blob: caa7c24dd8ed3d98b7f0cbc057ae72cb930d4d14 [file] [log] [blame]
 /* * 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 "CubicUtilities.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 parameterzied form. * * Given x = a*t*t*t + b*t*t + c*t + d (the parameterized form) * y = e*t*t*t + f*t*t + g*t + h * * we want to find an equation of the implicit form: * * A*x^3 + B*x*x*y + C*x*y*y + D*y^3 + E*x*x + F*x*y + G*y*y + H*x + I*y + J = 0 * * The implicit form can be expressed as a 6x6 determinant, as shown. * * The resultant obtained by Syvester's method is * * | a b c (d - x) 0 0 | * | 0 a b c (d - x) 0 | * | 0 0 a b c (d - x) | * | e f g (h - y) 0 0 | * | 0 e f g (h - y) 0 | * | 0 0 e f g (h - y) | * * which, according to Mathematica, expands as shown below. * * Resultant[a*t^3 + b*t^2 + c*t + d - x, e*t^3 + f*t^2 + g*t + h - y, t] * * -d^3 e^3 + c d^2 e^2 f - b d^2 e f^2 + a d^2 f^3 - c^2 d e^2 g + * 2 b d^2 e^2 g + b c d e f g - 3 a d^2 e f g - a c d f^2 g - * b^2 d e g^2 + 2 a c d e g^2 + a b d f g^2 - a^2 d g^3 + c^3 e^2 h - * 3 b c d e^2 h + 3 a d^2 e^2 h - b c^2 e f h + 2 b^2 d e f h + * a c d e f h + a c^2 f^2 h - 2 a b d f^2 h + b^2 c e g h - * 2 a c^2 e g h - a b d e g h - a b c f g h + 3 a^2 d f g h + * a^2 c g^2 h - b^3 e h^2 + 3 a b c e h^2 - 3 a^2 d e h^2 + * a b^2 f h^2 - 2 a^2 c f h^2 - a^2 b g h^2 + a^3 h^3 + 3 d^2 e^3 x - * 2 c d e^2 f x + 2 b d e f^2 x - 2 a d f^3 x + c^2 e^2 g x - * 4 b d e^2 g x - b c e f g x + 6 a d e f g x + a c f^2 g x + * b^2 e g^2 x - 2 a c e g^2 x - a b f g^2 x + a^2 g^3 x + * 3 b c e^2 h x - 6 a d e^2 h x - 2 b^2 e f h x - a c e f h x + * 2 a b f^2 h x + a b e g h x - 3 a^2 f g h x + 3 a^2 e h^2 x - * 3 d e^3 x^2 + c e^2 f x^2 - b e f^2 x^2 + a f^3 x^2 + * 2 b e^2 g x^2 - 3 a e f g x^2 + 3 a e^2 h x^2 + e^3 x^3 - * c^3 e^2 y + 3 b c d e^2 y - 3 a d^2 e^2 y + b c^2 e f y - * 2 b^2 d e f y - a c d e f y - a c^2 f^2 y + 2 a b d f^2 y - * b^2 c e g y + 2 a c^2 e g y + a b d e g y + a b c f g y - * 3 a^2 d f g y - a^2 c g^2 y + 2 b^3 e h y - 6 a b c e h y + * 6 a^2 d e h y - 2 a b^2 f h y + 4 a^2 c f h y + 2 a^2 b g h y - * 3 a^3 h^2 y - 3 b c e^2 x y + 6 a d e^2 x y + 2 b^2 e f x y + * a c e f x y - 2 a b f^2 x y - a b e g x y + 3 a^2 f g x y - * 6 a^2 e h x y - 3 a e^2 x^2 y - b^3 e y^2 + 3 a b c e y^2 - * 3 a^2 d e y^2 + a b^2 f y^2 - 2 a^2 c f y^2 - a^2 b g y^2 + * 3 a^3 h y^2 + 3 a^2 e x y^2 - a^3 y^3 */ enum { xxx_coeff, // A xxy_coeff, // B xyy_coeff, // C yyy_coeff, // D xx_coeff, xy_coeff, yy_coeff, x_coeff, y_coeff, c_coeff, coeff_count }; #define USE_SYVESTER 0 // if 0, use control-point base parametric form #if USE_SYVESTER // FIXME: factoring version unwritten // static bool straight_forward = true; /* from CubicParameterizationCode.cpp output: * double A = e * e * e; * double B = -3 * a * e * e; * double C = 3 * a * a * e; * double D = -a * a * a; */ static void calc_ABCD(double a, double e, double p[coeff_count]) { double ee = e * e; p[xxx_coeff] = e * ee; p[xxy_coeff] = -3 * a * ee; double aa = a * a; p[xyy_coeff] = 3 * aa * e; p[yyy_coeff] = -aa * a; } /* CubicParameterizationCode.cpp turns Mathematica output into C. * Rather than edit the lines below, please edit the code there instead. */ // start of generated code static double calc_xx(double a, double b, double c, double d, double e, double f, double g, double h) { return -3 * d * e * e * e + c * e * e * f - b * e * f * f + a * f * f * f + 2 * b * e * e * g - 3 * a * e * f * g + 3 * a * e * e * h; } static double calc_xy(double a, double b, double c, double d, double e, double f, double g, double h) { return -3 * b * c * e * e + 6 * a * d * e * e + 2 * b * b * e * f + a * c * e * f - 2 * a * b * f * f - a * b * e * g + 3 * a * a * f * g - 6 * a * a * e * h; } static double calc_yy(double a, double b, double c, double d, double e, double f, double g, double h) { return -b * b * b * e + 3 * a * b * c * e - 3 * a * a * d * e + a * b * b * f - 2 * a * a * c * f - a * a * b * g + 3 * a * a * a * h; } static double calc_x(double a, double b, double c, double d, double e, double f, double g, double h) { return 3 * d * d * e * e * e - 2 * c * d * e * e * f + 2 * b * d * e * f * f - 2 * a * d * f * f * f + c * c * e * e * g - 4 * b * d * e * e * g - b * c * e * f * g + 6 * a * d * e * f * g + a * c * f * f * g + b * b * e * g * g - 2 * a * c * e * g * g - a * b * f * g * g + a * a * g * g * g + 3 * b * c * e * e * h - 6 * a * d * e * e * h - 2 * b * b * e * f * h - a * c * e * f * h + 2 * a * b * f * f * h + a * b * e * g * h - 3 * a * a * f * g * h + 3 * a * a * e * h * h; } static double calc_y(double a, double b, double c, double d, double e, double f, double g, double h) { return -c * c * c * e * e + 3 * b * c * d * e * e - 3 * a * d * d * e * e + b * c * c * e * f - 2 * b * b * d * e * f - a * c * d * e * f - a * c * c * f * f + 2 * a * b * d * f * f - b * b * c * e * g + 2 * a * c * c * e * g + a * b * d * e * g + a * b * c * f * g - 3 * a * a * d * f * g - a * a * c * g * g + 2 * b * b * b * e * h - 6 * a * b * c * e * h + 6 * a * a * d * e * h - 2 * a * b * b * f * h + 4 * a * a * c * f * h + 2 * a * a * b * g * h - 3 * a * a * a * h * h; } static double calc_c(double a, double b, double c, double d, double e, double f, double g, double h) { return -d * d * d * e * e * e + c * d * d * e * e * f - b * d * d * e * f * f + a * d * d * f * f * f - c * c * d * e * e * g + 2 * b * d * d * e * e * g + b * c * d * e * f * g - 3 * a * d * d * e * f * g - a * c * d * f * f * g - b * b * d * e * g * g + 2 * a * c * d * e * g * g + a * b * d * f * g * g - a * a * d * g * g * g + c * c * c * e * e * h - 3 * b * c * d * e * e * h + 3 * a * d * d * e * e * h - b * c * c * e * f * h + 2 * b * b * d * e * f * h + a * c * d * e * f * h + a * c * c * f * f * h - 2 * a * b * d * f * f * h + b * b * c * e * g * h - 2 * a * c * c * e * g * h - a * b * d * e * g * h - a * b * c * f * g * h + 3 * a * a * d * f * g * h + a * a * c * g * g * h - b * b * b * e * h * h + 3 * a * b * c * e * h * h - 3 * a * a * d * e * h * h + a * b * b * f * h * h - 2 * a * a * c * f * h * h - a * a * b * g * h * h + a * a * a * h * h * h; } // end of generated code #else /* more Mathematica generated code. This takes a different tack, starting with the control-point based parametric formulas. The C code is unoptimized -- in this form, this is a proof of concept (since the other code didn't work) */ static double calc_c(double a, double b, double c, double d, double e, double f, double g, double h) { return d*d*d*e*e*e - 3*d*d*(3*c*e*e*f + 3*b*e*(-3*f*f + 2*e*g) + a*(9*f*f*f - 9*e*f*g + e*e*h)) - h*(27*c*c*c*e*e - 27*c*c*(3*b*e*f - 3*a*f*f + 2*a*e*g) + h*(-27*b*b*b*e + 27*a*b*b*f - 9*a*a*b*g + a*a*a*h) + 9*c*(9*b*b*e*g + a*b*(-9*f*g + 3*e*h) + a*a*(3*g*g - 2*f*h))) + 3*d*(9*c*c*e*e*g + 9*b*b*e*(3*g*g - 2*f*h) + 3*a*b*(-9*f*g*g + 6*f*f*h + e*g*h) + a*a*(9*g*g*g - 9*f*g*h + e*h*h) + 3*c*(3*b*e*(-3*f*g + e*h) + a*(9*f*f*g - 6*e*g*g - e*f*h))) ; } // - Power(e - 3*f + 3*g - h,3)*Power(x,3) static double calc_xxx(double e3f3gh) { return -e3f3gh * e3f3gh * e3f3gh; } static double calc_y(double a, double b, double c, double d, double e, double f, double g, double h) { return + 3*(6*b*d*d*e*e - d*d*d*e*e + 18*b*b*d*e*f - 18*b*d*d*e*f - 9*b*d*d*f*f - 54*b*b*d*e*g + 12*b*d*d*e*g - 27*b*b*d*g*g - 18*b*b*b*e*h + 18*b*b*d*e*h + 18*b*b*d*f*h + a*a*a*h*h - 9*b*b*b*h*h + 9*c*c*c*e*(e + 2*h) + a*a*(-3*b*h*(2*g + h) + d*(-27*g*g + 9*g*h - h*(2*e + h) + 9*f*(g + h))) + a*(9*b*b*h*(2*f + h) - 3*b*d*(6*f*f - 6*f*(3*g - 2*h) + g*(-9*g + h) + e*(g + h)) + d*d*(e*e + 9*f*(3*f - g) + e*(-9*f - 9*g + 2*h))) - 9*c*c*(d*e*(e + 2*g) + 3*b*(f*h + e*(f + h)) + a*(-3*f*f - 6*f*h + 2*(g*h + e*(g + h)))) + 3*c*(d*d*e*(e + 2*f) + a*a*(3*g*g + 6*g*h - 2*h*(2*f + h)) + 9*b*b*(g*h + e*(g + h)) + a*d*(-9*f*f - 18*f*g + 6*g*g + f*h + e*(f + 12*g + h)) + b*(d*(-3*e*e + 9*f*g + e*(9*f + 9*g - 6*h)) + 3*a*(h*(2*e - 3*g + h) - 3*f*(g + h))))) // *y ; } static double calc_yy(double a, double b, double c, double d, double e, double f, double g, double h) { return - 3*(18*c*c*c*e - 18*c*c*d*e + 6*c*d*d*e - d*d*d*e + 3*c*d*d*f - 9*c*c*d*g + a*a*a*h + 9*c*c*c*h - 9*b*b*b*(e + 2*h) - a*a*(d*(e - 9*f + 18*g - 7*h) + 3*c*(2*f - 6*g + h)) + a*(-9*c*c*(2*e - 6*f + 2*g - h) + d*d*(-7*e + 18*f - 9*g + h) + 3*c*d*(7*e - 17*f + 3*g + h)) + 9*b*b*(3*c*(e + g + h) + a*(f + 2*h) - d*(e - 2*(f - 3*g + h))) - 3*b*(-(d*d*(e - 6*f + 2*g)) - 3*c*d*(e + 3*f + 3*g - h) + 9*c*c*(e + f + h) + a*a*(g + 2*h) + a*(c*(-3*e + 9*f + 9*g + 3*h) + d*(e + 3*f - 17*g + 7*h)))) // *Power(y,2) ; } // + Power(a - 3*b + 3*c - d,3)*Power(y,3) static double calc_yyy(double a3b3cd) { return a3b3cd * a3b3cd * a3b3cd; } static double calc_xx(double a, double b, double c, double d, double e, double f, double g, double h) { return // + Power(x,2)* (-3*(-9*b*e*f*f + 9*a*f*f*f + 6*b*e*e*g - 9*a*e*f*g + 27*b*e*f*g - 27*a*f*f*g + 18*a*e*g*g - 54*b*e*g*g + 27*a*f*g*g + 27*b*f*g*g - 18*a*g*g*g + a*e*e*h - 9*b*e*e*h + 3*a*e*f*h + 9*b*e*f*h + 9*a*f*f*h - 18*b*f*f*h - 21*a*e*g*h + 51*b*e*g*h - 9*a*f*g*h - 27*b*f*g*h + 18*a*g*g*h + 7*a*e*h*h - 18*b*e*h*h - 3*a*f*h*h + 18*b*f*h*h - 6*a*g*h*h - 3*b*g*h*h + a*h*h*h + 3*c*(-9*f*f*(g - 2*h) + 3*g*g*h - f*h*(9*g + 2*h) + e*e*(f - 6*g + 6*h) + e*(9*f*g + 6*g*g - 17*f*h - 3*g*h + 3*h*h)) - d*(e*e*e + e*e*(-6*f - 3*g + 7*h) - 9*(2*f - g)*(f*f + g*g - f*(g + h)) + e*(18*f*f + 9*g*g + 3*g*h + h*h - 3*f*(3*g + 7*h)))) ) ; } // + Power(x,2)*(3*(a - 3*b + 3*c - d)*Power(e - 3*f + 3*g - h,2)*y) static double calc_xxy(double a3b3cd, double e3f3gh) { return 3 * a3b3cd * e3f3gh * e3f3gh; } static double calc_x(double a, double b, double c, double d, double e, double f, double g, double h) { return // + x* (-3*(27*b*b*e*g*g - 27*a*b*f*g*g + 9*a*a*g*g*g - 18*b*b*e*f*h + 18*a*b*f*f*h + 3*a*b*e*g*h - 27*b*b*e*g*h - 9*a*a*f*g*h + 27*a*b*f*g*h - 9*a*a*g*g*h + a*a*e*h*h - 9*a*b*e*h*h + 27*b*b*e*h*h + 6*a*a*f*h*h - 18*a*b*f*h*h - 9*b*b*f*h*h + 3*a*a*g*h*h + 6*a*b*g*h*h - a*a*h*h*h + 9*c*c*(e*e*(g - 3*h) - 3*f*f*h + e*(3*f + 2*g)*h) + d*d*(e*e*e - 9*f*f*f + 9*e*f*(f + g) - e*e*(3*f + 6*g + h)) + d*(-3*c*(-9*f*f*g + e*e*(2*f - 6*g - 3*h) + e*(9*f*g + 6*g*g + f*h)) + a*(-18*f*f*f - 18*e*g*g + 18*g*g*g - 2*e*e*h + 3*e*g*h + 2*e*h*h + 9*f*f*(3*g + 2*h) + 3*f*(6*e*g - 9*g*g - e*h - 6*g*h)) - 3*b*(9*f*g*g + e*e*(4*g - 3*h) - 6*f*f*h - e*(6*f*f + g*(18*g + h) - 3*f*(3*g + 4*h)))) + 3*c*(3*b*(e*e*h + 3*f*g*h - e*(3*f*g - 6*f*h + 6*g*h + h*h)) + a*(9*f*f*(g - 2*h) + f*h*(-e + 9*g + 4*h) - 3*(2*g*g*h + e*(2*g*g - 4*g*h + h*h))))) ) ; } static double calc_xy(double a, double b, double c, double d, double e, double f, double g, double h) { return // + x*3* (-2*a*d*e*e - 7*d*d*e*e + 15*a*d*e*f + 21*d*d*e*f - 9*a*d*f*f - 18*d*d*f*f - 15*a*d*e*g - 3*d*d*e*g - 9*a*a*f*g + 9*d*d*f*g + 18*a*a*g*g + 9*a*d*g*g + 2*a*a*e*h - 2*d*d*e*h + 3*a*a*f*h + 15*a*d*f*h - 21*a*a*g*h - 15*a*d*g*h + 7*a*a*h*h + 2*a*d*h*h - 9*c*c*(2*e*e + 3*f*f + 3*f*h - 2*g*h + e*(-3*f - 4*g + h)) + 9*b*b*(3*g*g - 3*g*h + 2*h*(-2*f + h) + e*(-2*f + 3*g + h)) + 3*b*(3*c*(e*e + 3*e*(f - 3*g) + (9*f - 3*g - h)*h) + a*(6*f*f + e*g - 9*f*g - 9*g*g - 5*e*h + 9*f*h + 14*g*h - 7*h*h) + d*(-e*e + 12*f*f - 27*f*g + e*(-9*f + 20*g - 5*h) + g*(9*g + h))) + 3*c*(a*(-(e*f) - 9*f*f + 27*f*g - 12*g*g + 5*e*h - 20*f*h + 9*g*h + h*h) + d*(7*e*e + 9*f*f + 9*f*g - 6*g*g - f*h + e*(-14*f - 9*g + 5*h)))) // *y ; } // - x*3*Power(a - 3*b + 3*c - d,2)*(e - 3*f + 3*g - h)*Power(y,2) static double calc_xyy(double a3b3cd, double e3f3gh) { return -3 * a3b3cd * a3b3cd * e3f3gh; } #endif static double (*calc_proc[])(double a, double b, double c, double d, double e, double f, double g, double h) = { calc_xx, calc_xy, calc_yy, calc_x, calc_y, calc_c }; #if USE_SYVESTER /* Control points to parametric coefficients s = 1 - t Attt + 3Btts + 3Ctss + Dsss == Attt + 3B(1 - t)tt + 3C(1 - t)(t - tt) + D(1 - t)(1 - 2t + tt) == Attt + 3B(tt - ttt) + 3C(t - tt - tt + ttt) + D(1-2t+tt-t+2tt-ttt) == Attt + 3Btt - 3Bttt + 3Ct - 6Ctt + 3Cttt + D - 3Dt + 3Dtt - Dttt == D + (3C - 3D)t + (3B - 6C + 3D)tt + (A - 3B + 3C - D)ttt a = A - 3*B + 3*C - D b = 3*B - 6*C + 3*D c = 3*C - 3*D d = D */ /* http://www.algorithmist.net/bezier3.html p = 3 * A q = 3 * B r = 3 * C a = A b = q - p c = p - 2 * q + r d = D - A + q - r B(t) = a + t * (b + t * (c + t * d)) so B(t) = a + t*b + t*t*(c + t*d) = a + t*b + t*t*c + t*t*t*d */ static void set_abcd(const double* cubic, double& a, double& b, double& c, double& d) { a = cubic[0]; // a = A b = 3 * cubic[2]; // b = 3*B (compute rest of b lazily) c = 3 * cubic[4]; // c = 3*C (compute rest of c lazily) d = cubic[6]; // d = D a += -b + c - d; // a = A - 3*B + 3*C - D } static void calc_bc(const double d, double& b, double& c) { b -= 3 * c; // b = 3*B - 3*C c -= 3 * d; // c = 3*C - 3*D b -= c; // b = 3*B - 6*C + 3*D } static void alt_set_abcd(const double* cubic, double& a, double& b, double& c, double& d) { a = cubic[0]; double p = 3 * a; double q = 3 * cubic[2]; double r = 3 * cubic[4]; b = q - p; c = p - 2 * q + r; d = cubic[6] - a + q - r; } const bool try_alt = true; #else static void calc_ABCD(double a, double b, double c, double d, double e, double f, double g, double h, double p[coeff_count]) { double a3b3cd = a - 3 * (b - c) - d; double e3f3gh = e - 3 * (f - g) - h; p[xxx_coeff] = calc_xxx(e3f3gh); p[xxy_coeff] = calc_xxy(a3b3cd, e3f3gh); p[xyy_coeff] = calc_xyy(a3b3cd, e3f3gh); p[yyy_coeff] = calc_yyy(a3b3cd); } #endif bool implicit_matches(const Cubic& one, const Cubic& two) { double p1[coeff_count]; // a'xxx , b'xxy , c'xyy , d'xx , e'xy , f'yy, etc. double p2[coeff_count]; #if USE_SYVESTER double a1, b1, c1, d1; if (try_alt) alt_set_abcd(&one[0].x, a1, b1, c1, d1); else set_abcd(&one[0].x, a1, b1, c1, d1); double e1, f1, g1, h1; if (try_alt) alt_set_abcd(&one[0].y, e1, f1, g1, h1); else set_abcd(&one[0].y, e1, f1, g1, h1); calc_ABCD(a1, e1, p1); double a2, b2, c2, d2; if (try_alt) alt_set_abcd(&two[0].x, a2, b2, c2, d2); else set_abcd(&two[0].x, a2, b2, c2, d2); double e2, f2, g2, h2; if (try_alt) alt_set_abcd(&two[0].y, e2, f2, g2, h2); else set_abcd(&two[0].y, e2, f2, g2, h2); calc_ABCD(a2, e2, p2); #else double a1 = one[0].x; double b1 = one[1].x; double c1 = one[2].x; double d1 = one[3].x; double e1 = one[0].y; double f1 = one[1].y; double g1 = one[2].y; double h1 = one[3].y; calc_ABCD(a1, b1, c1, d1, e1, f1, g1, h1, p1); double a2 = two[0].x; double b2 = two[1].x; double c2 = two[2].x; double d2 = two[3].x; double e2 = two[0].y; double f2 = two[1].y; double g2 = two[2].y; double h2 = two[3].y; calc_ABCD(a2, b2, c2, d2, e2, f2, g2, h2, p2); #endif int first = 0; for (int index = 0; index < coeff_count; ++index) { #if USE_SYVESTER if (!try_alt && index == xx_coeff) { calc_bc(d1, b1, c1); calc_bc(h1, f1, g1); calc_bc(d2, b2, c2); calc_bc(h2, f2, g2); } #endif if (index >= xx_coeff) { int procIndex = index - xx_coeff; p1[index] = (*calc_proc[procIndex])(a1, b1, c1, d1, e1, f1, g1, h1); p2[index] = (*calc_proc[procIndex])(a2, b2, c2, d2, e2, f2, g2, h2); } if (approximately_zero(p1[index]) || approximately_zero(p2[index])) { first += first == index; continue; } if (first == index) { continue; } if (!AlmostEqualUlps(p1[index] * p2[first], p1[first] * p2[index])) { return false; } } return true; } static double tangent(const double* cubic, double t) { double a, b, c, d; #if USE_SYVESTER set_abcd(cubic, a, b, c, d); calc_bc(d, b, c); #else coefficients(cubic, a, b, c, d); #endif return 3 * a * t * t + 2 * b * t + c; } void tangent(const Cubic& cubic, double t, _Point& result) { result.x = tangent(&cubic[0].x, t); result.y = tangent(&cubic[0].y, t); } // unit test to return and validate parametric coefficients #include "CubicParameterization_TestUtility.cpp"