SkGeometry.cpp revision b6474dd1a530a543ae799c3822e8bc60180761c0
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 12static SkVector to_vector(const Sk2s& x) { 13 SkVector vector; 14 x.store(&vector.fX); 15 return vector; 16} 17 18/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 19 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 20 May also introduce overflow of fixed when we compute our setup. 21*/ 22// #define DIRECT_EVAL_OF_POLYNOMIALS 23 24//////////////////////////////////////////////////////////////////////// 25 26static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 27 SkScalar ab = a - b; 28 SkScalar bc = b - c; 29 if (ab < 0) { 30 bc = -bc; 31 } 32 return ab == 0 || bc < 0; 33} 34 35//////////////////////////////////////////////////////////////////////// 36 37static bool is_unit_interval(SkScalar x) { 38 return x > 0 && x < SK_Scalar1; 39} 40 41static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 42 SkASSERT(ratio); 43 44 if (numer < 0) { 45 numer = -numer; 46 denom = -denom; 47 } 48 49 if (denom == 0 || numer == 0 || numer >= denom) { 50 return 0; 51 } 52 53 SkScalar r = numer / denom; 54 if (SkScalarIsNaN(r)) { 55 return 0; 56 } 57 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 58 if (r == 0) { // catch underflow if numer <<<< denom 59 return 0; 60 } 61 *ratio = r; 62 return 1; 63} 64 65/** From Numerical Recipes in C. 66 67 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 68 x1 = Q / A 69 x2 = C / Q 70*/ 71int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 72 SkASSERT(roots); 73 74 if (A == 0) { 75 return valid_unit_divide(-C, B, roots); 76 } 77 78 SkScalar* r = roots; 79 80 SkScalar R = B*B - 4*A*C; 81 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 82 // if R is infinite, it's possible that it may still produce 83 // useful results if the operation was repeated in doubles 84 // the flipside is determining if the more precise answer 85 // isn't useful because surrounding machinery (e.g., subtracting 86 // the axis offset from C) already discards the extra precision 87 // more investigation and unit tests required... 88 return 0; 89 } 90 R = SkScalarSqrt(R); 91 92 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 93 r += valid_unit_divide(Q, A, r); 94 r += valid_unit_divide(C, Q, r); 95 if (r - roots == 2) { 96 if (roots[0] > roots[1]) 97 SkTSwap<SkScalar>(roots[0], roots[1]); 98 else if (roots[0] == roots[1]) // nearly-equal? 99 r -= 1; // skip the double root 100 } 101 return (int)(r - roots); 102} 103 104/////////////////////////////////////////////////////////////////////////////// 105/////////////////////////////////////////////////////////////////////////////// 106 107static Sk2s quad_poly_eval(const Sk2s& A, const Sk2s& B, const Sk2s& C, const Sk2s& t) { 108 return (A * t + B) * t + C; 109} 110 111static SkScalar eval_quad(const SkScalar src[], SkScalar t) { 112 SkASSERT(src); 113 SkASSERT(t >= 0 && t <= SK_Scalar1); 114 115#ifdef DIRECT_EVAL_OF_POLYNOMIALS 116 SkScalar C = src[0]; 117 SkScalar A = src[4] - 2 * src[2] + C; 118 SkScalar B = 2 * (src[2] - C); 119 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 120#else 121 SkScalar ab = SkScalarInterp(src[0], src[2], t); 122 SkScalar bc = SkScalarInterp(src[2], src[4], t); 123 return SkScalarInterp(ab, bc, t); 124#endif 125} 126 127void SkQuadToCoeff(const SkPoint pts[3], SkPoint coeff[3]) { 128 Sk2s p0 = from_point(pts[0]); 129 Sk2s p1 = from_point(pts[1]); 130 Sk2s p2 = from_point(pts[2]); 131 132 Sk2s p1minus2 = p1 - p0; 133 134 coeff[0] = to_point(p2 - p1 - p1 + p0); // A * t^2 135 coeff[1] = to_point(p1minus2 + p1minus2); // B * t 136 coeff[2] = pts[0]; // C 137} 138 139void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 140 SkASSERT(src); 141 SkASSERT(t >= 0 && t <= SK_Scalar1); 142 143 if (pt) { 144 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 145 } 146 if (tangent) { 147 *tangent = SkEvalQuadTangentAt(src, t); 148 } 149} 150 151SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 152 SkASSERT(src); 153 SkASSERT(t >= 0 && t <= SK_Scalar1); 154 155 const Sk2s t2(t); 156 157 Sk2s P0 = from_point(src[0]); 158 Sk2s P1 = from_point(src[1]); 159 Sk2s P2 = from_point(src[2]); 160 161 Sk2s B = P1 - P0; 162 Sk2s A = P2 - P1 - B; 163 164 return to_point((A * t2 + B+B) * t2 + P0); 165} 166 167SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 168 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 169 // zero tangent vector when t is 0 or 1, and the control point is equal 170 // to the end point. In this case, use the quad end points to compute the tangent. 171 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 172 return src[2] - src[0]; 173 } 174 SkASSERT(src); 175 SkASSERT(t >= 0 && t <= SK_Scalar1); 176 177 Sk2s P0 = from_point(src[0]); 178 Sk2s P1 = from_point(src[1]); 179 Sk2s P2 = from_point(src[2]); 180 181 Sk2s B = P1 - P0; 182 Sk2s A = P2 - P1 - B; 183 Sk2s T = A * Sk2s(t) + B; 184 185 return to_vector(T + T); 186} 187 188static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 189 return v0 + (v1 - v0) * t; 190} 191 192void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 193 SkASSERT(t > 0 && t < SK_Scalar1); 194 195 Sk2s p0 = from_point(src[0]); 196 Sk2s p1 = from_point(src[1]); 197 Sk2s p2 = from_point(src[2]); 198 Sk2s tt(t); 199 200 Sk2s p01 = interp(p0, p1, tt); 201 Sk2s p12 = interp(p1, p2, tt); 202 203 dst[0] = to_point(p0); 204 dst[1] = to_point(p01); 205 dst[2] = to_point(interp(p01, p12, tt)); 206 dst[3] = to_point(p12); 207 dst[4] = to_point(p2); 208} 209 210void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 211 SkChopQuadAt(src, dst, 0.5f); 212} 213 214/** Quad'(t) = At + B, where 215 A = 2(a - 2b + c) 216 B = 2(b - a) 217 Solve for t, only if it fits between 0 < t < 1 218*/ 219int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 220 /* At + B == 0 221 t = -B / A 222 */ 223 return valid_unit_divide(a - b, a - b - b + c, tValue); 224} 225 226static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 227 coords[2] = coords[6] = coords[4]; 228} 229 230/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 231 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 232 */ 233int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 234 SkASSERT(src); 235 SkASSERT(dst); 236 237 SkScalar a = src[0].fY; 238 SkScalar b = src[1].fY; 239 SkScalar c = src[2].fY; 240 241 if (is_not_monotonic(a, b, c)) { 242 SkScalar tValue; 243 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 244 SkChopQuadAt(src, dst, tValue); 245 flatten_double_quad_extrema(&dst[0].fY); 246 return 1; 247 } 248 // if we get here, we need to force dst to be monotonic, even though 249 // we couldn't compute a unit_divide value (probably underflow). 250 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 251 } 252 dst[0].set(src[0].fX, a); 253 dst[1].set(src[1].fX, b); 254 dst[2].set(src[2].fX, c); 255 return 0; 256} 257 258/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 259 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 260 */ 261int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 262 SkASSERT(src); 263 SkASSERT(dst); 264 265 SkScalar a = src[0].fX; 266 SkScalar b = src[1].fX; 267 SkScalar c = src[2].fX; 268 269 if (is_not_monotonic(a, b, c)) { 270 SkScalar tValue; 271 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 272 SkChopQuadAt(src, dst, tValue); 273 flatten_double_quad_extrema(&dst[0].fX); 274 return 1; 275 } 276 // if we get here, we need to force dst to be monotonic, even though 277 // we couldn't compute a unit_divide value (probably underflow). 278 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 279 } 280 dst[0].set(a, src[0].fY); 281 dst[1].set(b, src[1].fY); 282 dst[2].set(c, src[2].fY); 283 return 0; 284} 285 286// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 287// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 288// F''(t) = 2 (a - 2b + c) 289// 290// A = 2 (b - a) 291// B = 2 (a - 2b + c) 292// 293// Maximum curvature for a quadratic means solving 294// Fx' Fx'' + Fy' Fy'' = 0 295// 296// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 297// 298SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 299 SkScalar Ax = src[1].fX - src[0].fX; 300 SkScalar Ay = src[1].fY - src[0].fY; 301 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 302 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 303 SkScalar t = 0; // 0 means don't chop 304 305 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 306 return t; 307} 308 309int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 310 SkScalar t = SkFindQuadMaxCurvature(src); 311 if (t == 0) { 312 memcpy(dst, src, 3 * sizeof(SkPoint)); 313 return 1; 314 } else { 315 SkChopQuadAt(src, dst, t); 316 return 2; 317 } 318} 319 320void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 321 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 322 Sk2s s0 = from_point(src[0]); 323 Sk2s s1 = from_point(src[1]); 324 Sk2s s2 = from_point(src[2]); 325 326 dst[0] = src[0]; 327 dst[1] = to_point(s0 + (s1 - s0) * scale); 328 dst[2] = to_point(s2 + (s1 - s2) * scale); 329 dst[3] = src[2]; 330} 331 332////////////////////////////////////////////////////////////////////////////// 333///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 334////////////////////////////////////////////////////////////////////////////// 335 336static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 337 SkASSERT(src); 338 SkASSERT(t >= 0 && t <= SK_Scalar1); 339 340 if (t == 0) { 341 return src[0]; 342 } 343 344#ifdef DIRECT_EVAL_OF_POLYNOMIALS 345 SkScalar D = src[0]; 346 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 347 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 348 SkScalar C = 3*(src[2] - D); 349 350 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 351#else 352 SkScalar ab = SkScalarInterp(src[0], src[2], t); 353 SkScalar bc = SkScalarInterp(src[2], src[4], t); 354 SkScalar cd = SkScalarInterp(src[4], src[6], t); 355 SkScalar abc = SkScalarInterp(ab, bc, t); 356 SkScalar bcd = SkScalarInterp(bc, cd, t); 357 return SkScalarInterp(abc, bcd, t); 358#endif 359} 360 361/** return At^2 + Bt + C 362*/ 363static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { 364 SkASSERT(t >= 0 && t <= SK_Scalar1); 365 366 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 367} 368 369static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { 370 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 371 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 372 SkScalar C = src[2] - src[0]; 373 374 return eval_quadratic(A, B, C, t); 375} 376 377static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { 378 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 379 SkScalar B = src[4] - 2 * src[2] + src[0]; 380 381 return SkScalarMulAdd(A, t, B); 382} 383 384void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 385 SkVector* tangent, SkVector* curvature) { 386 SkASSERT(src); 387 SkASSERT(t >= 0 && t <= SK_Scalar1); 388 389 if (loc) { 390 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 391 } 392 if (tangent) { 393 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 394 // adjacent control point is equal to the end point. In this case, use the 395 // next control point or the end points to compute the tangent. 396 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 397 if (t == 0) { 398 *tangent = src[2] - src[0]; 399 } else { 400 *tangent = src[3] - src[1]; 401 } 402 if (!tangent->fX && !tangent->fY) { 403 *tangent = src[3] - src[0]; 404 } 405 } else { 406 tangent->set(eval_cubic_derivative(&src[0].fX, t), 407 eval_cubic_derivative(&src[0].fY, t)); 408 } 409 } 410 if (curvature) { 411 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 412 eval_cubic_2ndDerivative(&src[0].fY, t)); 413 } 414} 415 416/** Cubic'(t) = At^2 + Bt + C, where 417 A = 3(-a + 3(b - c) + d) 418 B = 6(a - 2b + c) 419 C = 3(b - a) 420 Solve for t, keeping only those that fit betwee 0 < t < 1 421*/ 422int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 423 SkScalar tValues[2]) { 424 // we divide A,B,C by 3 to simplify 425 SkScalar A = d - a + 3*(b - c); 426 SkScalar B = 2*(a - b - b + c); 427 SkScalar C = b - a; 428 429 return SkFindUnitQuadRoots(A, B, C, tValues); 430} 431 432void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 433 SkASSERT(t > 0 && t < SK_Scalar1); 434 435 Sk2s p0 = from_point(src[0]); 436 Sk2s p1 = from_point(src[1]); 437 Sk2s p2 = from_point(src[2]); 438 Sk2s p3 = from_point(src[3]); 439 Sk2s tt(t); 440 441 Sk2s ab = interp(p0, p1, tt); 442 Sk2s bc = interp(p1, p2, tt); 443 Sk2s cd = interp(p2, p3, tt); 444 Sk2s abc = interp(ab, bc, tt); 445 Sk2s bcd = interp(bc, cd, tt); 446 Sk2s abcd = interp(abc, bcd, tt); 447 448 dst[0] = src[0]; 449 dst[1] = to_point(ab); 450 dst[2] = to_point(abc); 451 dst[3] = to_point(abcd); 452 dst[4] = to_point(bcd); 453 dst[5] = to_point(cd); 454 dst[6] = src[3]; 455} 456 457void SkCubicToCoeff(const SkPoint pts[4], SkPoint coeff[4]) { 458 Sk2s p0 = from_point(pts[0]); 459 Sk2s p1 = from_point(pts[1]); 460 Sk2s p2 = from_point(pts[2]); 461 Sk2s p3 = from_point(pts[3]); 462 463 const Sk2s three(3); 464 Sk2s p1minusp2 = p1 - p2; 465 466 Sk2s D = p0; 467 Sk2s A = p3 + three * p1minusp2 - D; 468 Sk2s B = three * (D - p1minusp2 - p1); 469 Sk2s C = three * (p1 - D); 470 471 coeff[0] = to_point(A); 472 coeff[1] = to_point(B); 473 coeff[2] = to_point(C); 474 coeff[3] = to_point(D); 475} 476 477/* http://code.google.com/p/skia/issues/detail?id=32 478 479 This test code would fail when we didn't check the return result of 480 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 481 that after the first chop, the parameters to valid_unit_divide are equal 482 (thanks to finite float precision and rounding in the subtracts). Thus 483 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 484 up with 1.0, hence the need to check and just return the last cubic as 485 a degenerate clump of 4 points in the sampe place. 486 487 static void test_cubic() { 488 SkPoint src[4] = { 489 { 556.25000, 523.03003 }, 490 { 556.23999, 522.96002 }, 491 { 556.21997, 522.89001 }, 492 { 556.21997, 522.82001 } 493 }; 494 SkPoint dst[10]; 495 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 496 SkChopCubicAt(src, dst, tval, 2); 497 } 498 */ 499 500void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 501 const SkScalar tValues[], int roots) { 502#ifdef SK_DEBUG 503 { 504 for (int i = 0; i < roots - 1; i++) 505 { 506 SkASSERT(is_unit_interval(tValues[i])); 507 SkASSERT(is_unit_interval(tValues[i+1])); 508 SkASSERT(tValues[i] < tValues[i+1]); 509 } 510 } 511#endif 512 513 if (dst) { 514 if (roots == 0) { // nothing to chop 515 memcpy(dst, src, 4*sizeof(SkPoint)); 516 } else { 517 SkScalar t = tValues[0]; 518 SkPoint tmp[4]; 519 520 for (int i = 0; i < roots; i++) { 521 SkChopCubicAt(src, dst, t); 522 if (i == roots - 1) { 523 break; 524 } 525 526 dst += 3; 527 // have src point to the remaining cubic (after the chop) 528 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 529 src = tmp; 530 531 // watch out in case the renormalized t isn't in range 532 if (!valid_unit_divide(tValues[i+1] - tValues[i], 533 SK_Scalar1 - tValues[i], &t)) { 534 // if we can't, just create a degenerate cubic 535 dst[4] = dst[5] = dst[6] = src[3]; 536 break; 537 } 538 } 539 } 540 } 541} 542 543void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 544 SkChopCubicAt(src, dst, 0.5f); 545} 546 547static void flatten_double_cubic_extrema(SkScalar coords[14]) { 548 coords[4] = coords[8] = coords[6]; 549} 550 551/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 552 the resulting beziers are monotonic in Y. This is called by the scan 553 converter. Depending on what is returned, dst[] is treated as follows: 554 0 dst[0..3] is the original cubic 555 1 dst[0..3] and dst[3..6] are the two new cubics 556 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 557 If dst == null, it is ignored and only the count is returned. 558*/ 559int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 560 SkScalar tValues[2]; 561 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 562 src[3].fY, tValues); 563 564 SkChopCubicAt(src, dst, tValues, roots); 565 if (dst && roots > 0) { 566 // we do some cleanup to ensure our Y extrema are flat 567 flatten_double_cubic_extrema(&dst[0].fY); 568 if (roots == 2) { 569 flatten_double_cubic_extrema(&dst[3].fY); 570 } 571 } 572 return roots; 573} 574 575int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 576 SkScalar tValues[2]; 577 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 578 src[3].fX, tValues); 579 580 SkChopCubicAt(src, dst, tValues, roots); 581 if (dst && roots > 0) { 582 // we do some cleanup to ensure our Y extrema are flat 583 flatten_double_cubic_extrema(&dst[0].fX); 584 if (roots == 2) { 585 flatten_double_cubic_extrema(&dst[3].fX); 586 } 587 } 588 return roots; 589} 590 591/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 592 593 Inflection means that curvature is zero. 594 Curvature is [F' x F''] / [F'^3] 595 So we solve F'x X F''y - F'y X F''y == 0 596 After some canceling of the cubic term, we get 597 A = b - a 598 B = c - 2b + a 599 C = d - 3c + 3b - a 600 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 601*/ 602int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 603 SkScalar Ax = src[1].fX - src[0].fX; 604 SkScalar Ay = src[1].fY - src[0].fY; 605 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 606 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 607 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 608 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 609 610 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 611 Ax*Cy - Ay*Cx, 612 Ax*By - Ay*Bx, 613 tValues); 614} 615 616int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 617 SkScalar tValues[2]; 618 int count = SkFindCubicInflections(src, tValues); 619 620 if (dst) { 621 if (count == 0) { 622 memcpy(dst, src, 4 * sizeof(SkPoint)); 623 } else { 624 SkChopCubicAt(src, dst, tValues, count); 625 } 626 } 627 return count + 1; 628} 629 630// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3) 631// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 632// Classification: 633// discr(I) > 0 Serpentine 634// discr(I) = 0 Cusp 635// discr(I) < 0 Loop 636// d0 = d1 = 0 Quadratic 637// d0 = d1 = d2 = 0 Line 638// p0 = p1 = p2 = p3 Point 639static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 640 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 641 return kPoint_SkCubicType; 642 } 643 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 644 if (discr > SK_ScalarNearlyZero) { 645 return kSerpentine_SkCubicType; 646 } else if (discr < -SK_ScalarNearlyZero) { 647 return kLoop_SkCubicType; 648 } else { 649 if (0.f == d[0] && 0.f == d[1]) { 650 return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType); 651 } else { 652 return kCusp_SkCubicType; 653 } 654 } 655} 656 657// Assumes the third component of points is 1. 658// Calcs p0 . (p1 x p2) 659static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 660 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 661 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 662 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 663 return (xComp + yComp + wComp); 664} 665 666// Calc coefficients of I(s,t) where roots of I are inflection points of curve 667// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 668// d0 = a1 - 2*a2+3*a3 669// d1 = -a2 + 3*a3 670// d2 = 3*a3 671// a1 = p0 . (p3 x p2) 672// a2 = p1 . (p0 x p3) 673// a3 = p2 . (p1 x p0) 674// Places the values of d1, d2, d3 in array d passed in 675static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 676 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 677 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 678 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 679 680 // need to scale a's or values in later calculations will grow to high 681 SkScalar max = SkScalarAbs(a1); 682 max = SkMaxScalar(max, SkScalarAbs(a2)); 683 max = SkMaxScalar(max, SkScalarAbs(a3)); 684 max = 1.f/max; 685 a1 = a1 * max; 686 a2 = a2 * max; 687 a3 = a3 * max; 688 689 d[2] = 3.f * a3; 690 d[1] = d[2] - a2; 691 d[0] = d[1] - a2 + a1; 692} 693 694SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) { 695 calc_cubic_inflection_func(src, d); 696 return classify_cubic(src, d); 697} 698 699template <typename T> void bubble_sort(T array[], int count) { 700 for (int i = count - 1; i > 0; --i) 701 for (int j = i; j > 0; --j) 702 if (array[j] < array[j-1]) 703 { 704 T tmp(array[j]); 705 array[j] = array[j-1]; 706 array[j-1] = tmp; 707 } 708} 709 710/** 711 * Given an array and count, remove all pair-wise duplicates from the array, 712 * keeping the existing sorting, and return the new count 713 */ 714static int collaps_duplicates(SkScalar array[], int count) { 715 for (int n = count; n > 1; --n) { 716 if (array[0] == array[1]) { 717 for (int i = 1; i < n; ++i) { 718 array[i - 1] = array[i]; 719 } 720 count -= 1; 721 } else { 722 array += 1; 723 } 724 } 725 return count; 726} 727 728#ifdef SK_DEBUG 729 730#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 731 732static void test_collaps_duplicates() { 733 static bool gOnce; 734 if (gOnce) { return; } 735 gOnce = true; 736 const SkScalar src0[] = { 0 }; 737 const SkScalar src1[] = { 0, 0 }; 738 const SkScalar src2[] = { 0, 1 }; 739 const SkScalar src3[] = { 0, 0, 0 }; 740 const SkScalar src4[] = { 0, 0, 1 }; 741 const SkScalar src5[] = { 0, 1, 1 }; 742 const SkScalar src6[] = { 0, 1, 2 }; 743 const struct { 744 const SkScalar* fData; 745 int fCount; 746 int fCollapsedCount; 747 } data[] = { 748 { TEST_COLLAPS_ENTRY(src0), 1 }, 749 { TEST_COLLAPS_ENTRY(src1), 1 }, 750 { TEST_COLLAPS_ENTRY(src2), 2 }, 751 { TEST_COLLAPS_ENTRY(src3), 1 }, 752 { TEST_COLLAPS_ENTRY(src4), 2 }, 753 { TEST_COLLAPS_ENTRY(src5), 2 }, 754 { TEST_COLLAPS_ENTRY(src6), 3 }, 755 }; 756 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 757 SkScalar dst[3]; 758 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 759 int count = collaps_duplicates(dst, data[i].fCount); 760 SkASSERT(data[i].fCollapsedCount == count); 761 for (int j = 1; j < count; ++j) { 762 SkASSERT(dst[j-1] < dst[j]); 763 } 764 } 765} 766#endif 767 768static SkScalar SkScalarCubeRoot(SkScalar x) { 769 return SkScalarPow(x, 0.3333333f); 770} 771 772/* Solve coeff(t) == 0, returning the number of roots that 773 lie withing 0 < t < 1. 774 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 775 776 Eliminates repeated roots (so that all tValues are distinct, and are always 777 in increasing order. 778*/ 779static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 780 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 781 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 782 } 783 784 SkScalar a, b, c, Q, R; 785 786 { 787 SkASSERT(coeff[0] != 0); 788 789 SkScalar inva = SkScalarInvert(coeff[0]); 790 a = coeff[1] * inva; 791 b = coeff[2] * inva; 792 c = coeff[3] * inva; 793 } 794 Q = (a*a - b*3) / 9; 795 R = (2*a*a*a - 9*a*b + 27*c) / 54; 796 797 SkScalar Q3 = Q * Q * Q; 798 SkScalar R2MinusQ3 = R * R - Q3; 799 SkScalar adiv3 = a / 3; 800 801 SkScalar* roots = tValues; 802 SkScalar r; 803 804 if (R2MinusQ3 < 0) { // we have 3 real roots 805 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 806 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 807 808 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 809 if (is_unit_interval(r)) { 810 *roots++ = r; 811 } 812 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 813 if (is_unit_interval(r)) { 814 *roots++ = r; 815 } 816 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 817 if (is_unit_interval(r)) { 818 *roots++ = r; 819 } 820 SkDEBUGCODE(test_collaps_duplicates();) 821 822 // now sort the roots 823 int count = (int)(roots - tValues); 824 SkASSERT((unsigned)count <= 3); 825 bubble_sort(tValues, count); 826 count = collaps_duplicates(tValues, count); 827 roots = tValues + count; // so we compute the proper count below 828 } else { // we have 1 real root 829 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 830 A = SkScalarCubeRoot(A); 831 if (R > 0) { 832 A = -A; 833 } 834 if (A != 0) { 835 A += Q / A; 836 } 837 r = A - adiv3; 838 if (is_unit_interval(r)) { 839 *roots++ = r; 840 } 841 } 842 843 return (int)(roots - tValues); 844} 845 846/* Looking for F' dot F'' == 0 847 848 A = b - a 849 B = c - 2b + a 850 C = d - 3c + 3b - a 851 852 F' = 3Ct^2 + 6Bt + 3A 853 F'' = 6Ct + 6B 854 855 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 856*/ 857static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 858 SkScalar a = src[2] - src[0]; 859 SkScalar b = src[4] - 2 * src[2] + src[0]; 860 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 861 862 coeff[0] = c * c; 863 coeff[1] = 3 * b * c; 864 coeff[2] = 2 * b * b + c * a; 865 coeff[3] = a * b; 866} 867 868/* Looking for F' dot F'' == 0 869 870 A = b - a 871 B = c - 2b + a 872 C = d - 3c + 3b - a 873 874 F' = 3Ct^2 + 6Bt + 3A 875 F'' = 6Ct + 6B 876 877 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 878*/ 879int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 880 SkScalar coeffX[4], coeffY[4]; 881 int i; 882 883 formulate_F1DotF2(&src[0].fX, coeffX); 884 formulate_F1DotF2(&src[0].fY, coeffY); 885 886 for (i = 0; i < 4; i++) { 887 coeffX[i] += coeffY[i]; 888 } 889 890 SkScalar t[3]; 891 int count = solve_cubic_poly(coeffX, t); 892 int maxCount = 0; 893 894 // now remove extrema where the curvature is zero (mins) 895 // !!!! need a test for this !!!! 896 for (i = 0; i < count; i++) { 897 // if (not_min_curvature()) 898 if (t[i] > 0 && t[i] < SK_Scalar1) { 899 tValues[maxCount++] = t[i]; 900 } 901 } 902 return maxCount; 903} 904 905int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 906 SkScalar tValues[3]) { 907 SkScalar t_storage[3]; 908 909 if (tValues == nullptr) { 910 tValues = t_storage; 911 } 912 913 int count = SkFindCubicMaxCurvature(src, tValues); 914 915 if (dst) { 916 if (count == 0) { 917 memcpy(dst, src, 4 * sizeof(SkPoint)); 918 } else { 919 SkChopCubicAt(src, dst, tValues, count); 920 } 921 } 922 return count + 1; 923} 924 925#include "../pathops/SkPathOpsCubic.h" 926 927typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 928 929static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 930 InterceptProc method) { 931 SkDCubic cubic; 932 double roots[3]; 933 int count = (cubic.set(src).*method)(intercept, roots); 934 if (count > 0) { 935 SkDCubicPair pair = cubic.chopAt(roots[0]); 936 for (int i = 0; i < 7; ++i) { 937 dst[i] = pair.pts[i].asSkPoint(); 938 } 939 return true; 940 } 941 return false; 942} 943 944bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 945 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 946} 947 948bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 949 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 950} 951 952/////////////////////////////////////////////////////////////////////////////// 953 954/* Find t value for quadratic [a, b, c] = d. 955 Return 0 if there is no solution within [0, 1) 956*/ 957static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { 958 // At^2 + Bt + C = d 959 SkScalar A = a - 2 * b + c; 960 SkScalar B = 2 * (b - a); 961 SkScalar C = a - d; 962 963 SkScalar roots[2]; 964 int count = SkFindUnitQuadRoots(A, B, C, roots); 965 966 SkASSERT(count <= 1); 967 return count == 1 ? roots[0] : 0; 968} 969 970/* given a quad-curve and a point (x,y), chop the quad at that point and place 971 the new off-curve point and endpoint into 'dest'. 972 Should only return false if the computed pos is the start of the curve 973 (i.e. root == 0) 974*/ 975static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, 976 SkPoint* dest) { 977 const SkScalar* base; 978 SkScalar value; 979 980 if (SkScalarAbs(x) < SkScalarAbs(y)) { 981 base = &quad[0].fX; 982 value = x; 983 } else { 984 base = &quad[0].fY; 985 value = y; 986 } 987 988 // note: this returns 0 if it thinks value is out of range, meaning the 989 // root might return something outside of [0, 1) 990 SkScalar t = quad_solve(base[0], base[2], base[4], value); 991 992 if (t > 0) { 993 SkPoint tmp[5]; 994 SkChopQuadAt(quad, tmp, t); 995 dest[0] = tmp[1]; 996 dest[1].set(x, y); 997 return true; 998 } else { 999 /* t == 0 means either the value triggered a root outside of [0, 1) 1000 For our purposes, we can ignore the <= 0 roots, but we want to 1001 catch the >= 1 roots (which given our caller, will basically mean 1002 a root of 1, give-or-take numerical instability). If we are in the 1003 >= 1 case, return the existing offCurve point. 1004 1005 The test below checks to see if we are close to the "end" of the 1006 curve (near base[4]). Rather than specifying a tolerance, I just 1007 check to see if value is on to the right/left of the middle point 1008 (depending on the direction/sign of the end points). 1009 */ 1010 if ((base[0] < base[4] && value > base[2]) || 1011 (base[0] > base[4] && value < base[2])) // should root have been 1 1012 { 1013 dest[0] = quad[1]; 1014 dest[1].set(x, y); 1015 return true; 1016 } 1017 } 1018 return false; 1019} 1020 1021static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1022// The mid point of the quadratic arc approximation is half way between the two 1023// control points. The float epsilon adjustment moves the on curve point out by 1024// two bits, distributing the convex test error between the round rect 1025// approximation and the convex cross product sign equality test. 1026#define SK_MID_RRECT_OFFSET \ 1027 (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1028 { SK_Scalar1, 0 }, 1029 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1030 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1031 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1032 1033 { 0, SK_Scalar1 }, 1034 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1035 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1036 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1037 1038 { -SK_Scalar1, 0 }, 1039 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1040 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1041 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1042 1043 { 0, -SK_Scalar1 }, 1044 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1045 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1046 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1047 1048 { SK_Scalar1, 0 } 1049#undef SK_MID_RRECT_OFFSET 1050}; 1051 1052int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1053 SkRotationDirection dir, const SkMatrix* userMatrix, 1054 SkPoint quadPoints[]) { 1055 // rotate by x,y so that uStart is (1.0) 1056 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1057 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1058 1059 SkScalar absX = SkScalarAbs(x); 1060 SkScalar absY = SkScalarAbs(y); 1061 1062 int pointCount; 1063 1064 // check for (effectively) coincident vectors 1065 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1066 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1067 if (absY <= SK_ScalarNearlyZero && x > 0 && 1068 ((y >= 0 && kCW_SkRotationDirection == dir) || 1069 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1070 1071 // just return the start-point 1072 quadPoints[0].set(SK_Scalar1, 0); 1073 pointCount = 1; 1074 } else { 1075 if (dir == kCCW_SkRotationDirection) { 1076 y = -y; 1077 } 1078 // what octant (quadratic curve) is [xy] in? 1079 int oct = 0; 1080 bool sameSign = true; 1081 1082 if (0 == y) { 1083 oct = 4; // 180 1084 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1085 } else if (0 == x) { 1086 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1087 oct = y > 0 ? 2 : 6; // 90 : 270 1088 } else { 1089 if (y < 0) { 1090 oct += 4; 1091 } 1092 if ((x < 0) != (y < 0)) { 1093 oct += 2; 1094 sameSign = false; 1095 } 1096 if ((absX < absY) == sameSign) { 1097 oct += 1; 1098 } 1099 } 1100 1101 int wholeCount = oct << 1; 1102 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1103 1104 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1105 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { 1106 wholeCount += 2; 1107 } 1108 pointCount = wholeCount + 1; 1109 } 1110 1111 // now handle counter-clockwise and the initial unitStart rotation 1112 SkMatrix matrix; 1113 matrix.setSinCos(uStart.fY, uStart.fX); 1114 if (dir == kCCW_SkRotationDirection) { 1115 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1116 } 1117 if (userMatrix) { 1118 matrix.postConcat(*userMatrix); 1119 } 1120 matrix.mapPoints(quadPoints, pointCount); 1121 return pointCount; 1122} 1123 1124 1125/////////////////////////////////////////////////////////////////////////////// 1126// 1127// NURB representation for conics. Helpful explanations at: 1128// 1129// http://citeseerx.ist.psu.edu/viewdoc/ 1130// download?doi=10.1.1.44.5740&rep=rep1&type=ps 1131// and 1132// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 1133// 1134// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1135// ------------------------------------------ 1136// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1137// 1138// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1139// ------------------------------------------------ 1140// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1141// 1142 1143// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1144// 1145// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1146// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1147// t^0 : -2 P0 w + 2 P1 w 1148// 1149// We disregard magnitude, so we can freely ignore the denominator of F', and 1150// divide the numerator by 2 1151// 1152// coeff[0] for t^2 1153// coeff[1] for t^1 1154// coeff[2] for t^0 1155// 1156static void conic_deriv_coeff(const SkScalar src[], 1157 SkScalar w, 1158 SkScalar coeff[3]) { 1159 const SkScalar P20 = src[4] - src[0]; 1160 const SkScalar P10 = src[2] - src[0]; 1161 const SkScalar wP10 = w * P10; 1162 coeff[0] = w * P20 - P20; 1163 coeff[1] = P20 - 2 * wP10; 1164 coeff[2] = wP10; 1165} 1166 1167static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1168 SkScalar coeff[3]; 1169 conic_deriv_coeff(src, w, coeff); 1170 1171 SkScalar tValues[2]; 1172 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1173 SkASSERT(0 == roots || 1 == roots); 1174 1175 if (1 == roots) { 1176 *t = tValues[0]; 1177 return true; 1178 } 1179 return false; 1180} 1181 1182struct SkP3D { 1183 SkScalar fX, fY, fZ; 1184 1185 void set(SkScalar x, SkScalar y, SkScalar z) { 1186 fX = x; fY = y; fZ = z; 1187 } 1188 1189 void projectDown(SkPoint* dst) const { 1190 dst->set(fX / fZ, fY / fZ); 1191 } 1192}; 1193 1194// We only interpolate one dimension at a time (the first, at +0, +3, +6). 1195static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 1196 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1197 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1198 dst[0] = ab; 1199 dst[3] = SkScalarInterp(ab, bc, t); 1200 dst[6] = bc; 1201} 1202 1203static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1204 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1205 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1206 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1207} 1208 1209void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1210 SkP3D tmp[3], tmp2[3]; 1211 1212 ratquad_mapTo3D(fPts, fW, tmp); 1213 1214 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1215 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1216 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1217 1218 dst[0].fPts[0] = fPts[0]; 1219 tmp2[0].projectDown(&dst[0].fPts[1]); 1220 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1221 tmp2[2].projectDown(&dst[1].fPts[1]); 1222 dst[1].fPts[2] = fPts[2]; 1223 1224 // to put in "standard form", where w0 and w2 are both 1, we compute the 1225 // new w1 as sqrt(w1*w1/w0*w2) 1226 // or 1227 // w1 /= sqrt(w0*w2) 1228 // 1229 // However, in our case, we know that for dst[0]: 1230 // w0 == 1, and for dst[1], w2 == 1 1231 // 1232 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1233 dst[0].fW = tmp2[0].fZ / root; 1234 dst[1].fW = tmp2[2].fZ / root; 1235} 1236 1237void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { 1238 if (0 == t1 || 1 == t2) { 1239 if (0 == t1 && 1 == t2) { 1240 *dst = *this; 1241 } else { 1242 SkConic pair[2]; 1243 this->chopAt(t1 ? t1 : t2, pair); 1244 *dst = pair[SkToBool(t1)]; 1245 } 1246 return; 1247 } 1248 SkConicCoeff coeff(*this); 1249 Sk2s tt1(t1); 1250 Sk2s aXY = coeff.fNumer.eval(tt1); 1251 Sk2s aZZ = coeff.fDenom.eval(tt1); 1252 Sk2s midTT((t1 + t2) / 2); 1253 Sk2s dXY = coeff.fNumer.eval(midTT); 1254 Sk2s dZZ = coeff.fDenom.eval(midTT); 1255 Sk2s tt2(t2); 1256 Sk2s cXY = coeff.fNumer.eval(tt2); 1257 Sk2s cZZ = coeff.fDenom.eval(tt2); 1258 Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); 1259 Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); 1260 dst->fPts[0] = to_point(aXY / aZZ); 1261 dst->fPts[1] = to_point(bXY / bZZ); 1262 dst->fPts[2] = to_point(cXY / cZZ); 1263 Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); 1264 dst->fW = ww.kth<0>(); 1265} 1266 1267SkPoint SkConic::evalAt(SkScalar t) const { 1268 Sk2s p0 = from_point(fPts[0]); 1269 Sk2s p1 = from_point(fPts[1]); 1270 Sk2s p2 = from_point(fPts[2]); 1271 Sk2s tt(t); 1272 Sk2s ww(fW); 1273 Sk2s one(1); 1274 1275 Sk2s p1w = p1 * ww; 1276 Sk2s C = p0; 1277 Sk2s A = p2 - times_2(p1w) + p0; 1278 Sk2s B = times_2(p1w - C); 1279 Sk2s numer = quad_poly_eval(A, B, C, tt); 1280 1281 B = times_2(ww - one); 1282 A = Sk2s(0)-B; 1283 Sk2s denom = quad_poly_eval(A, B, one, tt); 1284 1285 return to_point(numer / denom); 1286} 1287 1288SkVector SkConic::evalTangentAt(SkScalar t) const { 1289 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1290 // and the control point is equal to the end point. 1291 // In this case, use the conic endpoints to compute the tangent. 1292 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1293 return fPts[2] - fPts[0]; 1294 } 1295 Sk2s p0 = from_point(fPts[0]); 1296 Sk2s p1 = from_point(fPts[1]); 1297 Sk2s p2 = from_point(fPts[2]); 1298 Sk2s ww(fW); 1299 1300 Sk2s p20 = p2 - p0; 1301 Sk2s p10 = p1 - p0; 1302 1303 Sk2s C = ww * p10; 1304 Sk2s A = ww * p20 - p20; 1305 Sk2s B = p20 - C - C; 1306 1307 return to_vector(quad_poly_eval(A, B, C, Sk2s(t))); 1308} 1309 1310void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1311 SkASSERT(t >= 0 && t <= SK_Scalar1); 1312 1313 if (pt) { 1314 *pt = this->evalAt(t); 1315 } 1316 if (tangent) { 1317 *tangent = this->evalTangentAt(t); 1318 } 1319} 1320 1321static SkScalar subdivide_w_value(SkScalar w) { 1322 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1323} 1324 1325static Sk2s twice(const Sk2s& value) { 1326 return value + value; 1327} 1328 1329void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1330 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1331 SkScalar newW = subdivide_w_value(fW); 1332 1333 Sk2s p0 = from_point(fPts[0]); 1334 Sk2s p1 = from_point(fPts[1]); 1335 Sk2s p2 = from_point(fPts[2]); 1336 Sk2s ww(fW); 1337 1338 Sk2s wp1 = ww * p1; 1339 Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f); 1340 1341 dst[0].fPts[0] = fPts[0]; 1342 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1343 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1344 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1345 dst[1].fPts[2] = fPts[2]; 1346 1347 dst[0].fW = dst[1].fW = newW; 1348} 1349 1350/* 1351 * "High order approximation of conic sections by quadratic splines" 1352 * by Michael Floater, 1993 1353 */ 1354#define AS_QUAD_ERROR_SETUP \ 1355 SkScalar a = fW - 1; \ 1356 SkScalar k = a / (4 * (2 + a)); \ 1357 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1358 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1359 1360void SkConic::computeAsQuadError(SkVector* err) const { 1361 AS_QUAD_ERROR_SETUP 1362 err->set(x, y); 1363} 1364 1365bool SkConic::asQuadTol(SkScalar tol) const { 1366 AS_QUAD_ERROR_SETUP 1367 return (x * x + y * y) <= tol * tol; 1368} 1369 1370// Limit the number of suggested quads to approximate a conic 1371#define kMaxConicToQuadPOW2 5 1372 1373int SkConic::computeQuadPOW2(SkScalar tol) const { 1374 if (tol < 0 || !SkScalarIsFinite(tol)) { 1375 return 0; 1376 } 1377 1378 AS_QUAD_ERROR_SETUP 1379 1380 SkScalar error = SkScalarSqrt(x * x + y * y); 1381 int pow2; 1382 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1383 if (error <= tol) { 1384 break; 1385 } 1386 error *= 0.25f; 1387 } 1388 // float version -- using ceil gives the same results as the above. 1389 if (false) { 1390 SkScalar err = SkScalarSqrt(x * x + y * y); 1391 if (err <= tol) { 1392 return 0; 1393 } 1394 SkScalar tol2 = tol * tol; 1395 if (tol2 == 0) { 1396 return kMaxConicToQuadPOW2; 1397 } 1398 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1399 int altPow2 = SkScalarCeilToInt(fpow2); 1400 if (altPow2 != pow2) { 1401 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1402 } 1403 pow2 = altPow2; 1404 } 1405 return pow2; 1406} 1407 1408static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1409 SkASSERT(level >= 0); 1410 1411 if (0 == level) { 1412 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1413 return pts + 2; 1414 } else { 1415 SkConic dst[2]; 1416 src.chop(dst); 1417 --level; 1418 pts = subdivide(dst[0], pts, level); 1419 return subdivide(dst[1], pts, level); 1420 } 1421} 1422 1423int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1424 SkASSERT(pow2 >= 0); 1425 *pts = fPts[0]; 1426 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1427 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1428 return 1 << pow2; 1429} 1430 1431bool SkConic::findXExtrema(SkScalar* t) const { 1432 return conic_find_extrema(&fPts[0].fX, fW, t); 1433} 1434 1435bool SkConic::findYExtrema(SkScalar* t) const { 1436 return conic_find_extrema(&fPts[0].fY, fW, t); 1437} 1438 1439bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1440 SkScalar t; 1441 if (this->findXExtrema(&t)) { 1442 this->chopAt(t, dst); 1443 // now clean-up the middle, since we know t was meant to be at 1444 // an X-extrema 1445 SkScalar value = dst[0].fPts[2].fX; 1446 dst[0].fPts[1].fX = value; 1447 dst[1].fPts[0].fX = value; 1448 dst[1].fPts[1].fX = value; 1449 return true; 1450 } 1451 return false; 1452} 1453 1454bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1455 SkScalar t; 1456 if (this->findYExtrema(&t)) { 1457 this->chopAt(t, dst); 1458 // now clean-up the middle, since we know t was meant to be at 1459 // an Y-extrema 1460 SkScalar value = dst[0].fPts[2].fY; 1461 dst[0].fPts[1].fY = value; 1462 dst[1].fPts[0].fY = value; 1463 dst[1].fPts[1].fY = value; 1464 return true; 1465 } 1466 return false; 1467} 1468 1469void SkConic::computeTightBounds(SkRect* bounds) const { 1470 SkPoint pts[4]; 1471 pts[0] = fPts[0]; 1472 pts[1] = fPts[2]; 1473 int count = 2; 1474 1475 SkScalar t; 1476 if (this->findXExtrema(&t)) { 1477 this->evalAt(t, &pts[count++]); 1478 } 1479 if (this->findYExtrema(&t)) { 1480 this->evalAt(t, &pts[count++]); 1481 } 1482 bounds->set(pts, count); 1483} 1484 1485void SkConic::computeFastBounds(SkRect* bounds) const { 1486 bounds->set(fPts, 3); 1487} 1488 1489#if 0 // unimplemented 1490bool SkConic::findMaxCurvature(SkScalar* t) const { 1491 // TODO: Implement me 1492 return false; 1493} 1494#endif 1495 1496SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1497 const SkMatrix& matrix) { 1498 if (!matrix.hasPerspective()) { 1499 return w; 1500 } 1501 1502 SkP3D src[3], dst[3]; 1503 1504 ratquad_mapTo3D(pts, w, src); 1505 1506 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1507 1508 // w' = sqrt(w1*w1/w0*w2) 1509 SkScalar w0 = dst[0].fZ; 1510 SkScalar w1 = dst[1].fZ; 1511 SkScalar w2 = dst[2].fZ; 1512 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1513 return w; 1514} 1515 1516int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1517 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1518 // rotate by x,y so that uStart is (1.0) 1519 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1520 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1521 1522 SkScalar absY = SkScalarAbs(y); 1523 1524 // check for (effectively) coincident vectors 1525 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1526 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1527 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1528 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1529 return 0; 1530 } 1531 1532 if (dir == kCCW_SkRotationDirection) { 1533 y = -y; 1534 } 1535 1536 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1537 // 0 == [0 .. 90) 1538 // 1 == [90 ..180) 1539 // 2 == [180..270) 1540 // 3 == [270..360) 1541 // 1542 int quadrant = 0; 1543 if (0 == y) { 1544 quadrant = 2; // 180 1545 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1546 } else if (0 == x) { 1547 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1548 quadrant = y > 0 ? 1 : 3; // 90 : 270 1549 } else { 1550 if (y < 0) { 1551 quadrant += 2; 1552 } 1553 if ((x < 0) != (y < 0)) { 1554 quadrant += 1; 1555 } 1556 } 1557 1558 const SkPoint quadrantPts[] = { 1559 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1560 }; 1561 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1562 1563 int conicCount = quadrant; 1564 for (int i = 0; i < conicCount; ++i) { 1565 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1566 } 1567 1568 // Now compute any remaing (sub-90-degree) arc for the last conic 1569 const SkPoint finalP = { x, y }; 1570 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1571 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1572 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1573 1574 if (dot < 1) { 1575 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1576 // compute the bisector vector, and then rescale to be the off-curve point. 1577 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1578 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1579 // This is nice, since our computed weight is cos(theta/2) as well! 1580 // 1581 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1582 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1583 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1584 conicCount += 1; 1585 } 1586 1587 // now handle counter-clockwise and the initial unitStart rotation 1588 SkMatrix matrix; 1589 matrix.setSinCos(uStart.fY, uStart.fX); 1590 if (dir == kCCW_SkRotationDirection) { 1591 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1592 } 1593 if (userMatrix) { 1594 matrix.postConcat(*userMatrix); 1595 } 1596 for (int i = 0; i < conicCount; ++i) { 1597 matrix.mapPoints(dst[i].fPts, 3); 1598 } 1599 return conicCount; 1600} 1601