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
64 enum {
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 */
calc_ABCD(double a,double e,double p[coeff_count])90 static 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
calc_xx(double a,double b,double c,double d,double e,double f,double g,double h)103 static 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
calc_xy(double a,double b,double c,double d,double e,double f,double g,double h)115 static 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
calc_yy(double a,double b,double c,double d,double e,double f,double g,double h)128 static 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
calc_x(double a,double b,double c,double d,double e,double f,double g,double h)140 static 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
calc_y(double a,double b,double c,double d,double e,double f,double g,double h)166 static 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
calc_c(double a,double b,double c,double d,double e,double f,double g,double h)192 static 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 */
calc_c(double a,double b,double c,double d,double e,double f,double g,double h)238 static double calc_c(double a, double b, double c, double d,
239 double e, double f, double g, double h) {
240 return
241 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)) -
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)
calc_xxx(double e3f3gh)251 static double calc_xxx(double e3f3gh) {
252 return -e3f3gh * e3f3gh * e3f3gh;
253 }
254
calc_y(double a,double b,double c,double d,double e,double f,double g,double h)255 static 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
calc_yy(double a,double b,double c,double d,double e,double f,double g,double h)271 static 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)
calc_yyy(double a3b3cd)284 static double calc_yyy(double a3b3cd) {
285 return a3b3cd * a3b3cd * a3b3cd;
286 }
287
calc_xx(double a,double b,double c,double d,double e,double f,double g,double h)288 static 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)
calc_xxy(double a3b3cd,double e3f3gh)304 static double calc_xxy(double a3b3cd, double e3f3gh) {
305 return 3 * a3b3cd * e3f3gh * e3f3gh;
306 }
307
calc_x(double a,double b,double c,double d,double e,double f,double g,double h)308 static 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
calc_xy(double a,double b,double c,double d,double e,double f,double g,double h)326 static 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)
calc_xyy(double a3b3cd,double e3f3gh)343 static double calc_xyy(double a3b3cd, double e3f3gh) {
344 return -3 * a3b3cd * a3b3cd * e3f3gh;
345 }
346
347 #endif
348
349 static 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 */
set_abcd(const double * cubic,double & a,double & b,double & c,double & d)384 static 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
calc_bc(const double d,double & b,double & c)393 static 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
alt_set_abcd(const double * cubic,double & a,double & b,double & c,double & d)399 static 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
410 const bool try_alt = true;
411
412 #else
413
calc_ABCD(double a,double b,double c,double d,double e,double f,double g,double h,double p[coeff_count])414 static 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
implicit_matches(const Cubic & one,const Cubic & two)426 bool 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
tangent(const double * cubic,double t)501 static 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
tangent(const Cubic & cubic,double t,_Point & result)512 void 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