1 /* 2 * Copyright 2006 The Android Open Source Project 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 8 #include "SkGeometry.h" 9 #include "SkMatrix.h" 10 #include "SkNx.h" 11 12 static SkVector to_vector(const Sk2s& x) { 13 SkVector vector; 14 x.store(&vector); 15 return vector; 16 } 17 18 //////////////////////////////////////////////////////////////////////// 19 20 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 21 SkScalar ab = a - b; 22 SkScalar bc = b - c; 23 if (ab < 0) { 24 bc = -bc; 25 } 26 return ab == 0 || bc < 0; 27 } 28 29 //////////////////////////////////////////////////////////////////////// 30 31 static bool is_unit_interval(SkScalar x) { 32 return x > 0 && x < SK_Scalar1; 33 } 34 35 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 36 SkASSERT(ratio); 37 38 if (numer < 0) { 39 numer = -numer; 40 denom = -denom; 41 } 42 43 if (denom == 0 || numer == 0 || numer >= denom) { 44 return 0; 45 } 46 47 SkScalar r = numer / denom; 48 if (SkScalarIsNaN(r)) { 49 return 0; 50 } 51 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 52 if (r == 0) { // catch underflow if numer <<<< denom 53 return 0; 54 } 55 *ratio = r; 56 return 1; 57 } 58 59 /** From Numerical Recipes in C. 60 61 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 62 x1 = Q / A 63 x2 = C / Q 64 */ 65 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 66 SkASSERT(roots); 67 68 if (A == 0) { 69 return valid_unit_divide(-C, B, roots); 70 } 71 72 SkScalar* r = roots; 73 74 SkScalar R = B*B - 4*A*C; 75 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 76 // if R is infinite, it's possible that it may still produce 77 // useful results if the operation was repeated in doubles 78 // the flipside is determining if the more precise answer 79 // isn't useful because surrounding machinery (e.g., subtracting 80 // the axis offset from C) already discards the extra precision 81 // more investigation and unit tests required... 82 return 0; 83 } 84 R = SkScalarSqrt(R); 85 86 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 87 r += valid_unit_divide(Q, A, r); 88 r += valid_unit_divide(C, Q, r); 89 if (r - roots == 2) { 90 if (roots[0] > roots[1]) 91 SkTSwap<SkScalar>(roots[0], roots[1]); 92 else if (roots[0] == roots[1]) // nearly-equal? 93 r -= 1; // skip the double root 94 } 95 return (int)(r - roots); 96 } 97 98 /////////////////////////////////////////////////////////////////////////////// 99 /////////////////////////////////////////////////////////////////////////////// 100 101 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 102 SkASSERT(src); 103 SkASSERT(t >= 0 && t <= SK_Scalar1); 104 105 if (pt) { 106 *pt = SkEvalQuadAt(src, t); 107 } 108 if (tangent) { 109 *tangent = SkEvalQuadTangentAt(src, t); 110 } 111 } 112 113 SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 114 return to_point(SkQuadCoeff(src).eval(t)); 115 } 116 117 SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 118 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 119 // zero tangent vector when t is 0 or 1, and the control point is equal 120 // to the end point. In this case, use the quad end points to compute the tangent. 121 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 122 return src[2] - src[0]; 123 } 124 SkASSERT(src); 125 SkASSERT(t >= 0 && t <= SK_Scalar1); 126 127 Sk2s P0 = from_point(src[0]); 128 Sk2s P1 = from_point(src[1]); 129 Sk2s P2 = from_point(src[2]); 130 131 Sk2s B = P1 - P0; 132 Sk2s A = P2 - P1 - B; 133 Sk2s T = A * Sk2s(t) + B; 134 135 return to_vector(T + T); 136 } 137 138 static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 139 return v0 + (v1 - v0) * t; 140 } 141 142 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 143 SkASSERT(t > 0 && t < SK_Scalar1); 144 145 Sk2s p0 = from_point(src[0]); 146 Sk2s p1 = from_point(src[1]); 147 Sk2s p2 = from_point(src[2]); 148 Sk2s tt(t); 149 150 Sk2s p01 = interp(p0, p1, tt); 151 Sk2s p12 = interp(p1, p2, tt); 152 153 dst[0] = to_point(p0); 154 dst[1] = to_point(p01); 155 dst[2] = to_point(interp(p01, p12, tt)); 156 dst[3] = to_point(p12); 157 dst[4] = to_point(p2); 158 } 159 160 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 161 SkChopQuadAt(src, dst, 0.5f); 162 } 163 164 /** Quad'(t) = At + B, where 165 A = 2(a - 2b + c) 166 B = 2(b - a) 167 Solve for t, only if it fits between 0 < t < 1 168 */ 169 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 170 /* At + B == 0 171 t = -B / A 172 */ 173 return valid_unit_divide(a - b, a - b - b + c, tValue); 174 } 175 176 static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 177 coords[2] = coords[6] = coords[4]; 178 } 179 180 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 181 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 182 */ 183 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 184 SkASSERT(src); 185 SkASSERT(dst); 186 187 SkScalar a = src[0].fY; 188 SkScalar b = src[1].fY; 189 SkScalar c = src[2].fY; 190 191 if (is_not_monotonic(a, b, c)) { 192 SkScalar tValue; 193 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 194 SkChopQuadAt(src, dst, tValue); 195 flatten_double_quad_extrema(&dst[0].fY); 196 return 1; 197 } 198 // if we get here, we need to force dst to be monotonic, even though 199 // we couldn't compute a unit_divide value (probably underflow). 200 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 201 } 202 dst[0].set(src[0].fX, a); 203 dst[1].set(src[1].fX, b); 204 dst[2].set(src[2].fX, c); 205 return 0; 206 } 207 208 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 209 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 210 */ 211 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 212 SkASSERT(src); 213 SkASSERT(dst); 214 215 SkScalar a = src[0].fX; 216 SkScalar b = src[1].fX; 217 SkScalar c = src[2].fX; 218 219 if (is_not_monotonic(a, b, c)) { 220 SkScalar tValue; 221 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 222 SkChopQuadAt(src, dst, tValue); 223 flatten_double_quad_extrema(&dst[0].fX); 224 return 1; 225 } 226 // if we get here, we need to force dst to be monotonic, even though 227 // we couldn't compute a unit_divide value (probably underflow). 228 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 229 } 230 dst[0].set(a, src[0].fY); 231 dst[1].set(b, src[1].fY); 232 dst[2].set(c, src[2].fY); 233 return 0; 234 } 235 236 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 237 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t 238 // F''(t) = 2 (a - 2b + c) 239 // 240 // A = 2 (b - a) 241 // B = 2 (a - 2b + c) 242 // 243 // Maximum curvature for a quadratic means solving 244 // Fx' Fx'' + Fy' Fy'' = 0 245 // 246 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 247 // 248 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 249 SkScalar Ax = src[1].fX - src[0].fX; 250 SkScalar Ay = src[1].fY - src[0].fY; 251 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 252 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 253 SkScalar t = 0; // 0 means don't chop 254 255 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 256 return t; 257 } 258 259 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 260 SkScalar t = SkFindQuadMaxCurvature(src); 261 if (t == 0) { 262 memcpy(dst, src, 3 * sizeof(SkPoint)); 263 return 1; 264 } else { 265 SkChopQuadAt(src, dst, t); 266 return 2; 267 } 268 } 269 270 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 271 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 272 Sk2s s0 = from_point(src[0]); 273 Sk2s s1 = from_point(src[1]); 274 Sk2s s2 = from_point(src[2]); 275 276 dst[0] = src[0]; 277 dst[1] = to_point(s0 + (s1 - s0) * scale); 278 dst[2] = to_point(s2 + (s1 - s2) * scale); 279 dst[3] = src[2]; 280 } 281 282 ////////////////////////////////////////////////////////////////////////////// 283 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 284 ////////////////////////////////////////////////////////////////////////////// 285 286 static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) { 287 SkQuadCoeff coeff; 288 Sk2s P0 = from_point(src[0]); 289 Sk2s P1 = from_point(src[1]); 290 Sk2s P2 = from_point(src[2]); 291 Sk2s P3 = from_point(src[3]); 292 293 coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0; 294 coeff.fB = times_2(P2 - times_2(P1) + P0); 295 coeff.fC = P1 - P0; 296 return to_vector(coeff.eval(t)); 297 } 298 299 static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) { 300 Sk2s P0 = from_point(src[0]); 301 Sk2s P1 = from_point(src[1]); 302 Sk2s P2 = from_point(src[2]); 303 Sk2s P3 = from_point(src[3]); 304 Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0; 305 Sk2s B = P2 - times_2(P1) + P0; 306 307 return to_vector(A * Sk2s(t) + B); 308 } 309 310 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 311 SkVector* tangent, SkVector* curvature) { 312 SkASSERT(src); 313 SkASSERT(t >= 0 && t <= SK_Scalar1); 314 315 if (loc) { 316 *loc = to_point(SkCubicCoeff(src).eval(t)); 317 } 318 if (tangent) { 319 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 320 // adjacent control point is equal to the end point. In this case, use the 321 // next control point or the end points to compute the tangent. 322 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 323 if (t == 0) { 324 *tangent = src[2] - src[0]; 325 } else { 326 *tangent = src[3] - src[1]; 327 } 328 if (!tangent->fX && !tangent->fY) { 329 *tangent = src[3] - src[0]; 330 } 331 } else { 332 *tangent = eval_cubic_derivative(src, t); 333 } 334 } 335 if (curvature) { 336 *curvature = eval_cubic_2ndDerivative(src, t); 337 } 338 } 339 340 /** Cubic'(t) = At^2 + Bt + C, where 341 A = 3(-a + 3(b - c) + d) 342 B = 6(a - 2b + c) 343 C = 3(b - a) 344 Solve for t, keeping only those that fit betwee 0 < t < 1 345 */ 346 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 347 SkScalar tValues[2]) { 348 // we divide A,B,C by 3 to simplify 349 SkScalar A = d - a + 3*(b - c); 350 SkScalar B = 2*(a - b - b + c); 351 SkScalar C = b - a; 352 353 return SkFindUnitQuadRoots(A, B, C, tValues); 354 } 355 356 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 357 SkASSERT(t > 0 && t < SK_Scalar1); 358 359 Sk2s p0 = from_point(src[0]); 360 Sk2s p1 = from_point(src[1]); 361 Sk2s p2 = from_point(src[2]); 362 Sk2s p3 = from_point(src[3]); 363 Sk2s tt(t); 364 365 Sk2s ab = interp(p0, p1, tt); 366 Sk2s bc = interp(p1, p2, tt); 367 Sk2s cd = interp(p2, p3, tt); 368 Sk2s abc = interp(ab, bc, tt); 369 Sk2s bcd = interp(bc, cd, tt); 370 Sk2s abcd = interp(abc, bcd, tt); 371 372 dst[0] = src[0]; 373 dst[1] = to_point(ab); 374 dst[2] = to_point(abc); 375 dst[3] = to_point(abcd); 376 dst[4] = to_point(bcd); 377 dst[5] = to_point(cd); 378 dst[6] = src[3]; 379 } 380 381 /* http://code.google.com/p/skia/issues/detail?id=32 382 383 This test code would fail when we didn't check the return result of 384 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 385 that after the first chop, the parameters to valid_unit_divide are equal 386 (thanks to finite float precision and rounding in the subtracts). Thus 387 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 388 up with 1.0, hence the need to check and just return the last cubic as 389 a degenerate clump of 4 points in the sampe place. 390 391 static void test_cubic() { 392 SkPoint src[4] = { 393 { 556.25000, 523.03003 }, 394 { 556.23999, 522.96002 }, 395 { 556.21997, 522.89001 }, 396 { 556.21997, 522.82001 } 397 }; 398 SkPoint dst[10]; 399 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 400 SkChopCubicAt(src, dst, tval, 2); 401 } 402 */ 403 404 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 405 const SkScalar tValues[], int roots) { 406 #ifdef SK_DEBUG 407 { 408 for (int i = 0; i < roots - 1; i++) 409 { 410 SkASSERT(is_unit_interval(tValues[i])); 411 SkASSERT(is_unit_interval(tValues[i+1])); 412 SkASSERT(tValues[i] < tValues[i+1]); 413 } 414 } 415 #endif 416 417 if (dst) { 418 if (roots == 0) { // nothing to chop 419 memcpy(dst, src, 4*sizeof(SkPoint)); 420 } else { 421 SkScalar t = tValues[0]; 422 SkPoint tmp[4]; 423 424 for (int i = 0; i < roots; i++) { 425 SkChopCubicAt(src, dst, t); 426 if (i == roots - 1) { 427 break; 428 } 429 430 dst += 3; 431 // have src point to the remaining cubic (after the chop) 432 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 433 src = tmp; 434 435 // watch out in case the renormalized t isn't in range 436 if (!valid_unit_divide(tValues[i+1] - tValues[i], 437 SK_Scalar1 - tValues[i], &t)) { 438 // if we can't, just create a degenerate cubic 439 dst[4] = dst[5] = dst[6] = src[3]; 440 break; 441 } 442 } 443 } 444 } 445 } 446 447 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 448 SkChopCubicAt(src, dst, 0.5f); 449 } 450 451 static void flatten_double_cubic_extrema(SkScalar coords[14]) { 452 coords[4] = coords[8] = coords[6]; 453 } 454 455 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 456 the resulting beziers are monotonic in Y. This is called by the scan 457 converter. Depending on what is returned, dst[] is treated as follows: 458 0 dst[0..3] is the original cubic 459 1 dst[0..3] and dst[3..6] are the two new cubics 460 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 461 If dst == null, it is ignored and only the count is returned. 462 */ 463 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 464 SkScalar tValues[2]; 465 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 466 src[3].fY, tValues); 467 468 SkChopCubicAt(src, dst, tValues, roots); 469 if (dst && roots > 0) { 470 // we do some cleanup to ensure our Y extrema are flat 471 flatten_double_cubic_extrema(&dst[0].fY); 472 if (roots == 2) { 473 flatten_double_cubic_extrema(&dst[3].fY); 474 } 475 } 476 return roots; 477 } 478 479 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 480 SkScalar tValues[2]; 481 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 482 src[3].fX, tValues); 483 484 SkChopCubicAt(src, dst, tValues, roots); 485 if (dst && roots > 0) { 486 // we do some cleanup to ensure our Y extrema are flat 487 flatten_double_cubic_extrema(&dst[0].fX); 488 if (roots == 2) { 489 flatten_double_cubic_extrema(&dst[3].fX); 490 } 491 } 492 return roots; 493 } 494 495 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 496 497 Inflection means that curvature is zero. 498 Curvature is [F' x F''] / [F'^3] 499 So we solve F'x X F''y - F'y X F''y == 0 500 After some canceling of the cubic term, we get 501 A = b - a 502 B = c - 2b + a 503 C = d - 3c + 3b - a 504 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 505 */ 506 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 507 SkScalar Ax = src[1].fX - src[0].fX; 508 SkScalar Ay = src[1].fY - src[0].fY; 509 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 510 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 511 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 512 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 513 514 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 515 Ax*Cy - Ay*Cx, 516 Ax*By - Ay*Bx, 517 tValues); 518 } 519 520 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 521 SkScalar tValues[2]; 522 int count = SkFindCubicInflections(src, tValues); 523 524 if (dst) { 525 if (count == 0) { 526 memcpy(dst, src, 4 * sizeof(SkPoint)); 527 } else { 528 SkChopCubicAt(src, dst, tValues, count); 529 } 530 } 531 return count + 1; 532 } 533 534 // Assumes the third component of points is 1. 535 // Calcs p0 . (p1 x p2) 536 static double calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 537 const double xComp = (double) p0.fX * (double) (p1.fY - p2.fY); 538 const double yComp = (double) p0.fY * (double) (p2.fX - p1.fX); 539 const double wComp = (double) p1.fX * (double) p2.fY - (double) p1.fY * (double) p2.fX; 540 return (xComp + yComp + wComp); 541 } 542 543 // Calc coefficients of I(s,t) where roots of I are inflection points of curve 544 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 545 // d0 = a1 - 2*a2+3*a3 546 // d1 = -a2 + 3*a3 547 // d2 = 3*a3 548 // a1 = p0 . (p3 x p2) 549 // a2 = p1 . (p0 x p3) 550 // a3 = p2 . (p1 x p0) 551 // Places the values of d1, d2, d3 in array d passed in 552 static void calc_cubic_inflection_func(const SkPoint p[4], double d[4]) { 553 const double a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 554 const double a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 555 const double a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 556 557 d[3] = 3 * a3; 558 d[2] = d[3] - a2; 559 d[1] = d[2] - a2 + a1; 560 d[0] = 0; 561 } 562 563 static void normalize_t_s(double t[], double s[], int count) { 564 // Keep the exponents at or below zero to avoid overflow down the road. 565 for (int i = 0; i < count; ++i) { 566 SkASSERT(0 != s[i]); 567 union { double value; int64_t bits; } tt, ss, norm; 568 tt.value = t[i]; 569 ss.value = s[i]; 570 int64_t expT = ((tt.bits >> 52) & 0x7ff) - 1023, 571 expS = ((ss.bits >> 52) & 0x7ff) - 1023; 572 int64_t expNorm = -SkTMax(expT, expS) + 1023; 573 SkASSERT(expNorm > 0 && expNorm < 2047); // ensure we have a valid non-zero exponent. 574 norm.bits = expNorm << 52; 575 t[i] *= norm.value; 576 s[i] *= norm.value; 577 } 578 } 579 580 static void sort_and_orient_t_s(double t[2], double s[2]) { 581 // This copysign/abs business orients the implicit function so positive values are always on the 582 // "left" side of the curve. 583 t[1] = -copysign(t[1], t[1] * s[1]); 584 s[1] = -fabs(s[1]); 585 586 // Ensure t[0]/s[0] <= t[1]/s[1] (s[1] is negative from above). 587 if (copysign(s[1], s[0]) * t[0] > -fabs(s[0]) * t[1]) { 588 std::swap(t[0], t[1]); 589 std::swap(s[0], s[1]); 590 } 591 } 592 593 // See "Resolution Independent Curve Rendering using Programmable Graphics Hardware" 594 // https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf 595 // discr(I) = 3*d2^2 - 4*d1*d3 596 // Classification: 597 // d1 != 0, discr(I) > 0 Serpentine 598 // d1 != 0, discr(I) < 0 Loop 599 // d1 != 0, discr(I) = 0 Cusp (with inflection at infinity) 600 // d1 = 0, d2 != 0 Cusp (with cusp at infinity) 601 // d1 = d2 = 0, d3 != 0 Quadratic 602 // d1 = d2 = d3 = 0 Line or Point 603 static SkCubicType classify_cubic(const double d[4], double t[2], double s[2]) { 604 // Check for degenerate cubics (quadratics, lines, and points). 605 // This also attempts to detect near-quadratics in a resolution independent fashion, however it 606 // is still up to the caller to check for almost-linear curves if needed. 607 if (fabs(d[1]) + fabs(d[2]) <= fabs(d[3]) * 1e-3) { 608 if (t && s) { 609 t[0] = t[1] = 1; 610 s[0] = s[1] = 0; // infinity 611 } 612 return 0 == d[3] ? SkCubicType::kLineOrPoint : SkCubicType::kQuadratic; 613 } 614 615 if (0 == d[1]) { 616 SkASSERT(0 != d[2]); // captured in check for degeneracy above. 617 if (t && s) { 618 t[0] = d[3]; 619 s[0] = 3 * d[2]; 620 normalize_t_s(t, s, 1); 621 t[1] = 1; 622 s[1] = 0; // infinity 623 } 624 return SkCubicType::kCuspAtInfinity; 625 } 626 627 const double discr = 3 * d[2] * d[2] - 4 * d[1] * d[3]; 628 if (discr > 0) { 629 if (t && s) { 630 const double q = 3 * d[2] + copysign(sqrt(3 * discr), d[2]); 631 t[0] = q; 632 s[0] = 6 * d[1]; 633 t[1] = 2 * d[3]; 634 s[1] = q; 635 normalize_t_s(t, s, 2); 636 sort_and_orient_t_s(t, s); 637 } 638 return SkCubicType::kSerpentine; 639 } else if (discr < 0) { 640 if (t && s) { 641 const double q = d[2] + copysign(sqrt(-discr), d[2]); 642 t[0] = q; 643 s[0] = 2 * d[1]; 644 t[1] = 2 * (d[2] * d[2] - d[3] * d[1]); 645 s[1] = d[1] * q; 646 normalize_t_s(t, s, 2); 647 sort_and_orient_t_s(t, s); 648 } 649 return SkCubicType::kLoop; 650 } else { 651 SkASSERT(0 == discr); // Detect NaN. 652 if (t && s) { 653 t[0] = d[2]; 654 s[0] = 2 * d[1]; 655 normalize_t_s(t, s, 1); 656 t[1] = t[0]; 657 s[1] = s[0]; 658 sort_and_orient_t_s(t, s); 659 } 660 return SkCubicType::kLocalCusp; 661 } 662 } 663 664 SkCubicType SkClassifyCubic(const SkPoint src[4], double t[2], double s[2], double d[4]) { 665 double localD[4]; 666 double* dd = d ? d : localD; 667 calc_cubic_inflection_func(src, dd); 668 return classify_cubic(dd, t, s); 669 } 670 671 template <typename T> void bubble_sort(T array[], int count) { 672 for (int i = count - 1; i > 0; --i) 673 for (int j = i; j > 0; --j) 674 if (array[j] < array[j-1]) 675 { 676 T tmp(array[j]); 677 array[j] = array[j-1]; 678 array[j-1] = tmp; 679 } 680 } 681 682 /** 683 * Given an array and count, remove all pair-wise duplicates from the array, 684 * keeping the existing sorting, and return the new count 685 */ 686 static int collaps_duplicates(SkScalar array[], int count) { 687 for (int n = count; n > 1; --n) { 688 if (array[0] == array[1]) { 689 for (int i = 1; i < n; ++i) { 690 array[i - 1] = array[i]; 691 } 692 count -= 1; 693 } else { 694 array += 1; 695 } 696 } 697 return count; 698 } 699 700 #ifdef SK_DEBUG 701 702 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 703 704 static void test_collaps_duplicates() { 705 static bool gOnce; 706 if (gOnce) { return; } 707 gOnce = true; 708 const SkScalar src0[] = { 0 }; 709 const SkScalar src1[] = { 0, 0 }; 710 const SkScalar src2[] = { 0, 1 }; 711 const SkScalar src3[] = { 0, 0, 0 }; 712 const SkScalar src4[] = { 0, 0, 1 }; 713 const SkScalar src5[] = { 0, 1, 1 }; 714 const SkScalar src6[] = { 0, 1, 2 }; 715 const struct { 716 const SkScalar* fData; 717 int fCount; 718 int fCollapsedCount; 719 } data[] = { 720 { TEST_COLLAPS_ENTRY(src0), 1 }, 721 { TEST_COLLAPS_ENTRY(src1), 1 }, 722 { TEST_COLLAPS_ENTRY(src2), 2 }, 723 { TEST_COLLAPS_ENTRY(src3), 1 }, 724 { TEST_COLLAPS_ENTRY(src4), 2 }, 725 { TEST_COLLAPS_ENTRY(src5), 2 }, 726 { TEST_COLLAPS_ENTRY(src6), 3 }, 727 }; 728 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 729 SkScalar dst[3]; 730 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 731 int count = collaps_duplicates(dst, data[i].fCount); 732 SkASSERT(data[i].fCollapsedCount == count); 733 for (int j = 1; j < count; ++j) { 734 SkASSERT(dst[j-1] < dst[j]); 735 } 736 } 737 } 738 #endif 739 740 static SkScalar SkScalarCubeRoot(SkScalar x) { 741 return SkScalarPow(x, 0.3333333f); 742 } 743 744 /* Solve coeff(t) == 0, returning the number of roots that 745 lie withing 0 < t < 1. 746 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 747 748 Eliminates repeated roots (so that all tValues are distinct, and are always 749 in increasing order. 750 */ 751 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 752 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 753 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 754 } 755 756 SkScalar a, b, c, Q, R; 757 758 { 759 SkASSERT(coeff[0] != 0); 760 761 SkScalar inva = SkScalarInvert(coeff[0]); 762 a = coeff[1] * inva; 763 b = coeff[2] * inva; 764 c = coeff[3] * inva; 765 } 766 Q = (a*a - b*3) / 9; 767 R = (2*a*a*a - 9*a*b + 27*c) / 54; 768 769 SkScalar Q3 = Q * Q * Q; 770 SkScalar R2MinusQ3 = R * R - Q3; 771 SkScalar adiv3 = a / 3; 772 773 SkScalar* roots = tValues; 774 SkScalar r; 775 776 if (R2MinusQ3 < 0) { // we have 3 real roots 777 // the divide/root can, due to finite precisions, be slightly outside of -1...1 778 SkScalar theta = SkScalarACos(SkScalarPin(R / SkScalarSqrt(Q3), -1, 1)); 779 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 780 781 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 782 if (is_unit_interval(r)) { 783 *roots++ = r; 784 } 785 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 786 if (is_unit_interval(r)) { 787 *roots++ = r; 788 } 789 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 790 if (is_unit_interval(r)) { 791 *roots++ = r; 792 } 793 SkDEBUGCODE(test_collaps_duplicates();) 794 795 // now sort the roots 796 int count = (int)(roots - tValues); 797 SkASSERT((unsigned)count <= 3); 798 bubble_sort(tValues, count); 799 count = collaps_duplicates(tValues, count); 800 roots = tValues + count; // so we compute the proper count below 801 } else { // we have 1 real root 802 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 803 A = SkScalarCubeRoot(A); 804 if (R > 0) { 805 A = -A; 806 } 807 if (A != 0) { 808 A += Q / A; 809 } 810 r = A - adiv3; 811 if (is_unit_interval(r)) { 812 *roots++ = r; 813 } 814 } 815 816 return (int)(roots - tValues); 817 } 818 819 /* Looking for F' dot F'' == 0 820 821 A = b - a 822 B = c - 2b + a 823 C = d - 3c + 3b - a 824 825 F' = 3Ct^2 + 6Bt + 3A 826 F'' = 6Ct + 6B 827 828 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 829 */ 830 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 831 SkScalar a = src[2] - src[0]; 832 SkScalar b = src[4] - 2 * src[2] + src[0]; 833 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 834 835 coeff[0] = c * c; 836 coeff[1] = 3 * b * c; 837 coeff[2] = 2 * b * b + c * a; 838 coeff[3] = a * b; 839 } 840 841 /* Looking for F' dot F'' == 0 842 843 A = b - a 844 B = c - 2b + a 845 C = d - 3c + 3b - a 846 847 F' = 3Ct^2 + 6Bt + 3A 848 F'' = 6Ct + 6B 849 850 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 851 */ 852 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 853 SkScalar coeffX[4], coeffY[4]; 854 int i; 855 856 formulate_F1DotF2(&src[0].fX, coeffX); 857 formulate_F1DotF2(&src[0].fY, coeffY); 858 859 for (i = 0; i < 4; i++) { 860 coeffX[i] += coeffY[i]; 861 } 862 863 SkScalar t[3]; 864 int count = solve_cubic_poly(coeffX, t); 865 int maxCount = 0; 866 867 // now remove extrema where the curvature is zero (mins) 868 // !!!! need a test for this !!!! 869 for (i = 0; i < count; i++) { 870 // if (not_min_curvature()) 871 if (t[i] > 0 && t[i] < SK_Scalar1) { 872 tValues[maxCount++] = t[i]; 873 } 874 } 875 return maxCount; 876 } 877 878 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 879 SkScalar tValues[3]) { 880 SkScalar t_storage[3]; 881 882 if (tValues == nullptr) { 883 tValues = t_storage; 884 } 885 886 int count = SkFindCubicMaxCurvature(src, tValues); 887 888 if (dst) { 889 if (count == 0) { 890 memcpy(dst, src, 4 * sizeof(SkPoint)); 891 } else { 892 SkChopCubicAt(src, dst, tValues, count); 893 } 894 } 895 return count + 1; 896 } 897 898 #include "../pathops/SkPathOpsCubic.h" 899 900 typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 901 902 static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 903 InterceptProc method) { 904 SkDCubic cubic; 905 double roots[3]; 906 int count = (cubic.set(src).*method)(intercept, roots); 907 if (count > 0) { 908 SkDCubicPair pair = cubic.chopAt(roots[0]); 909 for (int i = 0; i < 7; ++i) { 910 dst[i] = pair.pts[i].asSkPoint(); 911 } 912 return true; 913 } 914 return false; 915 } 916 917 bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 918 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 919 } 920 921 bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 922 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 923 } 924 925 /////////////////////////////////////////////////////////////////////////////// 926 // 927 // NURB representation for conics. Helpful explanations at: 928 // 929 // http://citeseerx.ist.psu.edu/viewdoc/ 930 // download?doi=10.1.1.44.5740&rep=rep1&type=ps 931 // and 932 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 933 // 934 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 935 // ------------------------------------------ 936 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 937 // 938 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 939 // ------------------------------------------------ 940 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 941 // 942 943 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 944 // 945 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 946 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 947 // t^0 : -2 P0 w + 2 P1 w 948 // 949 // We disregard magnitude, so we can freely ignore the denominator of F', and 950 // divide the numerator by 2 951 // 952 // coeff[0] for t^2 953 // coeff[1] for t^1 954 // coeff[2] for t^0 955 // 956 static void conic_deriv_coeff(const SkScalar src[], 957 SkScalar w, 958 SkScalar coeff[3]) { 959 const SkScalar P20 = src[4] - src[0]; 960 const SkScalar P10 = src[2] - src[0]; 961 const SkScalar wP10 = w * P10; 962 coeff[0] = w * P20 - P20; 963 coeff[1] = P20 - 2 * wP10; 964 coeff[2] = wP10; 965 } 966 967 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 968 SkScalar coeff[3]; 969 conic_deriv_coeff(src, w, coeff); 970 971 SkScalar tValues[2]; 972 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 973 SkASSERT(0 == roots || 1 == roots); 974 975 if (1 == roots) { 976 *t = tValues[0]; 977 return true; 978 } 979 return false; 980 } 981 982 struct SkP3D { 983 SkScalar fX, fY, fZ; 984 985 void set(SkScalar x, SkScalar y, SkScalar z) { 986 fX = x; fY = y; fZ = z; 987 } 988 989 void projectDown(SkPoint* dst) const { 990 dst->set(fX / fZ, fY / fZ); 991 } 992 }; 993 994 // We only interpolate one dimension at a time (the first, at +0, +3, +6). 995 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 996 SkScalar ab = SkScalarInterp(src[0], src[3], t); 997 SkScalar bc = SkScalarInterp(src[3], src[6], t); 998 dst[0] = ab; 999 dst[3] = SkScalarInterp(ab, bc, t); 1000 dst[6] = bc; 1001 } 1002 1003 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1004 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1005 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1006 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1007 } 1008 1009 // return false if infinity or NaN is generated; caller must check 1010 bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1011 SkP3D tmp[3], tmp2[3]; 1012 1013 ratquad_mapTo3D(fPts, fW, tmp); 1014 1015 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1016 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1017 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1018 1019 dst[0].fPts[0] = fPts[0]; 1020 tmp2[0].projectDown(&dst[0].fPts[1]); 1021 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1022 tmp2[2].projectDown(&dst[1].fPts[1]); 1023 dst[1].fPts[2] = fPts[2]; 1024 1025 // to put in "standard form", where w0 and w2 are both 1, we compute the 1026 // new w1 as sqrt(w1*w1/w0*w2) 1027 // or 1028 // w1 /= sqrt(w0*w2) 1029 // 1030 // However, in our case, we know that for dst[0]: 1031 // w0 == 1, and for dst[1], w2 == 1 1032 // 1033 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1034 dst[0].fW = tmp2[0].fZ / root; 1035 dst[1].fW = tmp2[2].fZ / root; 1036 SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7); 1037 SkASSERT(0 == offsetof(SkConic, fPts[0].fX)); 1038 return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2); 1039 } 1040 1041 void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { 1042 if (0 == t1 || 1 == t2) { 1043 if (0 == t1 && 1 == t2) { 1044 *dst = *this; 1045 return; 1046 } else { 1047 SkConic pair[2]; 1048 if (this->chopAt(t1 ? t1 : t2, pair)) { 1049 *dst = pair[SkToBool(t1)]; 1050 return; 1051 } 1052 } 1053 } 1054 SkConicCoeff coeff(*this); 1055 Sk2s tt1(t1); 1056 Sk2s aXY = coeff.fNumer.eval(tt1); 1057 Sk2s aZZ = coeff.fDenom.eval(tt1); 1058 Sk2s midTT((t1 + t2) / 2); 1059 Sk2s dXY = coeff.fNumer.eval(midTT); 1060 Sk2s dZZ = coeff.fDenom.eval(midTT); 1061 Sk2s tt2(t2); 1062 Sk2s cXY = coeff.fNumer.eval(tt2); 1063 Sk2s cZZ = coeff.fDenom.eval(tt2); 1064 Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); 1065 Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); 1066 dst->fPts[0] = to_point(aXY / aZZ); 1067 dst->fPts[1] = to_point(bXY / bZZ); 1068 dst->fPts[2] = to_point(cXY / cZZ); 1069 Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); 1070 dst->fW = ww[0]; 1071 } 1072 1073 SkPoint SkConic::evalAt(SkScalar t) const { 1074 return to_point(SkConicCoeff(*this).eval(t)); 1075 } 1076 1077 SkVector SkConic::evalTangentAt(SkScalar t) const { 1078 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1079 // and the control point is equal to the end point. 1080 // In this case, use the conic endpoints to compute the tangent. 1081 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1082 return fPts[2] - fPts[0]; 1083 } 1084 Sk2s p0 = from_point(fPts[0]); 1085 Sk2s p1 = from_point(fPts[1]); 1086 Sk2s p2 = from_point(fPts[2]); 1087 Sk2s ww(fW); 1088 1089 Sk2s p20 = p2 - p0; 1090 Sk2s p10 = p1 - p0; 1091 1092 Sk2s C = ww * p10; 1093 Sk2s A = ww * p20 - p20; 1094 Sk2s B = p20 - C - C; 1095 1096 return to_vector(SkQuadCoeff(A, B, C).eval(t)); 1097 } 1098 1099 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1100 SkASSERT(t >= 0 && t <= SK_Scalar1); 1101 1102 if (pt) { 1103 *pt = this->evalAt(t); 1104 } 1105 if (tangent) { 1106 *tangent = this->evalTangentAt(t); 1107 } 1108 } 1109 1110 static SkScalar subdivide_w_value(SkScalar w) { 1111 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1112 } 1113 1114 void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1115 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1116 SkScalar newW = subdivide_w_value(fW); 1117 1118 Sk2s p0 = from_point(fPts[0]); 1119 Sk2s p1 = from_point(fPts[1]); 1120 Sk2s p2 = from_point(fPts[2]); 1121 Sk2s ww(fW); 1122 1123 Sk2s wp1 = ww * p1; 1124 Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f); 1125 1126 dst[0].fPts[0] = fPts[0]; 1127 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1128 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1129 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1130 dst[1].fPts[2] = fPts[2]; 1131 1132 dst[0].fW = dst[1].fW = newW; 1133 } 1134 1135 /* 1136 * "High order approximation of conic sections by quadratic splines" 1137 * by Michael Floater, 1993 1138 */ 1139 #define AS_QUAD_ERROR_SETUP \ 1140 SkScalar a = fW - 1; \ 1141 SkScalar k = a / (4 * (2 + a)); \ 1142 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1143 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1144 1145 void SkConic::computeAsQuadError(SkVector* err) const { 1146 AS_QUAD_ERROR_SETUP 1147 err->set(x, y); 1148 } 1149 1150 bool SkConic::asQuadTol(SkScalar tol) const { 1151 AS_QUAD_ERROR_SETUP 1152 return (x * x + y * y) <= tol * tol; 1153 } 1154 1155 // Limit the number of suggested quads to approximate a conic 1156 #define kMaxConicToQuadPOW2 5 1157 1158 int SkConic::computeQuadPOW2(SkScalar tol) const { 1159 if (tol < 0 || !SkScalarIsFinite(tol)) { 1160 return 0; 1161 } 1162 1163 AS_QUAD_ERROR_SETUP 1164 1165 SkScalar error = SkScalarSqrt(x * x + y * y); 1166 int pow2; 1167 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1168 if (error <= tol) { 1169 break; 1170 } 1171 error *= 0.25f; 1172 } 1173 // float version -- using ceil gives the same results as the above. 1174 if (false) { 1175 SkScalar err = SkScalarSqrt(x * x + y * y); 1176 if (err <= tol) { 1177 return 0; 1178 } 1179 SkScalar tol2 = tol * tol; 1180 if (tol2 == 0) { 1181 return kMaxConicToQuadPOW2; 1182 } 1183 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1184 int altPow2 = SkScalarCeilToInt(fpow2); 1185 if (altPow2 != pow2) { 1186 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1187 } 1188 pow2 = altPow2; 1189 } 1190 return pow2; 1191 } 1192 1193 // This was originally developed and tested for pathops: see SkOpTypes.h 1194 // returns true if (a <= b <= c) || (a >= b >= c) 1195 static bool between(SkScalar a, SkScalar b, SkScalar c) { 1196 return (a - b) * (c - b) <= 0; 1197 } 1198 1199 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1200 SkASSERT(level >= 0); 1201 1202 if (0 == level) { 1203 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1204 return pts + 2; 1205 } else { 1206 SkConic dst[2]; 1207 src.chop(dst); 1208 const SkScalar startY = src.fPts[0].fY; 1209 const SkScalar endY = src.fPts[2].fY; 1210 if (between(startY, src.fPts[1].fY, endY)) { 1211 // If the input is monotonic and the output is not, the scan converter hangs. 1212 // Ensure that the chopped conics maintain their y-order. 1213 SkScalar midY = dst[0].fPts[2].fY; 1214 if (!between(startY, midY, endY)) { 1215 // If the computed midpoint is outside the ends, move it to the closer one. 1216 SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY; 1217 dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY; 1218 } 1219 if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) { 1220 // If the 1st control is not between the start and end, put it at the start. 1221 // This also reduces the quad to a line. 1222 dst[0].fPts[1].fY = startY; 1223 } 1224 if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) { 1225 // If the 2nd control is not between the start and end, put it at the end. 1226 // This also reduces the quad to a line. 1227 dst[1].fPts[1].fY = endY; 1228 } 1229 // Verify that all five points are in order. 1230 SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)); 1231 SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY)); 1232 SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY)); 1233 } 1234 --level; 1235 pts = subdivide(dst[0], pts, level); 1236 return subdivide(dst[1], pts, level); 1237 } 1238 } 1239 1240 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1241 SkASSERT(pow2 >= 0); 1242 *pts = fPts[0]; 1243 SkDEBUGCODE(SkPoint* endPts); 1244 if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ... 1245 SkConic dst[2]; 1246 this->chop(dst); 1247 // check to see if the first chop generates a pair of lines 1248 if (dst[0].fPts[1].equalsWithinTolerance(dst[0].fPts[2]) 1249 && dst[1].fPts[0].equalsWithinTolerance(dst[1].fPts[1])) { 1250 pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines 1251 pts[4] = dst[1].fPts[2]; 1252 pow2 = 1; 1253 SkDEBUGCODE(endPts = &pts[5]); 1254 goto commonFinitePtCheck; 1255 } 1256 } 1257 SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2); 1258 commonFinitePtCheck: 1259 const int quadCount = 1 << pow2; 1260 const int ptCount = 2 * quadCount + 1; 1261 SkASSERT(endPts - pts == ptCount); 1262 if (!SkPointsAreFinite(pts, ptCount)) { 1263 // if we generated a non-finite, pin ourselves to the middle of the hull, 1264 // as our first and last are already on the first/last pts of the hull. 1265 for (int i = 1; i < ptCount - 1; ++i) { 1266 pts[i] = fPts[1]; 1267 } 1268 } 1269 return 1 << pow2; 1270 } 1271 1272 bool SkConic::findXExtrema(SkScalar* t) const { 1273 return conic_find_extrema(&fPts[0].fX, fW, t); 1274 } 1275 1276 bool SkConic::findYExtrema(SkScalar* t) const { 1277 return conic_find_extrema(&fPts[0].fY, fW, t); 1278 } 1279 1280 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1281 SkScalar t; 1282 if (this->findXExtrema(&t)) { 1283 if (!this->chopAt(t, dst)) { 1284 // if chop can't return finite values, don't chop 1285 return false; 1286 } 1287 // now clean-up the middle, since we know t was meant to be at 1288 // an X-extrema 1289 SkScalar value = dst[0].fPts[2].fX; 1290 dst[0].fPts[1].fX = value; 1291 dst[1].fPts[0].fX = value; 1292 dst[1].fPts[1].fX = value; 1293 return true; 1294 } 1295 return false; 1296 } 1297 1298 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1299 SkScalar t; 1300 if (this->findYExtrema(&t)) { 1301 if (!this->chopAt(t, dst)) { 1302 // if chop can't return finite values, don't chop 1303 return false; 1304 } 1305 // now clean-up the middle, since we know t was meant to be at 1306 // an Y-extrema 1307 SkScalar value = dst[0].fPts[2].fY; 1308 dst[0].fPts[1].fY = value; 1309 dst[1].fPts[0].fY = value; 1310 dst[1].fPts[1].fY = value; 1311 return true; 1312 } 1313 return false; 1314 } 1315 1316 void SkConic::computeTightBounds(SkRect* bounds) const { 1317 SkPoint pts[4]; 1318 pts[0] = fPts[0]; 1319 pts[1] = fPts[2]; 1320 int count = 2; 1321 1322 SkScalar t; 1323 if (this->findXExtrema(&t)) { 1324 this->evalAt(t, &pts[count++]); 1325 } 1326 if (this->findYExtrema(&t)) { 1327 this->evalAt(t, &pts[count++]); 1328 } 1329 bounds->set(pts, count); 1330 } 1331 1332 void SkConic::computeFastBounds(SkRect* bounds) const { 1333 bounds->set(fPts, 3); 1334 } 1335 1336 #if 0 // unimplemented 1337 bool SkConic::findMaxCurvature(SkScalar* t) const { 1338 // TODO: Implement me 1339 return false; 1340 } 1341 #endif 1342 1343 SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1344 const SkMatrix& matrix) { 1345 if (!matrix.hasPerspective()) { 1346 return w; 1347 } 1348 1349 SkP3D src[3], dst[3]; 1350 1351 ratquad_mapTo3D(pts, w, src); 1352 1353 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1354 1355 // w' = sqrt(w1*w1/w0*w2) 1356 SkScalar w0 = dst[0].fZ; 1357 SkScalar w1 = dst[1].fZ; 1358 SkScalar w2 = dst[2].fZ; 1359 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1360 return w; 1361 } 1362 1363 int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1364 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1365 // rotate by x,y so that uStart is (1.0) 1366 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1367 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1368 1369 SkScalar absY = SkScalarAbs(y); 1370 1371 // check for (effectively) coincident vectors 1372 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1373 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1374 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1375 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1376 return 0; 1377 } 1378 1379 if (dir == kCCW_SkRotationDirection) { 1380 y = -y; 1381 } 1382 1383 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1384 // 0 == [0 .. 90) 1385 // 1 == [90 ..180) 1386 // 2 == [180..270) 1387 // 3 == [270..360) 1388 // 1389 int quadrant = 0; 1390 if (0 == y) { 1391 quadrant = 2; // 180 1392 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1393 } else if (0 == x) { 1394 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1395 quadrant = y > 0 ? 1 : 3; // 90 : 270 1396 } else { 1397 if (y < 0) { 1398 quadrant += 2; 1399 } 1400 if ((x < 0) != (y < 0)) { 1401 quadrant += 1; 1402 } 1403 } 1404 1405 const SkPoint quadrantPts[] = { 1406 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1407 }; 1408 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1409 1410 int conicCount = quadrant; 1411 for (int i = 0; i < conicCount; ++i) { 1412 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1413 } 1414 1415 // Now compute any remaing (sub-90-degree) arc for the last conic 1416 const SkPoint finalP = { x, y }; 1417 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1418 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1419 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1420 1421 if (dot < 1) { 1422 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1423 // compute the bisector vector, and then rescale to be the off-curve point. 1424 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1425 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1426 // This is nice, since our computed weight is cos(theta/2) as well! 1427 // 1428 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1429 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1430 if (!lastQ.equalsWithinTolerance(offCurve)) { 1431 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1432 conicCount += 1; 1433 } 1434 } 1435 1436 // now handle counter-clockwise and the initial unitStart rotation 1437 SkMatrix matrix; 1438 matrix.setSinCos(uStart.fY, uStart.fX); 1439 if (dir == kCCW_SkRotationDirection) { 1440 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1441 } 1442 if (userMatrix) { 1443 matrix.postConcat(*userMatrix); 1444 } 1445 for (int i = 0; i < conicCount; ++i) { 1446 matrix.mapPoints(dst[i].fPts, 3); 1447 } 1448 return conicCount; 1449 } 1450