysize,&dist1);

side2 = GetSide(commonverts[i].id2,commonverts[j].id2,patch2->xsize,patch2->ysize,&dist2);

if (side1 == SIDE_INVALID || side2 == SIDE_INVALID)

continue;

if(dist1 != dist2)

{

// no patch welding if the resolutions mismatch

continue;

}

// Update every lod level

for (k=0;k<PATCH_LODS_NUM;++k)

{

tess1 = side1 == SIDE_X ? &patch1->lods[k].xtess : &patch1->lods[k].ytess;

tess2 = side2 == SIDE_X ? &patch2->lods[k].xtess : &patch2->lods[k].ytess;

if (*tess1 != *tess2)

{

if (*tess1 < *tess2)

*tess1 = *tess2;

else

*tess2 = *tess1;

modified = true;

}

}

}

return modified;

}

#undef SIDE_INVALID

#undef SIDE_X

#undef SIDE_Y

// calculates elements for a grid of vertices

// (such as those produced by Q3PatchTesselate)

// (note: width and height are the actual vertex size, this produces

// (width-1)*(height-1)*2 triangles, 3 elements each)

void Q3PatchTriangleElements(int *elements, int width, int height, int firstvertex)

{

int x, y, row0, row1;

for (y = 0;y < height - 1;y++)

{

if(y % 2)

{

// swap the triangle order in odd rows as optimization for collision stride

row0 = firstvertex + (y + 0) * width + width - 2;

row1 = firstvertex + (y + 1) * width + width - 2;

for (x = 0;x < width - 1;x++)

{

*elements++ = row1;

*elements++ = row1 + 1;

*elements++ = row0 + 1;

*elements++ = row0;

*elements++ = row1;

*elements++ = row0 + 1;

row0--;

row1--;

}

}

else

{

row0 = firstvertex + (y + 0) * width;

row1 = firstvertex + (y + 1) * width;

for (x = 0;x < width - 1;x++)

{

*elements++ = row0;

*elements++ = row1;

*elements++ = row0 + 1;

*elements++ = row1;

*elements++ = row1 + 1;

*elements++ = row0 + 1;

row0++;

row1++;

}

}

}

}