SkGeometry.cpp revision dbaec7323f20c3a7e8a234dac9dfb8a9446a2717
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#if 0 13static Sk2s from_point(const SkPoint& point) { 14 return Sk2s::Load(&point.fX); 15} 16 17static SkPoint to_point(const Sk2s& x) { 18 SkPoint point; 19 x.store(&point.fX); 20 return point; 21} 22#endif 23 24static SkVector to_vector(const Sk2s& x) { 25 SkVector vector; 26 x.store(&vector.fX); 27 return vector; 28} 29 30/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 31 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 32 May also introduce overflow of fixed when we compute our setup. 33*/ 34// #define DIRECT_EVAL_OF_POLYNOMIALS 35 36//////////////////////////////////////////////////////////////////////// 37 38static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 39 SkScalar ab = a - b; 40 SkScalar bc = b - c; 41 if (ab < 0) { 42 bc = -bc; 43 } 44 return ab == 0 || bc < 0; 45} 46 47//////////////////////////////////////////////////////////////////////// 48 49static bool is_unit_interval(SkScalar x) { 50 return x > 0 && x < SK_Scalar1; 51} 52 53static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 54 SkASSERT(ratio); 55 56 if (numer < 0) { 57 numer = -numer; 58 denom = -denom; 59 } 60 61 if (denom == 0 || numer == 0 || numer >= denom) { 62 return 0; 63 } 64 65 SkScalar r = numer / denom; 66 if (SkScalarIsNaN(r)) { 67 return 0; 68 } 69 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 70 if (r == 0) { // catch underflow if numer <<<< denom 71 return 0; 72 } 73 *ratio = r; 74 return 1; 75} 76 77/** From Numerical Recipes in C. 78 79 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 80 x1 = Q / A 81 x2 = C / Q 82*/ 83int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 84 SkASSERT(roots); 85 86 if (A == 0) { 87 return valid_unit_divide(-C, B, roots); 88 } 89 90 SkScalar* r = roots; 91 92 SkScalar R = B*B - 4*A*C; 93 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 94 return 0; 95 } 96 R = SkScalarSqrt(R); 97 98 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 99 r += valid_unit_divide(Q, A, r); 100 r += valid_unit_divide(C, Q, r); 101 if (r - roots == 2) { 102 if (roots[0] > roots[1]) 103 SkTSwap<SkScalar>(roots[0], roots[1]); 104 else if (roots[0] == roots[1]) // nearly-equal? 105 r -= 1; // skip the double root 106 } 107 return (int)(r - roots); 108} 109 110/////////////////////////////////////////////////////////////////////////////// 111/////////////////////////////////////////////////////////////////////////////// 112 113static Sk2s quad_poly_eval(const Sk2s& A, const Sk2s& B, const Sk2s& C, const Sk2s& t) { 114 return (A * t + B) * t + C; 115} 116 117static SkScalar eval_quad(const SkScalar src[], SkScalar t) { 118 SkASSERT(src); 119 SkASSERT(t >= 0 && t <= SK_Scalar1); 120 121#ifdef DIRECT_EVAL_OF_POLYNOMIALS 122 SkScalar C = src[0]; 123 SkScalar A = src[4] - 2 * src[2] + C; 124 SkScalar B = 2 * (src[2] - C); 125 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 126#else 127 SkScalar ab = SkScalarInterp(src[0], src[2], t); 128 SkScalar bc = SkScalarInterp(src[2], src[4], t); 129 return SkScalarInterp(ab, bc, t); 130#endif 131} 132 133void SkQuadToCoeff(const SkPoint pts[3], SkPoint coeff[3]) { 134 Sk2s p0 = from_point(pts[0]); 135 Sk2s p1 = from_point(pts[1]); 136 Sk2s p2 = from_point(pts[2]); 137 138 Sk2s p1minus2 = p1 - p0; 139 140 coeff[0] = to_point(p2 - p1 - p1 + p0); // A * t^2 141 coeff[1] = to_point(p1minus2 + p1minus2); // B * t 142 coeff[2] = pts[0]; // C 143} 144 145void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 146 SkASSERT(src); 147 SkASSERT(t >= 0 && t <= SK_Scalar1); 148 149 if (pt) { 150 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 151 } 152 if (tangent) { 153 *tangent = SkEvalQuadTangentAt(src, t); 154 } 155} 156 157SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 158 SkASSERT(src); 159 SkASSERT(t >= 0 && t <= SK_Scalar1); 160 161 const Sk2s t2(t); 162 163 Sk2s P0 = from_point(src[0]); 164 Sk2s P1 = from_point(src[1]); 165 Sk2s P2 = from_point(src[2]); 166 167 Sk2s B = P1 - P0; 168 Sk2s A = P2 - P1 - B; 169 170 return to_point((A * t2 + B+B) * t2 + P0); 171} 172 173SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 174 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 175 // zero tangent vector when t is 0 or 1, and the control point is equal 176 // to the end point. In this case, use the quad end points to compute the tangent. 177 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 178 return src[2] - src[0]; 179 } 180 SkASSERT(src); 181 SkASSERT(t >= 0 && t <= SK_Scalar1); 182 183 Sk2s P0 = from_point(src[0]); 184 Sk2s P1 = from_point(src[1]); 185 Sk2s P2 = from_point(src[2]); 186 187 Sk2s B = P1 - P0; 188 Sk2s A = P2 - P1 - B; 189 Sk2s T = A * Sk2s(t) + B; 190 191 return to_vector(T + T); 192} 193 194static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 195 return v0 + (v1 - v0) * t; 196} 197 198void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 199 SkASSERT(t > 0 && t < SK_Scalar1); 200 201 Sk2s p0 = from_point(src[0]); 202 Sk2s p1 = from_point(src[1]); 203 Sk2s p2 = from_point(src[2]); 204 Sk2s tt(t); 205 206 Sk2s p01 = interp(p0, p1, tt); 207 Sk2s p12 = interp(p1, p2, tt); 208 209 dst[0] = to_point(p0); 210 dst[1] = to_point(p01); 211 dst[2] = to_point(interp(p01, p12, tt)); 212 dst[3] = to_point(p12); 213 dst[4] = to_point(p2); 214} 215 216void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 217 SkChopQuadAt(src, dst, 0.5f); return; 218} 219 220/** Quad'(t) = At + B, where 221 A = 2(a - 2b + c) 222 B = 2(b - a) 223 Solve for t, only if it fits between 0 < t < 1 224*/ 225int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 226 /* At + B == 0 227 t = -B / A 228 */ 229 return valid_unit_divide(a - b, a - b - b + c, tValue); 230} 231 232static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 233 coords[2] = coords[6] = coords[4]; 234} 235 236/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 237 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 238 */ 239int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 240 SkASSERT(src); 241 SkASSERT(dst); 242 243 SkScalar a = src[0].fY; 244 SkScalar b = src[1].fY; 245 SkScalar c = src[2].fY; 246 247 if (is_not_monotonic(a, b, c)) { 248 SkScalar tValue; 249 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 250 SkChopQuadAt(src, dst, tValue); 251 flatten_double_quad_extrema(&dst[0].fY); 252 return 1; 253 } 254 // if we get here, we need to force dst to be monotonic, even though 255 // we couldn't compute a unit_divide value (probably underflow). 256 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 257 } 258 dst[0].set(src[0].fX, a); 259 dst[1].set(src[1].fX, b); 260 dst[2].set(src[2].fX, c); 261 return 0; 262} 263 264/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 265 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 266 */ 267int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 268 SkASSERT(src); 269 SkASSERT(dst); 270 271 SkScalar a = src[0].fX; 272 SkScalar b = src[1].fX; 273 SkScalar c = src[2].fX; 274 275 if (is_not_monotonic(a, b, c)) { 276 SkScalar tValue; 277 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 278 SkChopQuadAt(src, dst, tValue); 279 flatten_double_quad_extrema(&dst[0].fX); 280 return 1; 281 } 282 // if we get here, we need to force dst to be monotonic, even though 283 // we couldn't compute a unit_divide value (probably underflow). 284 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 285 } 286 dst[0].set(a, src[0].fY); 287 dst[1].set(b, src[1].fY); 288 dst[2].set(c, src[2].fY); 289 return 0; 290} 291 292// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 293// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 294// F''(t) = 2 (a - 2b + c) 295// 296// A = 2 (b - a) 297// B = 2 (a - 2b + c) 298// 299// Maximum curvature for a quadratic means solving 300// Fx' Fx'' + Fy' Fy'' = 0 301// 302// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 303// 304SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 305 SkScalar Ax = src[1].fX - src[0].fX; 306 SkScalar Ay = src[1].fY - src[0].fY; 307 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 308 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 309 SkScalar t = 0; // 0 means don't chop 310 311 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 312 return t; 313} 314 315int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 316 SkScalar t = SkFindQuadMaxCurvature(src); 317 if (t == 0) { 318 memcpy(dst, src, 3 * sizeof(SkPoint)); 319 return 1; 320 } else { 321 SkChopQuadAt(src, dst, t); 322 return 2; 323 } 324} 325 326void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 327 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 328 Sk2s s0 = from_point(src[0]); 329 Sk2s s1 = from_point(src[1]); 330 Sk2s s2 = from_point(src[2]); 331 332 dst[0] = src[0]; 333 dst[1] = to_point(s0 + (s1 - s0) * scale); 334 dst[2] = to_point(s2 + (s1 - s2) * scale); 335 dst[3] = src[2]; 336} 337 338////////////////////////////////////////////////////////////////////////////// 339///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 340////////////////////////////////////////////////////////////////////////////// 341 342static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 343 SkASSERT(src); 344 SkASSERT(t >= 0 && t <= SK_Scalar1); 345 346 if (t == 0) { 347 return src[0]; 348 } 349 350#ifdef DIRECT_EVAL_OF_POLYNOMIALS 351 SkScalar D = src[0]; 352 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 353 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 354 SkScalar C = 3*(src[2] - D); 355 356 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 357#else 358 SkScalar ab = SkScalarInterp(src[0], src[2], t); 359 SkScalar bc = SkScalarInterp(src[2], src[4], t); 360 SkScalar cd = SkScalarInterp(src[4], src[6], t); 361 SkScalar abc = SkScalarInterp(ab, bc, t); 362 SkScalar bcd = SkScalarInterp(bc, cd, t); 363 return SkScalarInterp(abc, bcd, t); 364#endif 365} 366 367/** return At^2 + Bt + C 368*/ 369static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { 370 SkASSERT(t >= 0 && t <= SK_Scalar1); 371 372 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 373} 374 375static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { 376 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 377 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 378 SkScalar C = src[2] - src[0]; 379 380 return eval_quadratic(A, B, C, t); 381} 382 383static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { 384 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 385 SkScalar B = src[4] - 2 * src[2] + src[0]; 386 387 return SkScalarMulAdd(A, t, B); 388} 389 390void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 391 SkVector* tangent, SkVector* curvature) { 392 SkASSERT(src); 393 SkASSERT(t >= 0 && t <= SK_Scalar1); 394 395 if (loc) { 396 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 397 } 398 if (tangent) { 399 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 400 // adjacent control point is equal to the end point. In this case, use the 401 // next control point or the end points to compute the tangent. 402 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 403 if (t == 0) { 404 *tangent = src[2] - src[0]; 405 } else { 406 *tangent = src[3] - src[1]; 407 } 408 if (!tangent->fX && !tangent->fY) { 409 *tangent = src[3] - src[0]; 410 } 411 } else { 412 tangent->set(eval_cubic_derivative(&src[0].fX, t), 413 eval_cubic_derivative(&src[0].fY, t)); 414 } 415 } 416 if (curvature) { 417 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 418 eval_cubic_2ndDerivative(&src[0].fY, t)); 419 } 420} 421 422/** Cubic'(t) = At^2 + Bt + C, where 423 A = 3(-a + 3(b - c) + d) 424 B = 6(a - 2b + c) 425 C = 3(b - a) 426 Solve for t, keeping only those that fit betwee 0 < t < 1 427*/ 428int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 429 SkScalar tValues[2]) { 430 // we divide A,B,C by 3 to simplify 431 SkScalar A = d - a + 3*(b - c); 432 SkScalar B = 2*(a - b - b + c); 433 SkScalar C = b - a; 434 435 return SkFindUnitQuadRoots(A, B, C, tValues); 436} 437 438void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 439 SkASSERT(t > 0 && t < SK_Scalar1); 440 441 Sk2s p0 = from_point(src[0]); 442 Sk2s p1 = from_point(src[1]); 443 Sk2s p2 = from_point(src[2]); 444 Sk2s p3 = from_point(src[3]); 445 Sk2s tt(t); 446 447 Sk2s ab = interp(p0, p1, tt); 448 Sk2s bc = interp(p1, p2, tt); 449 Sk2s cd = interp(p2, p3, tt); 450 Sk2s abc = interp(ab, bc, tt); 451 Sk2s bcd = interp(bc, cd, tt); 452 Sk2s abcd = interp(abc, bcd, tt); 453 454 dst[0] = src[0]; 455 dst[1] = to_point(ab); 456 dst[2] = to_point(abc); 457 dst[3] = to_point(abcd); 458 dst[4] = to_point(bcd); 459 dst[5] = to_point(cd); 460 dst[6] = src[3]; 461} 462 463void SkCubicToCoeff(const SkPoint pts[4], SkPoint coeff[4]) { 464 Sk2s p0 = from_point(pts[0]); 465 Sk2s p1 = from_point(pts[1]); 466 Sk2s p2 = from_point(pts[2]); 467 Sk2s p3 = from_point(pts[3]); 468 469 const Sk2s three(3); 470 Sk2s p1minusp2 = p1 - p2; 471 472 Sk2s D = p0; 473 Sk2s A = p3 + three * p1minusp2 - D; 474 Sk2s B = three * (D - p1minusp2 - p1); 475 Sk2s C = three * (p1 - D); 476 477 coeff[0] = to_point(A); 478 coeff[1] = to_point(B); 479 coeff[2] = to_point(C); 480 coeff[3] = to_point(D); 481} 482 483/* http://code.google.com/p/skia/issues/detail?id=32 484 485 This test code would fail when we didn't check the return result of 486 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 487 that after the first chop, the parameters to valid_unit_divide are equal 488 (thanks to finite float precision and rounding in the subtracts). Thus 489 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 490 up with 1.0, hence the need to check and just return the last cubic as 491 a degenerate clump of 4 points in the sampe place. 492 493 static void test_cubic() { 494 SkPoint src[4] = { 495 { 556.25000, 523.03003 }, 496 { 556.23999, 522.96002 }, 497 { 556.21997, 522.89001 }, 498 { 556.21997, 522.82001 } 499 }; 500 SkPoint dst[10]; 501 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 502 SkChopCubicAt(src, dst, tval, 2); 503 } 504 */ 505 506void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 507 const SkScalar tValues[], int roots) { 508#ifdef SK_DEBUG 509 { 510 for (int i = 0; i < roots - 1; i++) 511 { 512 SkASSERT(is_unit_interval(tValues[i])); 513 SkASSERT(is_unit_interval(tValues[i+1])); 514 SkASSERT(tValues[i] < tValues[i+1]); 515 } 516 } 517#endif 518 519 if (dst) { 520 if (roots == 0) { // nothing to chop 521 memcpy(dst, src, 4*sizeof(SkPoint)); 522 } else { 523 SkScalar t = tValues[0]; 524 SkPoint tmp[4]; 525 526 for (int i = 0; i < roots; i++) { 527 SkChopCubicAt(src, dst, t); 528 if (i == roots - 1) { 529 break; 530 } 531 532 dst += 3; 533 // have src point to the remaining cubic (after the chop) 534 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 535 src = tmp; 536 537 // watch out in case the renormalized t isn't in range 538 if (!valid_unit_divide(tValues[i+1] - tValues[i], 539 SK_Scalar1 - tValues[i], &t)) { 540 // if we can't, just create a degenerate cubic 541 dst[4] = dst[5] = dst[6] = src[3]; 542 break; 543 } 544 } 545 } 546 } 547} 548 549void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 550 SkChopCubicAt(src, dst, 0.5f); 551} 552 553static void flatten_double_cubic_extrema(SkScalar coords[14]) { 554 coords[4] = coords[8] = coords[6]; 555} 556 557/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 558 the resulting beziers are monotonic in Y. This is called by the scan 559 converter. Depending on what is returned, dst[] is treated as follows: 560 0 dst[0..3] is the original cubic 561 1 dst[0..3] and dst[3..6] are the two new cubics 562 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 563 If dst == null, it is ignored and only the count is returned. 564*/ 565int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 566 SkScalar tValues[2]; 567 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 568 src[3].fY, tValues); 569 570 SkChopCubicAt(src, dst, tValues, roots); 571 if (dst && roots > 0) { 572 // we do some cleanup to ensure our Y extrema are flat 573 flatten_double_cubic_extrema(&dst[0].fY); 574 if (roots == 2) { 575 flatten_double_cubic_extrema(&dst[3].fY); 576 } 577 } 578 return roots; 579} 580 581int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 582 SkScalar tValues[2]; 583 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 584 src[3].fX, tValues); 585 586 SkChopCubicAt(src, dst, tValues, roots); 587 if (dst && roots > 0) { 588 // we do some cleanup to ensure our Y extrema are flat 589 flatten_double_cubic_extrema(&dst[0].fX); 590 if (roots == 2) { 591 flatten_double_cubic_extrema(&dst[3].fX); 592 } 593 } 594 return roots; 595} 596 597/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 598 599 Inflection means that curvature is zero. 600 Curvature is [F' x F''] / [F'^3] 601 So we solve F'x X F''y - F'y X F''y == 0 602 After some canceling of the cubic term, we get 603 A = b - a 604 B = c - 2b + a 605 C = d - 3c + 3b - a 606 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 607*/ 608int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 609 SkScalar Ax = src[1].fX - src[0].fX; 610 SkScalar Ay = src[1].fY - src[0].fY; 611 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 612 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 613 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 614 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 615 616 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 617 Ax*Cy - Ay*Cx, 618 Ax*By - Ay*Bx, 619 tValues); 620} 621 622int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 623 SkScalar tValues[2]; 624 int count = SkFindCubicInflections(src, tValues); 625 626 if (dst) { 627 if (count == 0) { 628 memcpy(dst, src, 4 * sizeof(SkPoint)); 629 } else { 630 SkChopCubicAt(src, dst, tValues, count); 631 } 632 } 633 return count + 1; 634} 635 636// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3) 637// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 638// Classification: 639// discr(I) > 0 Serpentine 640// discr(I) = 0 Cusp 641// discr(I) < 0 Loop 642// d0 = d1 = 0 Quadratic 643// d0 = d1 = d2 = 0 Line 644// p0 = p1 = p2 = p3 Point 645static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 646 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 647 return kPoint_SkCubicType; 648 } 649 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 650 if (discr > SK_ScalarNearlyZero) { 651 return kSerpentine_SkCubicType; 652 } else if (discr < -SK_ScalarNearlyZero) { 653 return kLoop_SkCubicType; 654 } else { 655 if (0.f == d[0] && 0.f == d[1]) { 656 return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType); 657 } else { 658 return kCusp_SkCubicType; 659 } 660 } 661} 662 663// Assumes the third component of points is 1. 664// Calcs p0 . (p1 x p2) 665static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 666 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 667 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 668 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 669 return (xComp + yComp + wComp); 670} 671 672// Calc coefficients of I(s,t) where roots of I are inflection points of curve 673// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 674// d0 = a1 - 2*a2+3*a3 675// d1 = -a2 + 3*a3 676// d2 = 3*a3 677// a1 = p0 . (p3 x p2) 678// a2 = p1 . (p0 x p3) 679// a3 = p2 . (p1 x p0) 680// Places the values of d1, d2, d3 in array d passed in 681static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 682 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 683 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 684 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 685 686 // need to scale a's or values in later calculations will grow to high 687 SkScalar max = SkScalarAbs(a1); 688 max = SkMaxScalar(max, SkScalarAbs(a2)); 689 max = SkMaxScalar(max, SkScalarAbs(a3)); 690 max = 1.f/max; 691 a1 = a1 * max; 692 a2 = a2 * max; 693 a3 = a3 * max; 694 695 d[2] = 3.f * a3; 696 d[1] = d[2] - a2; 697 d[0] = d[1] - a2 + a1; 698} 699 700SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) { 701 calc_cubic_inflection_func(src, d); 702 return classify_cubic(src, d); 703} 704 705template <typename T> void bubble_sort(T array[], int count) { 706 for (int i = count - 1; i > 0; --i) 707 for (int j = i; j > 0; --j) 708 if (array[j] < array[j-1]) 709 { 710 T tmp(array[j]); 711 array[j] = array[j-1]; 712 array[j-1] = tmp; 713 } 714} 715 716/** 717 * Given an array and count, remove all pair-wise duplicates from the array, 718 * keeping the existing sorting, and return the new count 719 */ 720static int collaps_duplicates(SkScalar array[], int count) { 721 for (int n = count; n > 1; --n) { 722 if (array[0] == array[1]) { 723 for (int i = 1; i < n; ++i) { 724 array[i - 1] = array[i]; 725 } 726 count -= 1; 727 } else { 728 array += 1; 729 } 730 } 731 return count; 732} 733 734#ifdef SK_DEBUG 735 736#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 737 738static void test_collaps_duplicates() { 739 static bool gOnce; 740 if (gOnce) { return; } 741 gOnce = true; 742 const SkScalar src0[] = { 0 }; 743 const SkScalar src1[] = { 0, 0 }; 744 const SkScalar src2[] = { 0, 1 }; 745 const SkScalar src3[] = { 0, 0, 0 }; 746 const SkScalar src4[] = { 0, 0, 1 }; 747 const SkScalar src5[] = { 0, 1, 1 }; 748 const SkScalar src6[] = { 0, 1, 2 }; 749 const struct { 750 const SkScalar* fData; 751 int fCount; 752 int fCollapsedCount; 753 } data[] = { 754 { TEST_COLLAPS_ENTRY(src0), 1 }, 755 { TEST_COLLAPS_ENTRY(src1), 1 }, 756 { TEST_COLLAPS_ENTRY(src2), 2 }, 757 { TEST_COLLAPS_ENTRY(src3), 1 }, 758 { TEST_COLLAPS_ENTRY(src4), 2 }, 759 { TEST_COLLAPS_ENTRY(src5), 2 }, 760 { TEST_COLLAPS_ENTRY(src6), 3 }, 761 }; 762 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 763 SkScalar dst[3]; 764 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 765 int count = collaps_duplicates(dst, data[i].fCount); 766 SkASSERT(data[i].fCollapsedCount == count); 767 for (int j = 1; j < count; ++j) { 768 SkASSERT(dst[j-1] < dst[j]); 769 } 770 } 771} 772#endif 773 774static SkScalar SkScalarCubeRoot(SkScalar x) { 775 return SkScalarPow(x, 0.3333333f); 776} 777 778/* Solve coeff(t) == 0, returning the number of roots that 779 lie withing 0 < t < 1. 780 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 781 782 Eliminates repeated roots (so that all tValues are distinct, and are always 783 in increasing order. 784*/ 785static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 786 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 787 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 788 } 789 790 SkScalar a, b, c, Q, R; 791 792 { 793 SkASSERT(coeff[0] != 0); 794 795 SkScalar inva = SkScalarInvert(coeff[0]); 796 a = coeff[1] * inva; 797 b = coeff[2] * inva; 798 c = coeff[3] * inva; 799 } 800 Q = (a*a - b*3) / 9; 801 R = (2*a*a*a - 9*a*b + 27*c) / 54; 802 803 SkScalar Q3 = Q * Q * Q; 804 SkScalar R2MinusQ3 = R * R - Q3; 805 SkScalar adiv3 = a / 3; 806 807 SkScalar* roots = tValues; 808 SkScalar r; 809 810 if (R2MinusQ3 < 0) { // we have 3 real roots 811 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 812 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 813 814 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 815 if (is_unit_interval(r)) { 816 *roots++ = r; 817 } 818 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 819 if (is_unit_interval(r)) { 820 *roots++ = r; 821 } 822 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 823 if (is_unit_interval(r)) { 824 *roots++ = r; 825 } 826 SkDEBUGCODE(test_collaps_duplicates();) 827 828 // now sort the roots 829 int count = (int)(roots - tValues); 830 SkASSERT((unsigned)count <= 3); 831 bubble_sort(tValues, count); 832 count = collaps_duplicates(tValues, count); 833 roots = tValues + count; // so we compute the proper count below 834 } else { // we have 1 real root 835 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 836 A = SkScalarCubeRoot(A); 837 if (R > 0) { 838 A = -A; 839 } 840 if (A != 0) { 841 A += Q / A; 842 } 843 r = A - adiv3; 844 if (is_unit_interval(r)) { 845 *roots++ = r; 846 } 847 } 848 849 return (int)(roots - tValues); 850} 851 852/* Looking for F' dot F'' == 0 853 854 A = b - a 855 B = c - 2b + a 856 C = d - 3c + 3b - a 857 858 F' = 3Ct^2 + 6Bt + 3A 859 F'' = 6Ct + 6B 860 861 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 862*/ 863static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 864 SkScalar a = src[2] - src[0]; 865 SkScalar b = src[4] - 2 * src[2] + src[0]; 866 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 867 868 coeff[0] = c * c; 869 coeff[1] = 3 * b * c; 870 coeff[2] = 2 * b * b + c * a; 871 coeff[3] = a * b; 872} 873 874/* Looking for F' dot F'' == 0 875 876 A = b - a 877 B = c - 2b + a 878 C = d - 3c + 3b - a 879 880 F' = 3Ct^2 + 6Bt + 3A 881 F'' = 6Ct + 6B 882 883 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 884*/ 885int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 886 SkScalar coeffX[4], coeffY[4]; 887 int i; 888 889 formulate_F1DotF2(&src[0].fX, coeffX); 890 formulate_F1DotF2(&src[0].fY, coeffY); 891 892 for (i = 0; i < 4; i++) { 893 coeffX[i] += coeffY[i]; 894 } 895 896 SkScalar t[3]; 897 int count = solve_cubic_poly(coeffX, t); 898 int maxCount = 0; 899 900 // now remove extrema where the curvature is zero (mins) 901 // !!!! need a test for this !!!! 902 for (i = 0; i < count; i++) { 903 // if (not_min_curvature()) 904 if (t[i] > 0 && t[i] < SK_Scalar1) { 905 tValues[maxCount++] = t[i]; 906 } 907 } 908 return maxCount; 909} 910 911int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 912 SkScalar tValues[3]) { 913 SkScalar t_storage[3]; 914 915 if (tValues == nullptr) { 916 tValues = t_storage; 917 } 918 919 int count = SkFindCubicMaxCurvature(src, tValues); 920 921 if (dst) { 922 if (count == 0) { 923 memcpy(dst, src, 4 * sizeof(SkPoint)); 924 } else { 925 SkChopCubicAt(src, dst, tValues, count); 926 } 927 } 928 return count + 1; 929} 930 931#include "../pathops/SkPathOpsCubic.h" 932 933typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 934 935static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 936 InterceptProc method) { 937 SkDCubic cubic; 938 double roots[3]; 939 int count = (cubic.set(src).*method)(intercept, roots); 940 if (count > 0) { 941 SkDCubicPair pair = cubic.chopAt(roots[0]); 942 for (int i = 0; i < 7; ++i) { 943 dst[i] = pair.pts[i].asSkPoint(); 944 } 945 return true; 946 } 947 return false; 948} 949 950bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 951 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 952} 953 954bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 955 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 956} 957 958/////////////////////////////////////////////////////////////////////////////// 959 960/* Find t value for quadratic [a, b, c] = d. 961 Return 0 if there is no solution within [0, 1) 962*/ 963static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { 964 // At^2 + Bt + C = d 965 SkScalar A = a - 2 * b + c; 966 SkScalar B = 2 * (b - a); 967 SkScalar C = a - d; 968 969 SkScalar roots[2]; 970 int count = SkFindUnitQuadRoots(A, B, C, roots); 971 972 SkASSERT(count <= 1); 973 return count == 1 ? roots[0] : 0; 974} 975 976/* given a quad-curve and a point (x,y), chop the quad at that point and place 977 the new off-curve point and endpoint into 'dest'. 978 Should only return false if the computed pos is the start of the curve 979 (i.e. root == 0) 980*/ 981static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, 982 SkPoint* dest) { 983 const SkScalar* base; 984 SkScalar value; 985 986 if (SkScalarAbs(x) < SkScalarAbs(y)) { 987 base = &quad[0].fX; 988 value = x; 989 } else { 990 base = &quad[0].fY; 991 value = y; 992 } 993 994 // note: this returns 0 if it thinks value is out of range, meaning the 995 // root might return something outside of [0, 1) 996 SkScalar t = quad_solve(base[0], base[2], base[4], value); 997 998 if (t > 0) { 999 SkPoint tmp[5]; 1000 SkChopQuadAt(quad, tmp, t); 1001 dest[0] = tmp[1]; 1002 dest[1].set(x, y); 1003 return true; 1004 } else { 1005 /* t == 0 means either the value triggered a root outside of [0, 1) 1006 For our purposes, we can ignore the <= 0 roots, but we want to 1007 catch the >= 1 roots (which given our caller, will basically mean 1008 a root of 1, give-or-take numerical instability). If we are in the 1009 >= 1 case, return the existing offCurve point. 1010 1011 The test below checks to see if we are close to the "end" of the 1012 curve (near base[4]). Rather than specifying a tolerance, I just 1013 check to see if value is on to the right/left of the middle point 1014 (depending on the direction/sign of the end points). 1015 */ 1016 if ((base[0] < base[4] && value > base[2]) || 1017 (base[0] > base[4] && value < base[2])) // should root have been 1 1018 { 1019 dest[0] = quad[1]; 1020 dest[1].set(x, y); 1021 return true; 1022 } 1023 } 1024 return false; 1025} 1026 1027static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1028// The mid point of the quadratic arc approximation is half way between the two 1029// control points. The float epsilon adjustment moves the on curve point out by 1030// two bits, distributing the convex test error between the round rect 1031// approximation and the convex cross product sign equality test. 1032#define SK_MID_RRECT_OFFSET \ 1033 (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1034 { SK_Scalar1, 0 }, 1035 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1036 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1037 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1038 1039 { 0, SK_Scalar1 }, 1040 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1041 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1042 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1043 1044 { -SK_Scalar1, 0 }, 1045 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1046 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1047 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1048 1049 { 0, -SK_Scalar1 }, 1050 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1051 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1052 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1053 1054 { SK_Scalar1, 0 } 1055#undef SK_MID_RRECT_OFFSET 1056}; 1057 1058int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1059 SkRotationDirection dir, const SkMatrix* userMatrix, 1060 SkPoint quadPoints[]) { 1061 // rotate by x,y so that uStart is (1.0) 1062 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1063 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1064 1065 SkScalar absX = SkScalarAbs(x); 1066 SkScalar absY = SkScalarAbs(y); 1067 1068 int pointCount; 1069 1070 // check for (effectively) coincident vectors 1071 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1072 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1073 if (absY <= SK_ScalarNearlyZero && x > 0 && 1074 ((y >= 0 && kCW_SkRotationDirection == dir) || 1075 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1076 1077 // just return the start-point 1078 quadPoints[0].set(SK_Scalar1, 0); 1079 pointCount = 1; 1080 } else { 1081 if (dir == kCCW_SkRotationDirection) { 1082 y = -y; 1083 } 1084 // what octant (quadratic curve) is [xy] in? 1085 int oct = 0; 1086 bool sameSign = true; 1087 1088 if (0 == y) { 1089 oct = 4; // 180 1090 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1091 } else if (0 == x) { 1092 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1093 oct = y > 0 ? 2 : 6; // 90 : 270 1094 } else { 1095 if (y < 0) { 1096 oct += 4; 1097 } 1098 if ((x < 0) != (y < 0)) { 1099 oct += 2; 1100 sameSign = false; 1101 } 1102 if ((absX < absY) == sameSign) { 1103 oct += 1; 1104 } 1105 } 1106 1107 int wholeCount = oct << 1; 1108 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1109 1110 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1111 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { 1112 wholeCount += 2; 1113 } 1114 pointCount = wholeCount + 1; 1115 } 1116 1117 // now handle counter-clockwise and the initial unitStart rotation 1118 SkMatrix matrix; 1119 matrix.setSinCos(uStart.fY, uStart.fX); 1120 if (dir == kCCW_SkRotationDirection) { 1121 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1122 } 1123 if (userMatrix) { 1124 matrix.postConcat(*userMatrix); 1125 } 1126 matrix.mapPoints(quadPoints, pointCount); 1127 return pointCount; 1128} 1129 1130 1131/////////////////////////////////////////////////////////////////////////////// 1132// 1133// NURB representation for conics. Helpful explanations at: 1134// 1135// http://citeseerx.ist.psu.edu/viewdoc/ 1136// download?doi=10.1.1.44.5740&rep=rep1&type=ps 1137// and 1138// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 1139// 1140// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1141// ------------------------------------------ 1142// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1143// 1144// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1145// ------------------------------------------------ 1146// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1147// 1148 1149// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1150// 1151// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1152// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1153// t^0 : -2 P0 w + 2 P1 w 1154// 1155// We disregard magnitude, so we can freely ignore the denominator of F', and 1156// divide the numerator by 2 1157// 1158// coeff[0] for t^2 1159// coeff[1] for t^1 1160// coeff[2] for t^0 1161// 1162static void conic_deriv_coeff(const SkScalar src[], 1163 SkScalar w, 1164 SkScalar coeff[3]) { 1165 const SkScalar P20 = src[4] - src[0]; 1166 const SkScalar P10 = src[2] - src[0]; 1167 const SkScalar wP10 = w * P10; 1168 coeff[0] = w * P20 - P20; 1169 coeff[1] = P20 - 2 * wP10; 1170 coeff[2] = wP10; 1171} 1172 1173static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1174 SkScalar coeff[3]; 1175 conic_deriv_coeff(src, w, coeff); 1176 1177 SkScalar tValues[2]; 1178 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1179 SkASSERT(0 == roots || 1 == roots); 1180 1181 if (1 == roots) { 1182 *t = tValues[0]; 1183 return true; 1184 } 1185 return false; 1186} 1187 1188struct SkP3D { 1189 SkScalar fX, fY, fZ; 1190 1191 void set(SkScalar x, SkScalar y, SkScalar z) { 1192 fX = x; fY = y; fZ = z; 1193 } 1194 1195 void projectDown(SkPoint* dst) const { 1196 dst->set(fX / fZ, fY / fZ); 1197 } 1198}; 1199 1200// We only interpolate one dimension at a time (the first, at +0, +3, +6). 1201static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 1202 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1203 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1204 dst[0] = ab; 1205 dst[3] = SkScalarInterp(ab, bc, t); 1206 dst[6] = bc; 1207} 1208 1209static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1210 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1211 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1212 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1213} 1214 1215void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1216 SkP3D tmp[3], tmp2[3]; 1217 1218 ratquad_mapTo3D(fPts, fW, tmp); 1219 1220 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1221 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1222 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1223 1224 dst[0].fPts[0] = fPts[0]; 1225 tmp2[0].projectDown(&dst[0].fPts[1]); 1226 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1227 tmp2[2].projectDown(&dst[1].fPts[1]); 1228 dst[1].fPts[2] = fPts[2]; 1229 1230 // to put in "standard form", where w0 and w2 are both 1, we compute the 1231 // new w1 as sqrt(w1*w1/w0*w2) 1232 // or 1233 // w1 /= sqrt(w0*w2) 1234 // 1235 // However, in our case, we know that for dst[0]: 1236 // w0 == 1, and for dst[1], w2 == 1 1237 // 1238 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1239 dst[0].fW = tmp2[0].fZ / root; 1240 dst[1].fW = tmp2[2].fZ / root; 1241} 1242 1243static Sk2s times_2(const Sk2s& value) { 1244 return value + value; 1245} 1246 1247SkPoint SkConic::evalAt(SkScalar t) const { 1248 Sk2s p0 = from_point(fPts[0]); 1249 Sk2s p1 = from_point(fPts[1]); 1250 Sk2s p2 = from_point(fPts[2]); 1251 Sk2s tt(t); 1252 Sk2s ww(fW); 1253 Sk2s one(1); 1254 1255 Sk2s p1w = p1 * ww; 1256 Sk2s C = p0; 1257 Sk2s A = p2 - times_2(p1w) + p0; 1258 Sk2s B = times_2(p1w - C); 1259 Sk2s numer = quad_poly_eval(A, B, C, tt); 1260 1261 B = times_2(ww - one); 1262 A = Sk2s(0)-B; 1263 Sk2s denom = quad_poly_eval(A, B, one, tt); 1264 1265 return to_point(numer / denom); 1266} 1267 1268SkVector SkConic::evalTangentAt(SkScalar t) const { 1269 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1270 // and the control point is equal to the end point. 1271 // In this case, use the conic endpoints to compute the tangent. 1272 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1273 return fPts[2] - fPts[0]; 1274 } 1275 Sk2s p0 = from_point(fPts[0]); 1276 Sk2s p1 = from_point(fPts[1]); 1277 Sk2s p2 = from_point(fPts[2]); 1278 Sk2s ww(fW); 1279 1280 Sk2s p20 = p2 - p0; 1281 Sk2s p10 = p1 - p0; 1282 1283 Sk2s C = ww * p10; 1284 Sk2s A = ww * p20 - p20; 1285 Sk2s B = p20 - C - C; 1286 1287 return to_vector(quad_poly_eval(A, B, C, Sk2s(t))); 1288} 1289 1290void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1291 SkASSERT(t >= 0 && t <= SK_Scalar1); 1292 1293 if (pt) { 1294 *pt = this->evalAt(t); 1295 } 1296 if (tangent) { 1297 *tangent = this->evalTangentAt(t); 1298 } 1299} 1300 1301static SkScalar subdivide_w_value(SkScalar w) { 1302 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1303} 1304 1305static Sk2s twice(const Sk2s& value) { 1306 return value + value; 1307} 1308 1309void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1310 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1311 SkScalar newW = subdivide_w_value(fW); 1312 1313 Sk2s p0 = from_point(fPts[0]); 1314 Sk2s p1 = from_point(fPts[1]); 1315 Sk2s p2 = from_point(fPts[2]); 1316 Sk2s ww(fW); 1317 1318 Sk2s wp1 = ww * p1; 1319 Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f); 1320 1321 dst[0].fPts[0] = fPts[0]; 1322 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1323 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1324 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1325 dst[1].fPts[2] = fPts[2]; 1326 1327 dst[0].fW = dst[1].fW = newW; 1328} 1329 1330/* 1331 * "High order approximation of conic sections by quadratic splines" 1332 * by Michael Floater, 1993 1333 */ 1334#define AS_QUAD_ERROR_SETUP \ 1335 SkScalar a = fW - 1; \ 1336 SkScalar k = a / (4 * (2 + a)); \ 1337 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1338 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1339 1340void SkConic::computeAsQuadError(SkVector* err) const { 1341 AS_QUAD_ERROR_SETUP 1342 err->set(x, y); 1343} 1344 1345bool SkConic::asQuadTol(SkScalar tol) const { 1346 AS_QUAD_ERROR_SETUP 1347 return (x * x + y * y) <= tol * tol; 1348} 1349 1350// Limit the number of suggested quads to approximate a conic 1351#define kMaxConicToQuadPOW2 5 1352 1353int SkConic::computeQuadPOW2(SkScalar tol) const { 1354 if (tol < 0 || !SkScalarIsFinite(tol)) { 1355 return 0; 1356 } 1357 1358 AS_QUAD_ERROR_SETUP 1359 1360 SkScalar error = SkScalarSqrt(x * x + y * y); 1361 int pow2; 1362 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1363 if (error <= tol) { 1364 break; 1365 } 1366 error *= 0.25f; 1367 } 1368 // float version -- using ceil gives the same results as the above. 1369 if (false) { 1370 SkScalar err = SkScalarSqrt(x * x + y * y); 1371 if (err <= tol) { 1372 return 0; 1373 } 1374 SkScalar tol2 = tol * tol; 1375 if (tol2 == 0) { 1376 return kMaxConicToQuadPOW2; 1377 } 1378 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1379 int altPow2 = SkScalarCeilToInt(fpow2); 1380 if (altPow2 != pow2) { 1381 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1382 } 1383 pow2 = altPow2; 1384 } 1385 return pow2; 1386} 1387 1388static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1389 SkASSERT(level >= 0); 1390 1391 if (0 == level) { 1392 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1393 return pts + 2; 1394 } else { 1395 SkConic dst[2]; 1396 src.chop(dst); 1397 --level; 1398 pts = subdivide(dst[0], pts, level); 1399 return subdivide(dst[1], pts, level); 1400 } 1401} 1402 1403int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1404 SkASSERT(pow2 >= 0); 1405 *pts = fPts[0]; 1406 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1407 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1408 return 1 << pow2; 1409} 1410 1411bool SkConic::findXExtrema(SkScalar* t) const { 1412 return conic_find_extrema(&fPts[0].fX, fW, t); 1413} 1414 1415bool SkConic::findYExtrema(SkScalar* t) const { 1416 return conic_find_extrema(&fPts[0].fY, fW, t); 1417} 1418 1419bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1420 SkScalar t; 1421 if (this->findXExtrema(&t)) { 1422 this->chopAt(t, dst); 1423 // now clean-up the middle, since we know t was meant to be at 1424 // an X-extrema 1425 SkScalar value = dst[0].fPts[2].fX; 1426 dst[0].fPts[1].fX = value; 1427 dst[1].fPts[0].fX = value; 1428 dst[1].fPts[1].fX = value; 1429 return true; 1430 } 1431 return false; 1432} 1433 1434bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1435 SkScalar t; 1436 if (this->findYExtrema(&t)) { 1437 this->chopAt(t, dst); 1438 // now clean-up the middle, since we know t was meant to be at 1439 // an Y-extrema 1440 SkScalar value = dst[0].fPts[2].fY; 1441 dst[0].fPts[1].fY = value; 1442 dst[1].fPts[0].fY = value; 1443 dst[1].fPts[1].fY = value; 1444 return true; 1445 } 1446 return false; 1447} 1448 1449void SkConic::computeTightBounds(SkRect* bounds) const { 1450 SkPoint pts[4]; 1451 pts[0] = fPts[0]; 1452 pts[1] = fPts[2]; 1453 int count = 2; 1454 1455 SkScalar t; 1456 if (this->findXExtrema(&t)) { 1457 this->evalAt(t, &pts[count++]); 1458 } 1459 if (this->findYExtrema(&t)) { 1460 this->evalAt(t, &pts[count++]); 1461 } 1462 bounds->set(pts, count); 1463} 1464 1465void SkConic::computeFastBounds(SkRect* bounds) const { 1466 bounds->set(fPts, 3); 1467} 1468 1469#if 0 // unimplemented 1470bool SkConic::findMaxCurvature(SkScalar* t) const { 1471 // TODO: Implement me 1472 return false; 1473} 1474#endif 1475 1476SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1477 const SkMatrix& matrix) { 1478 if (!matrix.hasPerspective()) { 1479 return w; 1480 } 1481 1482 SkP3D src[3], dst[3]; 1483 1484 ratquad_mapTo3D(pts, w, src); 1485 1486 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1487 1488 // w' = sqrt(w1*w1/w0*w2) 1489 SkScalar w0 = dst[0].fZ; 1490 SkScalar w1 = dst[1].fZ; 1491 SkScalar w2 = dst[2].fZ; 1492 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1493 return w; 1494} 1495 1496int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1497 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1498 // rotate by x,y so that uStart is (1.0) 1499 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1500 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1501 1502 SkScalar absY = SkScalarAbs(y); 1503 1504 // check for (effectively) coincident vectors 1505 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1506 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1507 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1508 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1509 return 0; 1510 } 1511 1512 if (dir == kCCW_SkRotationDirection) { 1513 y = -y; 1514 } 1515 1516 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1517 // 0 == [0 .. 90) 1518 // 1 == [90 ..180) 1519 // 2 == [180..270) 1520 // 3 == [270..360) 1521 // 1522 int quadrant = 0; 1523 if (0 == y) { 1524 quadrant = 2; // 180 1525 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1526 } else if (0 == x) { 1527 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1528 quadrant = y > 0 ? 1 : 3; // 90 : 270 1529 } else { 1530 if (y < 0) { 1531 quadrant += 2; 1532 } 1533 if ((x < 0) != (y < 0)) { 1534 quadrant += 1; 1535 } 1536 } 1537 1538 const SkPoint quadrantPts[] = { 1539 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1540 }; 1541 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1542 1543 int conicCount = quadrant; 1544 for (int i = 0; i < conicCount; ++i) { 1545 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1546 } 1547 1548 // Now compute any remaing (sub-90-degree) arc for the last conic 1549 const SkPoint finalP = { x, y }; 1550 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1551 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1552 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1553 1554 if (dot < 1) { 1555 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1556 // compute the bisector vector, and then rescale to be the off-curve point. 1557 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1558 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1559 // This is nice, since our computed weight is cos(theta/2) as well! 1560 // 1561 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1562 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1563 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1564 conicCount += 1; 1565 } 1566 1567 // now handle counter-clockwise and the initial unitStart rotation 1568 SkMatrix matrix; 1569 matrix.setSinCos(uStart.fY, uStart.fX); 1570 if (dir == kCCW_SkRotationDirection) { 1571 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1572 } 1573 if (userMatrix) { 1574 matrix.postConcat(*userMatrix); 1575 } 1576 for (int i = 0; i < conicCount; ++i) { 1577 matrix.mapPoints(dst[i].fPts, 3); 1578 } 1579 return conicCount; 1580} 1581