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 /* * Copyright 2012 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "CurveIntersection.h" #include "QuadraticParameterization.h" #include "QuadraticUtilities.h" /* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1 * * This paper proves that Syvester's method can compute the implicit form of * the quadratic from the parameterized form. * * Given x = a*t*t + b*t + c (the parameterized form) * y = d*t*t + e*t + f * * we want to find an equation of the implicit form: * * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0 * * The implicit form can be expressed as a 4x4 determinant, as shown. * * The resultant obtained by Syvester's method is * * | a b (c - x) 0 | * | 0 a b (c - x) | * | d e (f - y) 0 | * | 0 d e (f - y) | * * which expands to * * d*d*x*x + -2*a*d*x*y + a*a*y*y * + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x * + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y * + * | a b c 0 | * | 0 a b c | == 0. * | d e f 0 | * | 0 d e f | * * Expanding the constant determinant results in * * | a b c | | b c 0 | * a*| e f 0 | + d*| a b c | == * | d e f | | d e f | * * 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) * */ static bool straight_forward = true; QuadImplicitForm::QuadImplicitForm(const Quadratic& q) { double a, b, c; set_abc(&q[0].x, a, b, c); double d, e, f; set_abc(&q[0].y, d, e, f); // compute the implicit coefficients if (straight_forward) { // 42 muls, 13 adds p[xx_coeff] = d * d; p[xy_coeff] = -2 * a * d; p[yy_coeff] = a * a; p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d; p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a; p[c_coeff] = 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); } else { // 26 muls, 11 adds double aa = a * a; double ad = a * d; double dd = d * d; p[xx_coeff] = dd; p[xy_coeff] = -2 * ad; p[yy_coeff] = aa; double be = b * e; double bde = be * d; double cdd = c * dd; double ee = e * e; p[x_coeff] = -2*cdd + bde - a*ee + 2*ad*f; double aaf = aa * f; double abe = a * be; double ac = a * c; double bb_2ac = b*b - 2*ac; p[y_coeff] = -2*aaf + abe - d*bb_2ac; p[c_coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde; } } /* Given a pair of quadratics, determine their parametric coefficients. * If the scaled coefficients are nearly equal, then the part of the quadratics * may be coincident. * FIXME: optimization -- since comparison short-circuits on no match, * lazily compute the coefficients, comparing the easiest to compute first. * xx and yy first; then xy; and so on. */ bool QuadImplicitForm::implicit_match(const QuadImplicitForm& p2) const { int first = 0; for (int index = 0; index < coeff_count; ++index) { if (approximately_zero(p[index]) && approximately_zero(p2.p[index])) { first += first == index; continue; } if (first == index) { continue; } if (!AlmostEqualUlps(p[index] * p2.p[first], p[first] * p2.p[index])) { return false; } } return true; } bool implicit_matches(const Quadratic& quad1, const Quadratic& quad2) { QuadImplicitForm i1(quad1); // a'xx , b'xy , c'yy , d'x , e'y , f QuadImplicitForm i2(quad2); return i1.implicit_match(i2); } static double tangent(const double* quadratic, double t) { double a, b, c; set_abc(quadratic, a, b, c); return 2 * a * t + b; } void tangent(const Quadratic& quadratic, double t, _Point& result) { result.x = tangent(&quadratic[0].x, t); result.y = tangent(&quadratic[0].y, t); } // unit test to return and validate parametric coefficients #include "QuadraticParameterization_TestUtility.cpp"