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" 136
