c682590538
git-svn-id: http://skia.googlecode.com/svn/trunk@3141 2bbb7eff-a529-9590-31e7-b0007b416f81
104 lines
3.4 KiB
C++
104 lines
3.4 KiB
C++
#include "CurveIntersection.h"
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#include "IntersectionUtilities.h"
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/*
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Given a cubic c, t1, and t2, find a small cubic segment.
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The new cubic is defined as points A, B, C, and D, where
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s1 = 1 - t1
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s2 = 1 - t2
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A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1
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D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2
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We don't have B or C. So We define two equations to isolate them.
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First, compute two reference T values 1/3 and 2/3 from t1 to t2:
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c(at (2*t1 + t2)/3) == E
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c(at (t1 + 2*t2)/3) == F
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Next, compute where those values must be if we know the values of B and C:
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_12 = A*2/3 + B*1/3
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12_ = A*1/3 + B*2/3
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_23 = B*2/3 + C*1/3
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23_ = B*1/3 + C*2/3
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_34 = C*2/3 + D*1/3
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34_ = C*1/3 + D*2/3
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_123 = (A*2/3 + B*1/3)*2/3 + (B*2/3 + C*1/3)*1/3 = A*4/9 + B*4/9 + C*1/9
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123_ = (A*1/3 + B*2/3)*1/3 + (B*1/3 + C*2/3)*2/3 = A*1/9 + B*4/9 + C*4/9
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_234 = (B*2/3 + C*1/3)*2/3 + (C*2/3 + D*1/3)*1/3 = B*4/9 + C*4/9 + D*1/9
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234_ = (B*1/3 + C*2/3)*1/3 + (C*1/3 + D*2/3)*2/3 = B*1/9 + C*4/9 + D*4/9
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_1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3
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= A*8/27 + B*12/27 + C*6/27 + D*1/27
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= E
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1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3
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= A*1/27 + B*6/27 + C*12/27 + D*8/27
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= F
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E*27 = A*8 + B*12 + C*6 + D
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F*27 = A + B*6 + C*12 + D*8
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Group the known values on one side:
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M = E*27 - A*8 - D = B*12 + C* 6
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N = F*27 - A - D*8 = B* 6 + C*12
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M*2 - N = B*18
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N*2 - M = C*18
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B = (M*2 - N)/18
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C = (N*2 - M)/18
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*/
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static double interp_cubic_coords(const double* src, double t)
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{
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double ab = interp(src[0], src[2], t);
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double bc = interp(src[2], src[4], t);
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double cd = interp(src[4], src[6], t);
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double abc = interp(ab, bc, t);
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double bcd = interp(bc, cd, t);
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double abcd = interp(abc, bcd, t);
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return abcd;
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}
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void sub_divide(const Cubic& src, double t1, double t2, Cubic& dst) {
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double ax = dst[0].x = interp_cubic_coords(&src[0].x, t1);
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double ay = dst[0].y = interp_cubic_coords(&src[0].y, t1);
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double ex = interp_cubic_coords(&src[0].x, (t1*2+t2)/3);
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double ey = interp_cubic_coords(&src[0].y, (t1*2+t2)/3);
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double fx = interp_cubic_coords(&src[0].x, (t1+t2*2)/3);
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double fy = interp_cubic_coords(&src[0].y, (t1+t2*2)/3);
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double dx = dst[3].x = interp_cubic_coords(&src[0].x, t2);
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double dy = dst[3].y = interp_cubic_coords(&src[0].y, t2);
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double mx = ex * 27 - ax * 8 - dx;
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double my = ey * 27 - ay * 8 - dy;
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double nx = fx * 27 - ax - dx * 8;
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double ny = fy * 27 - ay - dy * 8;
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/* bx = */ dst[1].x = (mx * 2 - nx) / 18;
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/* by = */ dst[1].y = (my * 2 - ny) / 18;
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/* cx = */ dst[2].x = (nx * 2 - mx) / 18;
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/* cy = */ dst[2].y = (ny * 2 - my) / 18;
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}
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/* classic one t subdivision */
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static void interp_cubic_coords(const double* src, double* dst, double t)
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{
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double ab = interp(src[0], src[2], t);
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double bc = interp(src[2], src[4], t);
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double cd = interp(src[4], src[6], t);
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double abc = interp(ab, bc, t);
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double bcd = interp(bc, cd, t);
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double abcd = interp(abc, bcd, t);
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dst[0] = src[0];
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dst[2] = ab;
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dst[4] = abc;
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dst[6] = abcd;
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dst[8] = bcd;
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dst[10] = cd;
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dst[12] = src[6];
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}
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void chop_at(const Cubic& src, CubicPair& dst, double t)
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{
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interp_cubic_coords(&src[0].x, &dst.pts[0].x, t);
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interp_cubic_coords(&src[0].y, &dst.pts[0].y, t);
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}
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