1 /* 2 * Copyright 2012 Google Inc. 3 * 4 * Use of this source code is governed by a BSD-style license that can be 5 * found in the LICENSE file. 6 */ 7 #include "CubicUtilities.h" 8 #include "IntersectionUtilities.h" 9 10 /* 11 Given a cubic c, t1, and t2, find a small cubic segment. 12 13 The new cubic is defined as points A, B, C, and D, where 14 s1 = 1 - t1 15 s2 = 1 - t2 16 A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1 17 D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2 18 19 We don't have B or C. So We define two equations to isolate them. 20 First, compute two reference T values 1/3 and 2/3 from t1 to t2: 21 22 c(at (2*t1 + t2)/3) == E 23 c(at (t1 + 2*t2)/3) == F 24 25 Next, compute where those values must be if we know the values of B and C: 26 27 _12 = A*2/3 + B*1/3 28 12_ = A*1/3 + B*2/3 29 _23 = B*2/3 + C*1/3 30 23_ = B*1/3 + C*2/3 31 _34 = C*2/3 + D*1/3 32 34_ = C*1/3 + D*2/3 33 _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 34 123_ = (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 35 _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 36 234_ = (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 37 _1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3 38 = A*8/27 + B*12/27 + C*6/27 + D*1/27 39 = E 40 1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3 41 = A*1/27 + B*6/27 + C*12/27 + D*8/27 42 = F 43 E*27 = A*8 + B*12 + C*6 + D 44 F*27 = A + B*6 + C*12 + D*8 45 46 Group the known values on one side: 47 48 M = E*27 - A*8 - D = B*12 + C* 6 49 N = F*27 - A - D*8 = B* 6 + C*12 50 M*2 - N = B*18 51 N*2 - M = C*18 52 B = (M*2 - N)/18 53 C = (N*2 - M)/18 54 */ 55 56 static double interp_cubic_coords(const double* src, double t) 57 { 58 double ab = interp(src[0], src[2], t); 59 double bc = interp(src[2], src[4], t); 60 double cd = interp(src[4], src[6], t); 61 double abc = interp(ab, bc, t); 62 double bcd = interp(bc, cd, t); 63 double abcd = interp(abc, bcd, t); 64 return abcd; 65 } 66 67 void sub_divide(const Cubic& src, double t1, double t2, Cubic& dst) { 68 if (t1 == 0 && t2 == 1) { 69 dst[0] = src[0]; 70 dst[1] = src[1]; 71 dst[2] = src[2]; 72 dst[3] = src[3]; 73 return; 74 } 75 double ax = dst[0].x = interp_cubic_coords(&src[0].x, t1); 76 double ay = dst[0].y = interp_cubic_coords(&src[0].y, t1); 77 double ex = interp_cubic_coords(&src[0].x, (t1*2+t2)/3); 78 double ey = interp_cubic_coords(&src[0].y, (t1*2+t2)/3); 79 double fx = interp_cubic_coords(&src[0].x, (t1+t2*2)/3); 80 double fy = interp_cubic_coords(&src[0].y, (t1+t2*2)/3); 81 double dx = dst[3].x = interp_cubic_coords(&src[0].x, t2); 82 double dy = dst[3].y = interp_cubic_coords(&src[0].y, t2); 83 double mx = ex * 27 - ax * 8 - dx; 84 double my = ey * 27 - ay * 8 - dy; 85 double nx = fx * 27 - ax - dx * 8; 86 double ny = fy * 27 - ay - dy * 8; 87 /* bx = */ dst[1].x = (mx * 2 - nx) / 18; 88 /* by = */ dst[1].y = (my * 2 - ny) / 18; 89 /* cx = */ dst[2].x = (nx * 2 - mx) / 18; 90 /* cy = */ dst[2].y = (ny * 2 - my) / 18; 91 } 92 93 void sub_divide(const Cubic& src, const _Point& a, const _Point& d, 94 double t1, double t2, _Point dst[2]) { 95 double ex = interp_cubic_coords(&src[0].x, (t1 * 2 + t2) / 3); 96 double ey = interp_cubic_coords(&src[0].y, (t1 * 2 + t2) / 3); 97 double fx = interp_cubic_coords(&src[0].x, (t1 + t2 * 2) / 3); 98 double fy = interp_cubic_coords(&src[0].y, (t1 + t2 * 2) / 3); 99 double mx = ex * 27 - a.x * 8 - d.x; 100 double my = ey * 27 - a.y * 8 - d.y; 101 double nx = fx * 27 - a.x - d.x * 8; 102 double ny = fy * 27 - a.y - d.y * 8; 103 /* bx = */ dst[0].x = (mx * 2 - nx) / 18; 104 /* by = */ dst[0].y = (my * 2 - ny) / 18; 105 /* cx = */ dst[1].x = (nx * 2 - mx) / 18; 106 /* cy = */ dst[1].y = (ny * 2 - my) / 18; 107 } 108 109 /* classic one t subdivision */ 110 static void interp_cubic_coords(const double* src, double* dst, double t) 111 { 112 double ab = interp(src[0], src[2], t); 113 double bc = interp(src[2], src[4], t); 114 double cd = interp(src[4], src[6], t); 115 double abc = interp(ab, bc, t); 116 double bcd = interp(bc, cd, t); 117 double abcd = interp(abc, bcd, t); 118 119 dst[0] = src[0]; 120 dst[2] = ab; 121 dst[4] = abc; 122 dst[6] = abcd; 123 dst[8] = bcd; 124 dst[10] = cd; 125 dst[12] = src[6]; 126 } 127 128 void chop_at(const Cubic& src, CubicPair& dst, double t) 129 { 130 if (t == 0.5) { 131 dst.pts[0] = src[0]; 132 dst.pts[1].x = (src[0].x + src[1].x) / 2; 133 dst.pts[1].y = (src[0].y + src[1].y) / 2; 134 dst.pts[2].x = (src[0].x + 2 * src[1].x + src[2].x) / 4; 135 dst.pts[2].y = (src[0].y + 2 * src[1].y + src[2].y) / 4; 136 dst.pts[3].x = (src[0].x + 3 * (src[1].x + src[2].x) + src[3].x) / 8; 137 dst.pts[3].y = (src[0].y + 3 * (src[1].y + src[2].y) + src[3].y) / 8; 138 dst.pts[4].x = (src[1].x + 2 * src[2].x + src[3].x) / 4; 139 dst.pts[4].y = (src[1].y + 2 * src[2].y + src[3].y) / 4; 140 dst.pts[5].x = (src[2].x + src[3].x) / 2; 141 dst.pts[5].y = (src[2].y + src[3].y) / 2; 142 dst.pts[6] = src[3]; 143 return; 144 } 145 interp_cubic_coords(&src[0].x, &dst.pts[0].x, t); 146 interp_cubic_coords(&src[0].y, &dst.pts[0].y, t); 147 } 148
