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 "CubicUtilities.h" 9 10/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1 11 * 12 * This paper proves that Syvester's method can compute the implicit form of 13 * the quadratic from the parameterzied form. 14 * 15 * Given x = a*t*t*t + b*t*t + c*t + d (the parameterized form) 16 * y = e*t*t*t + f*t*t + g*t + h 17 * 18 * we want to find an equation of the implicit form: 19 * 20 * 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 21 * 22 * The implicit form can be expressed as a 6x6 determinant, as shown. 23 * 24 * The resultant obtained by Syvester's method is 25 * 26 * | a b c (d - x) 0 0 | 27 * | 0 a b c (d - x) 0 | 28 * | 0 0 a b c (d - x) | 29 * | e f g (h - y) 0 0 | 30 * | 0 e f g (h - y) 0 | 31 * | 0 0 e f g (h - y) | 32 * 33 * which, according to Mathematica, expands as shown below. 34 * 35 * Resultant[a*t^3 + b*t^2 + c*t + d - x, e*t^3 + f*t^2 + g*t + h - y, t] 36 * 37 * -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 + 38 * 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 - 39 * 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 - 40 * 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 + 41 * 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 - 42 * 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 + 43 * 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 + 44 * 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 - 45 * 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 - 46 * 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 + 47 * b^2 e g^2 x - 2 a c e g^2 x - a b f g^2 x + a^2 g^3 x + 48 * 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 + 49 * 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 - 50 * 3 d e^3 x^2 + c e^2 f x^2 - b e f^2 x^2 + a f^3 x^2 + 51 * 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 - 52 * 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 - 53 * 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 - 54 * 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 - 55 * 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 + 56 * 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 - 57 * 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 + 58 * 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 - 59 * 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 - 60 * 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 + 61 * 3 a^3 h y^2 + 3 a^2 e x y^2 - a^3 y^3 62 */ 63 64enum { 65 xxx_coeff, // A 66 xxy_coeff, // B 67 xyy_coeff, // C 68 yyy_coeff, // D 69 xx_coeff, 70 xy_coeff, 71 yy_coeff, 72 x_coeff, 73 y_coeff, 74 c_coeff, 75 coeff_count 76}; 77 78#define USE_SYVESTER 0 // if 0, use control-point base parametric form 79#if USE_SYVESTER 80 81// FIXME: factoring version unwritten 82// static bool straight_forward = true; 83 84/* from CubicParameterizationCode.cpp output: 85 * double A = e * e * e; 86 * double B = -3 * a * e * e; 87 * double C = 3 * a * a * e; 88 * double D = -a * a * a; 89 */ 90static void calc_ABCD(double a, double e, double p[coeff_count]) { 91 double ee = e * e; 92 p[xxx_coeff] = e * ee; 93 p[xxy_coeff] = -3 * a * ee; 94 double aa = a * a; 95 p[xyy_coeff] = 3 * aa * e; 96 p[yyy_coeff] = -aa * a; 97} 98 99/* CubicParameterizationCode.cpp turns Mathematica output into C. 100 * Rather than edit the lines below, please edit the code there instead. 101 */ 102// start of generated code 103static double calc_xx(double a, double b, double c, double d, 104 double e, double f, double g, double h) { 105 return 106 -3 * d * e * e * e 107 + c * e * e * f 108 - b * e * f * f 109 + a * f * f * f 110 + 2 * b * e * e * g 111 - 3 * a * e * f * g 112 + 3 * a * e * e * h; 113} 114 115static double calc_xy(double a, double b, double c, double d, 116 double e, double f, double g, double h) { 117 return 118 -3 * b * c * e * e 119 + 6 * a * d * e * e 120 + 2 * b * b * e * f 121 + a * c * e * f 122 - 2 * a * b * f * f 123 - a * b * e * g 124 + 3 * a * a * f * g 125 - 6 * a * a * e * h; 126} 127 128static double calc_yy(double a, double b, double c, double d, 129 double e, double f, double g, double h) { 130 return 131 -b * b * b * e 132 + 3 * a * b * c * e 133 - 3 * a * a * d * e 134 + a * b * b * f 135 - 2 * a * a * c * f 136 - a * a * b * g 137 + 3 * a * a * a * h; 138} 139 140static double calc_x(double a, double b, double c, double d, 141 double e, double f, double g, double h) { 142 return 143 3 * d * d * e * e * e 144 - 2 * c * d * e * e * f 145 + 2 * b * d * e * f * f 146 - 2 * a * d * f * f * f 147 + c * c * e * e * g 148 - 4 * b * d * e * e * g 149 - b * c * e * f * g 150 + 6 * a * d * e * f * g 151 + a * c * f * f * g 152 + b * b * e * g * g 153 - 2 * a * c * e * g * g 154 - a * b * f * g * g 155 + a * a * g * g * g 156 + 3 * b * c * e * e * h 157 - 6 * a * d * e * e * h 158 - 2 * b * b * e * f * h 159 - a * c * e * f * h 160 + 2 * a * b * f * f * h 161 + a * b * e * g * h 162 - 3 * a * a * f * g * h 163 + 3 * a * a * e * h * h; 164} 165 166static double calc_y(double a, double b, double c, double d, 167 double e, double f, double g, double h) { 168 return 169 -c * c * c * e * e 170 + 3 * b * c * d * e * e 171 - 3 * a * d * d * e * e 172 + b * c * c * e * f 173 - 2 * b * b * d * e * f 174 - a * c * d * e * f 175 - a * c * c * f * f 176 + 2 * a * b * d * f * f 177 - b * b * c * e * g 178 + 2 * a * c * c * e * g 179 + a * b * d * e * g 180 + a * b * c * f * g 181 - 3 * a * a * d * f * g 182 - a * a * c * g * g 183 + 2 * b * b * b * e * h 184 - 6 * a * b * c * e * h 185 + 6 * a * a * d * e * h 186 - 2 * a * b * b * f * h 187 + 4 * a * a * c * f * h 188 + 2 * a * a * b * g * h 189 - 3 * a * a * a * h * h; 190} 191 192static double calc_c(double a, double b, double c, double d, 193 double e, double f, double g, double h) { 194 return 195 -d * d * d * e * e * e 196 + c * d * d * e * e * f 197 - b * d * d * e * f * f 198 + a * d * d * f * f * f 199 - c * c * d * e * e * g 200 + 2 * b * d * d * e * e * g 201 + b * c * d * e * f * g 202 - 3 * a * d * d * e * f * g 203 - a * c * d * f * f * g 204 - b * b * d * e * g * g 205 + 2 * a * c * d * e * g * g 206 + a * b * d * f * g * g 207 - a * a * d * g * g * g 208 + c * c * c * e * e * h 209 - 3 * b * c * d * e * e * h 210 + 3 * a * d * d * e * e * h 211 - b * c * c * e * f * h 212 + 2 * b * b * d * e * f * h 213 + a * c * d * e * f * h 214 + a * c * c * f * f * h 215 - 2 * a * b * d * f * f * h 216 + b * b * c * e * g * h 217 - 2 * a * c * c * e * g * h 218 - a * b * d * e * g * h 219 - a * b * c * f * g * h 220 + 3 * a * a * d * f * g * h 221 + a * a * c * g * g * h 222 - b * b * b * e * h * h 223 + 3 * a * b * c * e * h * h 224 - 3 * a * a * d * e * h * h 225 + a * b * b * f * h * h 226 - 2 * a * a * c * f * h * h 227 - a * a * b * g * h * h 228 + a * a * a * h * h * h; 229} 230// end of generated code 231 232#else 233 234/* more Mathematica generated code. This takes a different tack, starting with 235 the control-point based parametric formulas. The C code is unoptimized -- 236 in this form, this is a proof of concept (since the other code didn't work) 237*/ 238static double calc_c(double a, double b, double c, double d, 239 double e, double f, double g, double h) { 240 return 241d*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)) - 242 h*(27*c*c*c*e*e - 27*c*c*(3*b*e*f - 3*a*f*f + 2*a*e*g) + 243 h*(-27*b*b*b*e + 27*a*b*b*f - 9*a*a*b*g + a*a*a*h) + 244 9*c*(9*b*b*e*g + a*b*(-9*f*g + 3*e*h) + a*a*(3*g*g - 2*f*h))) + 245 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) + 246 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))) 247 ; 248} 249 250// - Power(e - 3*f + 3*g - h,3)*Power(x,3) 251static double calc_xxx(double e3f3gh) { 252 return -e3f3gh * e3f3gh * e3f3gh; 253} 254 255static double calc_y(double a, double b, double c, double d, 256 double e, double f, double g, double h) { 257 return 258+ 3*(6*b*d*d*e*e - d*d*d*e*e + 18*b*b*d*e*f - 18*b*d*d*e*f - 259 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 + 260 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) + 261 a*a*(-3*b*h*(2*g + h) + d*(-27*g*g + 9*g*h - h*(2*e + h) + 9*f*(g + h))) + 262 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)) + 263 d*d*(e*e + 9*f*(3*f - g) + e*(-9*f - 9*g + 2*h))) - 264 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)))) + 265 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)) + 266 a*d*(-9*f*f - 18*f*g + 6*g*g + f*h + e*(f + 12*g + h)) + 267 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 268 ; 269} 270 271static double calc_yy(double a, double b, double c, double d, 272 double e, double f, double g, double h) { 273 return 274- 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 - 275 9*b*b*b*(e + 2*h) - a*a*(d*(e - 9*f + 18*g - 7*h) + 3*c*(2*f - 6*g + h)) + 276 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)) + 277 9*b*b*(3*c*(e + g + h) + a*(f + 2*h) - d*(e - 2*(f - 3*g + h))) - 278 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) + 279 a*(c*(-3*e + 9*f + 9*g + 3*h) + d*(e + 3*f - 17*g + 7*h)))) // *Power(y,2) 280 ; 281} 282 283// + Power(a - 3*b + 3*c - d,3)*Power(y,3) 284static double calc_yyy(double a3b3cd) { 285 return a3b3cd * a3b3cd * a3b3cd; 286} 287 288static double calc_xx(double a, double b, double c, double d, 289 double e, double f, double g, double h) { 290 return 291// + Power(x,2)* 292(-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 + 293 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 - 294 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 + 295 18*b*f*h*h - 6*a*g*h*h - 3*b*g*h*h + a*h*h*h + 296 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) + 297 e*(9*f*g + 6*g*g - 17*f*h - 3*g*h + 3*h*h)) - 298 d*(e*e*e + e*e*(-6*f - 3*g + 7*h) - 9*(2*f - g)*(f*f + g*g - f*(g + h)) + 299 e*(18*f*f + 9*g*g + 3*g*h + h*h - 3*f*(3*g + 7*h)))) ) 300 ; 301} 302 303// + Power(x,2)*(3*(a - 3*b + 3*c - d)*Power(e - 3*f + 3*g - h,2)*y) 304static double calc_xxy(double a3b3cd, double e3f3gh) { 305 return 3 * a3b3cd * e3f3gh * e3f3gh; 306} 307 308static double calc_x(double a, double b, double c, double d, 309 double e, double f, double g, double h) { 310 return 311// + x* 312(-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 - 313 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 + 314 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 + 315 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) + 316 d*d*(e*e*e - 9*f*f*f + 9*e*f*(f + g) - e*e*(3*f + 6*g + h)) + 317 d*(-3*c*(-9*f*f*g + e*e*(2*f - 6*g - 3*h) + e*(9*f*g + 6*g*g + f*h)) + 318 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) + 319 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 - 320 e*(6*f*f + g*(18*g + h) - 3*f*(3*g + 4*h)))) + 321 3*c*(3*b*(e*e*h + 3*f*g*h - e*(3*f*g - 6*f*h + 6*g*h + h*h)) + 322 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))))) ) 323 ; 324} 325 326static double calc_xy(double a, double b, double c, double d, 327 double e, double f, double g, double h) { 328 return 329// + x*3* 330(-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 - 331 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 + 332 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 - 333 9*c*c*(2*e*e + 3*f*f + 3*f*h - 2*g*h + e*(-3*f - 4*g + h)) + 334 9*b*b*(3*g*g - 3*g*h + 2*h*(-2*f + h) + e*(-2*f + 3*g + h)) + 335 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) + 336 d*(-e*e + 12*f*f - 27*f*g + e*(-9*f + 20*g - 5*h) + g*(9*g + h))) + 337 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) + 338 d*(7*e*e + 9*f*f + 9*f*g - 6*g*g - f*h + e*(-14*f - 9*g + 5*h)))) // *y 339 ; 340} 341 342// - x*3*Power(a - 3*b + 3*c - d,2)*(e - 3*f + 3*g - h)*Power(y,2) 343static double calc_xyy(double a3b3cd, double e3f3gh) { 344 return -3 * a3b3cd * a3b3cd * e3f3gh; 345} 346 347#endif 348 349static double (*calc_proc[])(double a, double b, double c, double d, 350 double e, double f, double g, double h) = { 351 calc_xx, calc_xy, calc_yy, calc_x, calc_y, calc_c 352}; 353 354#if USE_SYVESTER 355/* Control points to parametric coefficients 356 s = 1 - t 357 Attt + 3Btts + 3Ctss + Dsss == 358 Attt + 3B(1 - t)tt + 3C(1 - t)(t - tt) + D(1 - t)(1 - 2t + tt) == 359 Attt + 3B(tt - ttt) + 3C(t - tt - tt + ttt) + D(1-2t+tt-t+2tt-ttt) == 360 Attt + 3Btt - 3Bttt + 3Ct - 6Ctt + 3Cttt + D - 3Dt + 3Dtt - Dttt == 361 D + (3C - 3D)t + (3B - 6C + 3D)tt + (A - 3B + 3C - D)ttt 362 a = A - 3*B + 3*C - D 363 b = 3*B - 6*C + 3*D 364 c = 3*C - 3*D 365 d = D 366 */ 367 368 /* http://www.algorithmist.net/bezier3.html 369 p = 3 * A 370 q = 3 * B 371 r = 3 * C 372 a = A 373 b = q - p 374 c = p - 2 * q + r 375 d = D - A + q - r 376 377 B(t) = a + t * (b + t * (c + t * d)) 378 379 so 380 381 B(t) = a + t*b + t*t*(c + t*d) 382 = a + t*b + t*t*c + t*t*t*d 383 */ 384static void set_abcd(const double* cubic, double& a, double& b, double& c, 385 double& d) { 386 a = cubic[0]; // a = A 387 b = 3 * cubic[2]; // b = 3*B (compute rest of b lazily) 388 c = 3 * cubic[4]; // c = 3*C (compute rest of c lazily) 389 d = cubic[6]; // d = D 390 a += -b + c - d; // a = A - 3*B + 3*C - D 391} 392 393static void calc_bc(const double d, double& b, double& c) { 394 b -= 3 * c; // b = 3*B - 3*C 395 c -= 3 * d; // c = 3*C - 3*D 396 b -= c; // b = 3*B - 6*C + 3*D 397} 398 399static void alt_set_abcd(const double* cubic, double& a, double& b, double& c, 400 double& d) { 401 a = cubic[0]; 402 double p = 3 * a; 403 double q = 3 * cubic[2]; 404 double r = 3 * cubic[4]; 405 b = q - p; 406 c = p - 2 * q + r; 407 d = cubic[6] - a + q - r; 408} 409 410const bool try_alt = true; 411 412#else 413 414static void calc_ABCD(double a, double b, double c, double d, 415 double e, double f, double g, double h, 416 double p[coeff_count]) { 417 double a3b3cd = a - 3 * (b - c) - d; 418 double e3f3gh = e - 3 * (f - g) - h; 419 p[xxx_coeff] = calc_xxx(e3f3gh); 420 p[xxy_coeff] = calc_xxy(a3b3cd, e3f3gh); 421 p[xyy_coeff] = calc_xyy(a3b3cd, e3f3gh); 422 p[yyy_coeff] = calc_yyy(a3b3cd); 423} 424#endif 425 426bool implicit_matches(const Cubic& one, const Cubic& two) { 427 double p1[coeff_count]; // a'xxx , b'xxy , c'xyy , d'xx , e'xy , f'yy, etc. 428 double p2[coeff_count]; 429#if USE_SYVESTER 430 double a1, b1, c1, d1; 431 if (try_alt) 432 alt_set_abcd(&one[0].x, a1, b1, c1, d1); 433 else 434 set_abcd(&one[0].x, a1, b1, c1, d1); 435 double e1, f1, g1, h1; 436 if (try_alt) 437 alt_set_abcd(&one[0].y, e1, f1, g1, h1); 438 else 439 set_abcd(&one[0].y, e1, f1, g1, h1); 440 calc_ABCD(a1, e1, p1); 441 double a2, b2, c2, d2; 442 if (try_alt) 443 alt_set_abcd(&two[0].x, a2, b2, c2, d2); 444 else 445 set_abcd(&two[0].x, a2, b2, c2, d2); 446 double e2, f2, g2, h2; 447 if (try_alt) 448 alt_set_abcd(&two[0].y, e2, f2, g2, h2); 449 else 450 set_abcd(&two[0].y, e2, f2, g2, h2); 451 calc_ABCD(a2, e2, p2); 452#else 453 double a1 = one[0].x; 454 double b1 = one[1].x; 455 double c1 = one[2].x; 456 double d1 = one[3].x; 457 double e1 = one[0].y; 458 double f1 = one[1].y; 459 double g1 = one[2].y; 460 double h1 = one[3].y; 461 calc_ABCD(a1, b1, c1, d1, e1, f1, g1, h1, p1); 462 double a2 = two[0].x; 463 double b2 = two[1].x; 464 double c2 = two[2].x; 465 double d2 = two[3].x; 466 double e2 = two[0].y; 467 double f2 = two[1].y; 468 double g2 = two[2].y; 469 double h2 = two[3].y; 470 calc_ABCD(a2, b2, c2, d2, e2, f2, g2, h2, p2); 471#endif 472 int first = 0; 473 for (int index = 0; index < coeff_count; ++index) { 474#if USE_SYVESTER 475 if (!try_alt && index == xx_coeff) { 476 calc_bc(d1, b1, c1); 477 calc_bc(h1, f1, g1); 478 calc_bc(d2, b2, c2); 479 calc_bc(h2, f2, g2); 480 } 481#endif 482 if (index >= xx_coeff) { 483 int procIndex = index - xx_coeff; 484 p1[index] = (*calc_proc[procIndex])(a1, b1, c1, d1, e1, f1, g1, h1); 485 p2[index] = (*calc_proc[procIndex])(a2, b2, c2, d2, e2, f2, g2, h2); 486 } 487 if (approximately_zero(p1[index]) || approximately_zero(p2[index])) { 488 first += first == index; 489 continue; 490 } 491 if (first == index) { 492 continue; 493 } 494 if (!AlmostEqualUlps(p1[index] * p2[first], p1[first] * p2[index])) { 495 return false; 496 } 497 } 498 return true; 499} 500 501static double tangent(const double* cubic, double t) { 502 double a, b, c, d; 503#if USE_SYVESTER 504 set_abcd(cubic, a, b, c, d); 505 calc_bc(d, b, c); 506#else 507 coefficients(cubic, a, b, c, d); 508#endif 509 return 3 * a * t * t + 2 * b * t + c; 510} 511 512void tangent(const Cubic& cubic, double t, _Point& result) { 513 result.x = tangent(&cubic[0].x, t); 514 result.y = tangent(&cubic[0].y, t); 515} 516 517// unit test to return and validate parametric coefficients 518#include "CubicParameterization_TestUtility.cpp" 519