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| 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 "CubicUtilities.h" | |
| 8 #include "Extrema.h" | |
| 9 #include "LineUtilities.h" | |
| 10 #include "QuadraticUtilities.h" | |
| 11 | |
| 12 const int gPrecisionUnit = 256; // FIXME: arbitrary -- should try different valu
es in test framework | |
| 13 | |
| 14 // FIXME: cache keep the bounds and/or precision with the caller? | |
| 15 double calcPrecision(const Cubic& cubic) { | |
| 16 _Rect dRect; | |
| 17 dRect.setBounds(cubic); // OPTIMIZATION: just use setRawBounds ? | |
| 18 double width = dRect.right - dRect.left; | |
| 19 double height = dRect.bottom - dRect.top; | |
| 20 return (width > height ? width : height) / gPrecisionUnit; | |
| 21 } | |
| 22 | |
| 23 #ifdef SK_DEBUG | |
| 24 double calcPrecision(const Cubic& cubic, double t, double scale) { | |
| 25 Cubic part; | |
| 26 sub_divide(cubic, SkTMax(0., t - scale), SkTMin(1., t + scale), part); | |
| 27 return calcPrecision(part); | |
| 28 } | |
| 29 #endif | |
| 30 | |
| 31 bool clockwise(const Cubic& c) { | |
| 32 double sum = (c[0].x - c[3].x) * (c[0].y + c[3].y); | |
| 33 for (int idx = 0; idx < 3; ++idx){ | |
| 34 sum += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); | |
| 35 } | |
| 36 return sum <= 0; | |
| 37 } | |
| 38 | |
| 39 void coefficients(const double* cubic, double& A, double& B, double& C, double&
D) { | |
| 40 A = cubic[6]; // d | |
| 41 B = cubic[4] * 3; // 3*c | |
| 42 C = cubic[2] * 3; // 3*b | |
| 43 D = cubic[0]; // a | |
| 44 A -= D - C + B; // A = -a + 3*b - 3*c + d | |
| 45 B += 3 * D - 2 * C; // B = 3*a - 6*b + 3*c | |
| 46 C -= 3 * D; // C = -3*a + 3*b | |
| 47 } | |
| 48 | |
| 49 bool controls_contained_by_ends(const Cubic& c) { | |
| 50 _Vector startTan = c[1] - c[0]; | |
| 51 if (startTan.x == 0 && startTan.y == 0) { | |
| 52 startTan = c[2] - c[0]; | |
| 53 } | |
| 54 _Vector endTan = c[2] - c[3]; | |
| 55 if (endTan.x == 0 && endTan.y == 0) { | |
| 56 endTan = c[1] - c[3]; | |
| 57 } | |
| 58 if (startTan.dot(endTan) >= 0) { | |
| 59 return false; | |
| 60 } | |
| 61 _Line startEdge = {c[0], c[0]}; | |
| 62 startEdge[1].x -= startTan.y; | |
| 63 startEdge[1].y += startTan.x; | |
| 64 _Line endEdge = {c[3], c[3]}; | |
| 65 endEdge[1].x -= endTan.y; | |
| 66 endEdge[1].y += endTan.x; | |
| 67 double leftStart1 = is_left(startEdge, c[1]); | |
| 68 if (leftStart1 * is_left(startEdge, c[2]) < 0) { | |
| 69 return false; | |
| 70 } | |
| 71 double leftEnd1 = is_left(endEdge, c[1]); | |
| 72 if (leftEnd1 * is_left(endEdge, c[2]) < 0) { | |
| 73 return false; | |
| 74 } | |
| 75 return leftStart1 * leftEnd1 >= 0; | |
| 76 } | |
| 77 | |
| 78 bool ends_are_extrema_in_x_or_y(const Cubic& c) { | |
| 79 return (between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x)) | |
| 80 || (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].
y)); | |
| 81 } | |
| 82 | |
| 83 bool monotonic_in_y(const Cubic& c) { | |
| 84 return between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y); | |
| 85 } | |
| 86 | |
| 87 bool serpentine(const Cubic& c) { | |
| 88 if (!controls_contained_by_ends(c)) { | |
| 89 return false; | |
| 90 } | |
| 91 double wiggle = (c[0].x - c[2].x) * (c[0].y + c[2].y); | |
| 92 for (int idx = 0; idx < 2; ++idx){ | |
| 93 wiggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); | |
| 94 } | |
| 95 double waggle = (c[1].x - c[3].x) * (c[1].y + c[3].y); | |
| 96 for (int idx = 1; idx < 3; ++idx){ | |
| 97 waggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); | |
| 98 } | |
| 99 return wiggle * waggle < 0; | |
| 100 } | |
| 101 | |
| 102 // cubic roots | |
| 103 | |
| 104 const double PI = 4 * atan(1); | |
| 105 | |
| 106 // from SkGeometry.cpp (and Numeric Solutions, 5.6) | |
| 107 int cubicRootsValidT(double A, double B, double C, double D, double t[3]) { | |
| 108 #if 0 | |
| 109 if (approximately_zero(A)) { // we're just a quadratic | |
| 110 return quadraticRootsValidT(B, C, D, t); | |
| 111 } | |
| 112 double a, b, c; | |
| 113 { | |
| 114 double invA = 1 / A; | |
| 115 a = B * invA; | |
| 116 b = C * invA; | |
| 117 c = D * invA; | |
| 118 } | |
| 119 double a2 = a * a; | |
| 120 double Q = (a2 - b * 3) / 9; | |
| 121 double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; | |
| 122 double Q3 = Q * Q * Q; | |
| 123 double R2MinusQ3 = R * R - Q3; | |
| 124 double adiv3 = a / 3; | |
| 125 double* roots = t; | |
| 126 double r; | |
| 127 | |
| 128 if (R2MinusQ3 < 0) // we have 3 real roots | |
| 129 { | |
| 130 double theta = acos(R / sqrt(Q3)); | |
| 131 double neg2RootQ = -2 * sqrt(Q); | |
| 132 | |
| 133 r = neg2RootQ * cos(theta / 3) - adiv3; | |
| 134 if (is_unit_interval(r)) | |
| 135 *roots++ = r; | |
| 136 | |
| 137 r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; | |
| 138 if (is_unit_interval(r)) | |
| 139 *roots++ = r; | |
| 140 | |
| 141 r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; | |
| 142 if (is_unit_interval(r)) | |
| 143 *roots++ = r; | |
| 144 } | |
| 145 else // we have 1 real root | |
| 146 { | |
| 147 double A = fabs(R) + sqrt(R2MinusQ3); | |
| 148 A = cube_root(A); | |
| 149 if (R > 0) { | |
| 150 A = -A; | |
| 151 } | |
| 152 if (A != 0) { | |
| 153 A += Q / A; | |
| 154 } | |
| 155 r = A - adiv3; | |
| 156 if (is_unit_interval(r)) | |
| 157 *roots++ = r; | |
| 158 } | |
| 159 return (int)(roots - t); | |
| 160 #else | |
| 161 double s[3]; | |
| 162 int realRoots = cubicRootsReal(A, B, C, D, s); | |
| 163 int foundRoots = add_valid_ts(s, realRoots, t); | |
| 164 return foundRoots; | |
| 165 #endif | |
| 166 } | |
| 167 | |
| 168 int cubicRootsReal(double A, double B, double C, double D, double s[3]) { | |
| 169 #ifdef SK_DEBUG | |
| 170 // create a string mathematica understands | |
| 171 // GDB set print repe 15 # if repeated digits is a bother | |
| 172 // set print elements 400 # if line doesn't fit | |
| 173 char str[1024]; | |
| 174 bzero(str, sizeof(str)); | |
| 175 sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A
, B, C, D); | |
| 176 mathematica_ize(str, sizeof(str)); | |
| 177 #if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA | |
| 178 SkDebugf("%s\n", str); | |
| 179 #endif | |
| 180 #endif | |
| 181 if (approximately_zero(A) | |
| 182 && approximately_zero_when_compared_to(A, B) | |
| 183 && approximately_zero_when_compared_to(A, C) | |
| 184 && approximately_zero_when_compared_to(A, D)) { // we're just a qua
dratic | |
| 185 return quadraticRootsReal(B, C, D, s); | |
| 186 } | |
| 187 if (approximately_zero_when_compared_to(D, A) | |
| 188 && approximately_zero_when_compared_to(D, B) | |
| 189 && approximately_zero_when_compared_to(D, C)) { // 0 is one root | |
| 190 int num = quadraticRootsReal(A, B, C, s); | |
| 191 for (int i = 0; i < num; ++i) { | |
| 192 if (approximately_zero(s[i])) { | |
| 193 return num; | |
| 194 } | |
| 195 } | |
| 196 s[num++] = 0; | |
| 197 return num; | |
| 198 } | |
| 199 if (approximately_zero(A + B + C + D)) { // 1 is one root | |
| 200 int num = quadraticRootsReal(A, A + B, -D, s); | |
| 201 for (int i = 0; i < num; ++i) { | |
| 202 if (AlmostEqualUlps(s[i], 1)) { | |
| 203 return num; | |
| 204 } | |
| 205 } | |
| 206 s[num++] = 1; | |
| 207 return num; | |
| 208 } | |
| 209 double a, b, c; | |
| 210 { | |
| 211 double invA = 1 / A; | |
| 212 a = B * invA; | |
| 213 b = C * invA; | |
| 214 c = D * invA; | |
| 215 } | |
| 216 double a2 = a * a; | |
| 217 double Q = (a2 - b * 3) / 9; | |
| 218 double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; | |
| 219 double R2 = R * R; | |
| 220 double Q3 = Q * Q * Q; | |
| 221 double R2MinusQ3 = R2 - Q3; | |
| 222 double adiv3 = a / 3; | |
| 223 double r; | |
| 224 double* roots = s; | |
| 225 #if 0 | |
| 226 if (approximately_zero_squared(R2MinusQ3) && AlmostEqualUlps(R2, Q3)) { | |
| 227 if (approximately_zero_squared(R)) {/* one triple solution */ | |
| 228 *roots++ = -adiv3; | |
| 229 } else { /* one single and one double solution */ | |
| 230 | |
| 231 double u = cube_root(-R); | |
| 232 *roots++ = 2 * u - adiv3; | |
| 233 *roots++ = -u - adiv3; | |
| 234 } | |
| 235 } | |
| 236 else | |
| 237 #endif | |
| 238 if (R2MinusQ3 < 0) // we have 3 real roots | |
| 239 { | |
| 240 double theta = acos(R / sqrt(Q3)); | |
| 241 double neg2RootQ = -2 * sqrt(Q); | |
| 242 | |
| 243 r = neg2RootQ * cos(theta / 3) - adiv3; | |
| 244 *roots++ = r; | |
| 245 | |
| 246 r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; | |
| 247 if (!AlmostEqualUlps(s[0], r)) { | |
| 248 *roots++ = r; | |
| 249 } | |
| 250 r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; | |
| 251 if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1
], r))) { | |
| 252 *roots++ = r; | |
| 253 } | |
| 254 } | |
| 255 else // we have 1 real root | |
| 256 { | |
| 257 double sqrtR2MinusQ3 = sqrt(R2MinusQ3); | |
| 258 double A = fabs(R) + sqrtR2MinusQ3; | |
| 259 A = cube_root(A); | |
| 260 if (R > 0) { | |
| 261 A = -A; | |
| 262 } | |
| 263 if (A != 0) { | |
| 264 A += Q / A; | |
| 265 } | |
| 266 r = A - adiv3; | |
| 267 *roots++ = r; | |
| 268 if (AlmostEqualUlps(R2, Q3)) { | |
| 269 r = -A / 2 - adiv3; | |
| 270 if (!AlmostEqualUlps(s[0], r)) { | |
| 271 *roots++ = r; | |
| 272 } | |
| 273 } | |
| 274 } | |
| 275 return (int)(roots - s); | |
| 276 } | |
| 277 | |
| 278 // from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf | |
| 279 // c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 | |
| 280 // c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 | |
| 281 // = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 | |
| 282 static double derivativeAtT(const double* cubic, double t) { | |
| 283 double one_t = 1 - t; | |
| 284 double a = cubic[0]; | |
| 285 double b = cubic[2]; | |
| 286 double c = cubic[4]; | |
| 287 double d = cubic[6]; | |
| 288 return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t
* t); | |
| 289 } | |
| 290 | |
| 291 double dx_at_t(const Cubic& cubic, double t) { | |
| 292 return derivativeAtT(&cubic[0].x, t); | |
| 293 } | |
| 294 | |
| 295 double dy_at_t(const Cubic& cubic, double t) { | |
| 296 return derivativeAtT(&cubic[0].y, t); | |
| 297 } | |
| 298 | |
| 299 // OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version o
f derivative at t? | |
| 300 _Vector dxdy_at_t(const Cubic& cubic, double t) { | |
| 301 _Vector result = { derivativeAtT(&cubic[0].x, t), derivativeAtT(&cubic[0].y,
t) }; | |
| 302 return result; | |
| 303 } | |
| 304 | |
| 305 // OPTIMIZE? share code with formulate_F1DotF2 | |
| 306 int find_cubic_inflections(const Cubic& src, double tValues[]) | |
| 307 { | |
| 308 double Ax = src[1].x - src[0].x; | |
| 309 double Ay = src[1].y - src[0].y; | |
| 310 double Bx = src[2].x - 2 * src[1].x + src[0].x; | |
| 311 double By = src[2].y - 2 * src[1].y + src[0].y; | |
| 312 double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x; | |
| 313 double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y; | |
| 314 return quadraticRootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By -
Ay * Bx, tValues); | |
| 315 } | |
| 316 | |
| 317 static void formulate_F1DotF2(const double src[], double coeff[4]) | |
| 318 { | |
| 319 double a = src[2] - src[0]; | |
| 320 double b = src[4] - 2 * src[2] + src[0]; | |
| 321 double c = src[6] + 3 * (src[2] - src[4]) - src[0]; | |
| 322 coeff[0] = c * c; | |
| 323 coeff[1] = 3 * b * c; | |
| 324 coeff[2] = 2 * b * b + c * a; | |
| 325 coeff[3] = a * b; | |
| 326 } | |
| 327 | |
| 328 /* from SkGeometry.cpp | |
| 329 Looking for F' dot F'' == 0 | |
| 330 | |
| 331 A = b - a | |
| 332 B = c - 2b + a | |
| 333 C = d - 3c + 3b - a | |
| 334 | |
| 335 F' = 3Ct^2 + 6Bt + 3A | |
| 336 F'' = 6Ct + 6B | |
| 337 | |
| 338 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB | |
| 339 */ | |
| 340 int find_cubic_max_curvature(const Cubic& src, double tValues[]) | |
| 341 { | |
| 342 double coeffX[4], coeffY[4]; | |
| 343 int i; | |
| 344 formulate_F1DotF2(&src[0].x, coeffX); | |
| 345 formulate_F1DotF2(&src[0].y, coeffY); | |
| 346 for (i = 0; i < 4; i++) { | |
| 347 coeffX[i] = coeffX[i] + coeffY[i]; | |
| 348 } | |
| 349 return cubicRootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues)
; | |
| 350 } | |
| 351 | |
| 352 | |
| 353 bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) { | |
| 354 double dy = cubic[index].y - cubic[zero].y; | |
| 355 double dx = cubic[index].x - cubic[zero].x; | |
| 356 if (approximately_zero(dy)) { | |
| 357 if (approximately_zero(dx)) { | |
| 358 return false; | |
| 359 } | |
| 360 memcpy(rotPath, cubic, sizeof(Cubic)); | |
| 361 return true; | |
| 362 } | |
| 363 for (int index = 0; index < 4; ++index) { | |
| 364 rotPath[index].x = cubic[index].x * dx + cubic[index].y * dy; | |
| 365 rotPath[index].y = cubic[index].y * dx - cubic[index].x * dy; | |
| 366 } | |
| 367 return true; | |
| 368 } | |
| 369 | |
| 370 #if 0 // unused for now | |
| 371 double secondDerivativeAtT(const double* cubic, double t) { | |
| 372 double a = cubic[0]; | |
| 373 double b = cubic[2]; | |
| 374 double c = cubic[4]; | |
| 375 double d = cubic[6]; | |
| 376 return (c - 2 * b + a) * (1 - t) + (d - 2 * c + b) * t; | |
| 377 } | |
| 378 #endif | |
| 379 | |
| 380 _Point top(const Cubic& cubic, double startT, double endT) { | |
| 381 Cubic sub; | |
| 382 sub_divide(cubic, startT, endT, sub); | |
| 383 _Point topPt = sub[0]; | |
| 384 if (topPt.y > sub[3].y || (topPt.y == sub[3].y && topPt.x > sub[3].x)) { | |
| 385 topPt = sub[3]; | |
| 386 } | |
| 387 double extremeTs[2]; | |
| 388 if (!monotonic_in_y(sub)) { | |
| 389 int roots = findExtrema(sub[0].y, sub[1].y, sub[2].y, sub[3].y, extremeT
s); | |
| 390 for (int index = 0; index < roots; ++index) { | |
| 391 _Point mid; | |
| 392 double t = startT + (endT - startT) * extremeTs[index]; | |
| 393 xy_at_t(cubic, t, mid.x, mid.y); | |
| 394 if (topPt.y > mid.y || (topPt.y == mid.y && topPt.x > mid.x)) { | |
| 395 topPt = mid; | |
| 396 } | |
| 397 } | |
| 398 } | |
| 399 return topPt; | |
| 400 } | |
| 401 | |
| 402 // OPTIMIZE: avoid computing the unused half | |
| 403 void xy_at_t(const Cubic& cubic, double t, double& x, double& y) { | |
| 404 _Point xy = xy_at_t(cubic, t); | |
| 405 if (&x) { | |
| 406 x = xy.x; | |
| 407 } | |
| 408 if (&y) { | |
| 409 y = xy.y; | |
| 410 } | |
| 411 } | |
| 412 | |
| 413 _Point xy_at_t(const Cubic& cubic, double t) { | |
| 414 double one_t = 1 - t; | |
| 415 double one_t2 = one_t * one_t; | |
| 416 double a = one_t2 * one_t; | |
| 417 double b = 3 * one_t2 * t; | |
| 418 double t2 = t * t; | |
| 419 double c = 3 * one_t * t2; | |
| 420 double d = t2 * t; | |
| 421 _Point result = {a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubi
c[3].x, | |
| 422 a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y}; | |
| 423 return result; | |
| 424 } | |
| OLD | NEW |
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Chromium Code Reviews
Issue 867213004:
remove prototype pathops code (Closed)
Base URL: https://skia.googlesource.com/skia.git@master