change names of functions, and add comments.

The code is mathematically correct now, I added a proof of the missing link:
quadratic spline area = exactly 2/3 * area of the control points triangle!

The actual subdivisions stayed invariant, just the names of the functions make
more sense now.

git-svn-id: svn://svn.icculus.org/twilight/trunk/darkplaces@8821 d7cf8633-e32d-0410-b094-e92efae38249

diff --git a/curves.c b/curves.c
index ea18cf47fed03d8aee60a5f3a1f93e15a3b34178..ad5bb63f74f51f50c9e92d025c19d90b48c571be 100644 (file)
--- a/curves.c
+++ b/curves.c
@@ -156,7 +156,7 @@ static int Q3PatchTesselation(float largestsquared2xcurvearea, float tolerance)
                // maps [4..8[ to 4
 }
 
-float Squared2xCurveArea(const float *a, const float *control, const float *b, int components)
+float Squared3xCurveArea(const float *a, const float *control, const float *b, int components)
 {
 #if 0
        // mimicing the old behaviour with the new code...
@@ -178,6 +178,13 @@ float Squared2xCurveArea(const float *a, const float *control, const float *b, i
        // 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:
        //
@@ -186,6 +193,8 @@ float Squared2xCurveArea(const float *a, const float *control, const float *b, i
        //
        //   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...
        //
@@ -214,7 +223,8 @@ float Squared2xCurveArea(const float *a, const float *control, const float *b, i
                bb += xb * xb;
        }
        // area is 0.5 * sqrt(aa*bb - ab*ab)
-       // 2x area is 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
 }
@@ -231,7 +241,7 @@ int Q3PatchTesselationOnX(int patchwidth, int patchheight, int components, const
                for (x = 0;x < patchwidth-1;x += 2)
                {
                        patch = in + ((y * patchwidth) + x) * components;
-                       squared2xcurvearea = Squared2xCurveArea(&patch[0], &patch[components], &patch[2*components], components);
+                       squared2xcurvearea = Squared3xCurveArea(&patch[0], &patch[components], &patch[2*components], components);
                        if (largestsquared2xcurvearea < squared2xcurvearea)
                                largestsquared2xcurvearea = squared2xcurvearea;
                }
@@ -251,7 +261,7 @@ int Q3PatchTesselationOnY(int patchwidth, int patchheight, int components, const
                for (x = 0;x < patchwidth;x++)
                {
                        patch = in + ((y * patchwidth) + x) * components;
-                       squared2xcurvearea = Squared2xCurveArea(&patch[0], &patch[patchwidth*components], &patch[2*patchwidth*components], components);
+                       squared2xcurvearea = Squared3xCurveArea(&patch[0], &patch[patchwidth*components], &patch[2*patchwidth*components], components);
                        if (largestsquared2xcurvearea < squared2xcurvearea)
                                largestsquared2xcurvearea = squared2xcurvearea;
                }