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 "CurveIntersection.h"
8#include "QuadraticParameterization.h"
9#include "QuadraticUtilities.h"
10
11/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
12 *
13 * This paper proves that Syvester's method can compute the implicit form of
14 * the quadratic from the parameterized form.
15 *
16 * Given x = a*t*t + b*t + c  (the parameterized form)
17 *       y = d*t*t + e*t + f
18 *
19 * we want to find an equation of the implicit form:
20 *
21 * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
22 *
23 * The implicit form can be expressed as a 4x4 determinant, as shown.
24 *
25 * The resultant obtained by Syvester's method is
26 *
27 * |   a   b   (c - x)     0     |
28 * |   0   a      b     (c - x)  |
29 * |   d   e   (f - y)     0     |
30 * |   0   d      e     (f - y)  |
31 *
32 * which expands to
33 *
34 * d*d*x*x + -2*a*d*x*y + a*a*y*y
35 *         + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
36 *         + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
37 *         +
38 * |   a   b   c   0   |
39 * |   0   a   b   c   | == 0.
40 * |   d   e   f   0   |
41 * |   0   d   e   f   |
42 *
43 * Expanding the constant determinant results in
44 *
45 *   | a b c |     | b c 0 |
46 * a*| e f 0 | + d*| a b c | ==
47 *   | d e f |     | d e f |
48 *
49 * 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)
50 *
51 */
52
53
54static bool straight_forward = true;
55
56QuadImplicitForm::QuadImplicitForm(const Quadratic& q) {
57    double a, b, c;
58    set_abc(&q[0].x, a, b, c);
59    double d, e, f;
60    set_abc(&q[0].y, d, e, f);
61    // compute the implicit coefficients
62    if (straight_forward) { // 42 muls, 13 adds
63        p[xx_coeff] = d * d;
64        p[xy_coeff] = -2 * a * d;
65        p[yy_coeff] = a * a;
66        p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
67        p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
68        p[c_coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
69                   + d*(b*b*f + c*c*d - c*a*f - c*e*b);
70    } else { // 26 muls, 11 adds
71        double aa = a * a;
72        double ad = a * d;
73        double dd = d * d;
74        p[xx_coeff] = dd;
75        p[xy_coeff] = -2 * ad;
76        p[yy_coeff] = aa;
77        double be = b * e;
78        double bde = be * d;
79        double cdd = c * dd;
80        double ee = e * e;
81        p[x_coeff] =  -2*cdd + bde - a*ee + 2*ad*f;
82        double aaf = aa * f;
83        double abe = a * be;
84        double ac = a * c;
85        double bb_2ac = b*b - 2*ac;
86        p[y_coeff] = -2*aaf + abe - d*bb_2ac;
87        p[c_coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
88    }
89}
90
91 /* Given a pair of quadratics, determine their parametric coefficients.
92  * If the scaled coefficients are nearly equal, then the part of the quadratics
93  * may be coincident.
94  * FIXME: optimization -- since comparison short-circuits on no match,
95  * lazily compute the coefficients, comparing the easiest to compute first.
96  * xx and yy first; then xy; and so on.
97  */
98bool QuadImplicitForm::implicit_match(const QuadImplicitForm& p2) const {
99    int first = 0;
100    for (int index = 0; index < coeff_count; ++index) {
101        if (approximately_zero(p[index]) && approximately_zero(p2.p[index])) {
102            first += first == index;
103            continue;
104        }
105        if (first == index) {
106            continue;
107        }
108        if (!AlmostEqualUlps(p[index] * p2.p[first], p[first] * p2.p[index])) {
109            return false;
110        }
111    }
112    return true;
113}
114
115bool implicit_matches(const Quadratic& quad1, const Quadratic& quad2) {
116    QuadImplicitForm i1(quad1);  // a'xx , b'xy , c'yy , d'x , e'y , f
117    QuadImplicitForm i2(quad2);
118    return i1.implicit_match(i2);
119}
120
121static double tangent(const double* quadratic, double t) {
122    double a, b, c;
123    set_abc(quadratic, a, b, c);
124    return 2 * a * t + b;
125}
126
127void tangent(const Quadratic& quadratic, double t, _Point& result) {
128    result.x = tangent(&quadratic[0].x, t);
129    result.y = tangent(&quadratic[0].y, t);
130}
131
132
133
134// unit test to return and validate parametric coefficients
135#include "QuadraticParameterization_TestUtility.cpp"
136