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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 | |
54 static bool straight_forward = true; | |
55 | |
56 QuadImplicitForm::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 */ | |
98 bool 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 | |
115 bool 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 | |
121 static 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 | |
127 void 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" | |
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