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
interp_cubic_coords(const double * src,double t)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
sub_divide(const Cubic & src,double t1,double t2,Cubic & dst)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
sub_divide(const Cubic & src,const _Point & a,const _Point & d,double t1,double t2,_Point dst[2])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 */
interp_cubic_coords(const double * src,double * dst,double t)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
chop_at(const Cubic & src,CubicPair & dst,double t)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