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
52static bool straight_forward = false;
53
54SkDQuadImplicit::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  */
96bool 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
113bool 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