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 "SkDQuadImplicit.h"
8
9 /* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
10 *
11 * This paper proves that Syvester's method can compute the implicit form of
12 * the quadratic from the parameterized form.
13 *
14 * Given x = a*t*t + b*t + c (the parameterized form)
15 * y = d*t*t + e*t + f
16 *
17 * we want to find an equation of the implicit form:
18 *
19 * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
20 *
21 * The implicit form can be expressed as a 4x4 determinant, as shown.
22 *
23 * The resultant obtained by Syvester's method is
24 *
25 * | a b (c - x) 0 |
26 * | 0 a b (c - x) |
27 * | d e (f - y) 0 |
28 * | 0 d e (f - y) |
29 *
30 * which expands to
31 *
32 * d*d*x*x + -2*a*d*x*y + a*a*y*y
33 * + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
34 * + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
35 * +
36 * | a b c 0 |
37 * | 0 a b c | == 0.
38 * | d e f 0 |
39 * | 0 d e f |
40 *
41 * Expanding the constant determinant results in
42 *
43 * | a b c | | b c 0 |
44 * a*| e f 0 | + d*| a b c | ==
45 * | d e f | | d e f |
46 *
47 * a*(a*f*f + c*e*e - c*f*d - b*e*f) + d*(b*b*f + c*c*d - c*a*f - c*e*b)
48 *
49 */
50
51 // use the tricky arithmetic path, but leave the original to compare just in case
52 static bool straight_forward = false;
53
SkDQuadImplicit(const SkDQuad & q)54 SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) {
55 double a, b, c;
56 SkDQuad::SetABC(&q[0].fX, &a, &b, &c);
57 double d, e, f;
58 SkDQuad::SetABC(&q[0].fY, &d, &e, &f);
59 // compute the implicit coefficients
60 if (straight_forward) { // 42 muls, 13 adds
61 fP[kXx_Coeff] = d * d;
62 fP[kXy_Coeff] = -2 * a * d;
63 fP[kYy_Coeff] = a * a;
64 fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
65 fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
66 fP[kC_Coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
67 + d*(b*b*f + c*c*d - c*a*f - c*e*b);
68 } else { // 26 muls, 11 adds
69 double aa = a * a;
70 double ad = a * d;
71 double dd = d * d;
72 fP[kXx_Coeff] = dd;
73 fP[kXy_Coeff] = -2 * ad;
74 fP[kYy_Coeff] = aa;
75 double be = b * e;
76 double bde = be * d;
77 double cdd = c * dd;
78 double ee = e * e;
79 fP[kX_Coeff] = -2*cdd + bde - a*ee + 2*ad*f;
80 double aaf = aa * f;
81 double abe = a * be;
82 double ac = a * c;
83 double bb_2ac = b*b - 2*ac;
84 fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac;
85 fP[kC_Coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
86 }
87 }
88
89 /* Given a pair of quadratics, determine their parametric coefficients.
90 * If the scaled coefficients are nearly equal, then the part of the quadratics
91 * may be coincident.
92 * OPTIMIZATION -- since comparison short-circuits on no match,
93 * lazily compute the coefficients, comparing the easiest to compute first.
94 * xx and yy first; then xy; and so on.
95 */
match(const SkDQuadImplicit & p2) const96 bool SkDQuadImplicit::match(const SkDQuadImplicit& p2) const {
97 int first = 0;
98 for (int index = 0; index <= kC_Coeff; ++index) {
99 if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) {
100 first += first == index;
101 continue;
102 }
103 if (first == index) {
104 continue;
105 }
106 if (!AlmostDequalUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) {
107 return false;
108 }
109 }
110 return true;
111 }
112
Match(const SkDQuad & quad1,const SkDQuad & quad2)113 bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) {
114 SkDQuadImplicit i1(quad1); // a'xx , b'xy , c'yy , d'x , e'y , f
115 SkDQuadImplicit i2(quad2);
116 return i1.match(i2);
117 }
118