SkGeometry.cpp revision ec3ed6a5ebf6f2c406d7bcf94b6bc34fcaeb976e
1 2/* 3 * Copyright 2006 The Android Open Source Project 4 * 5 * Use of this source code is governed by a BSD-style license that can be 6 * found in the LICENSE file. 7 */ 8 9 10#include "SkGeometry.h" 11#include "Sk64.h" 12#include "SkMatrix.h" 13 14bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) { 15 if (ambiguous) { 16 *ambiguous = false; 17 } 18 // Determine quick discards. 19 // Consider query line going exactly through point 0 to not 20 // intersect, for symmetry with SkXRayCrossesMonotonicCubic. 21 if (pt.fY == pts[0].fY) { 22 if (ambiguous) { 23 *ambiguous = true; 24 } 25 return false; 26 } 27 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) 28 return false; 29 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) 30 return false; 31 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) 32 return false; 33 // Determine degenerate cases 34 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) 35 return false; 36 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { 37 // We've already determined the query point lies within the 38 // vertical range of the line segment. 39 if (pt.fX <= pts[0].fX) { 40 if (ambiguous) { 41 *ambiguous = (pt.fY == pts[1].fY); 42 } 43 return true; 44 } 45 return false; 46 } 47 // Ambiguity check 48 if (pt.fY == pts[1].fY) { 49 if (pt.fX <= pts[1].fX) { 50 if (ambiguous) { 51 *ambiguous = true; 52 } 53 return true; 54 } 55 return false; 56 } 57 // Full line segment evaluation 58 SkScalar delta_y = pts[1].fY - pts[0].fY; 59 SkScalar delta_x = pts[1].fX - pts[0].fX; 60 SkScalar slope = SkScalarDiv(delta_y, delta_x); 61 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); 62 // Solve for x coordinate at y = pt.fY 63 SkScalar x = SkScalarDiv(pt.fY - b, slope); 64 return pt.fX <= x; 65} 66 67/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 68 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 69 May also introduce overflow of fixed when we compute our setup. 70*/ 71#ifdef SK_SCALAR_IS_FIXED 72 #define DIRECT_EVAL_OF_POLYNOMIALS 73#endif 74 75//////////////////////////////////////////////////////////////////////// 76 77#ifdef SK_SCALAR_IS_FIXED 78 static int is_not_monotonic(int a, int b, int c, int d) 79 { 80 return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31; 81 } 82 83 static int is_not_monotonic(int a, int b, int c) 84 { 85 return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31; 86 } 87#else 88 static int is_not_monotonic(float a, float b, float c) 89 { 90 float ab = a - b; 91 float bc = b - c; 92 if (ab < 0) 93 bc = -bc; 94 return ab == 0 || bc < 0; 95 } 96#endif 97 98//////////////////////////////////////////////////////////////////////// 99 100static bool is_unit_interval(SkScalar x) 101{ 102 return x > 0 && x < SK_Scalar1; 103} 104 105static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) 106{ 107 SkASSERT(ratio); 108 109 if (numer < 0) 110 { 111 numer = -numer; 112 denom = -denom; 113 } 114 115 if (denom == 0 || numer == 0 || numer >= denom) 116 return 0; 117 118 SkScalar r = SkScalarDiv(numer, denom); 119 if (SkScalarIsNaN(r)) { 120 return 0; 121 } 122 SkASSERT(r >= 0 && r < SK_Scalar1); 123 if (r == 0) // catch underflow if numer <<<< denom 124 return 0; 125 *ratio = r; 126 return 1; 127} 128 129/** From Numerical Recipes in C. 130 131 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 132 x1 = Q / A 133 x2 = C / Q 134*/ 135int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) 136{ 137 SkASSERT(roots); 138 139 if (A == 0) 140 return valid_unit_divide(-C, B, roots); 141 142 SkScalar* r = roots; 143 144#ifdef SK_SCALAR_IS_FLOAT 145 float R = B*B - 4*A*C; 146 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 147 return 0; 148 } 149 R = sk_float_sqrt(R); 150#else 151 Sk64 RR, tmp; 152 153 RR.setMul(B,B); 154 tmp.setMul(A,C); 155 tmp.shiftLeft(2); 156 RR.sub(tmp); 157 if (RR.isNeg()) 158 return 0; 159 SkFixed R = RR.getSqrt(); 160#endif 161 162 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 163 r += valid_unit_divide(Q, A, r); 164 r += valid_unit_divide(C, Q, r); 165 if (r - roots == 2) 166 { 167 if (roots[0] > roots[1]) 168 SkTSwap<SkScalar>(roots[0], roots[1]); 169 else if (roots[0] == roots[1]) // nearly-equal? 170 r -= 1; // skip the double root 171 } 172 return (int)(r - roots); 173} 174 175#ifdef SK_SCALAR_IS_FIXED 176/** Trim A/B/C down so that they are all <= 32bits 177 and then call SkFindUnitQuadRoots() 178*/ 179static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2]) 180{ 181 int na = A.shiftToMake32(); 182 int nb = B.shiftToMake32(); 183 int nc = C.shiftToMake32(); 184 185 int shift = SkMax32(na, SkMax32(nb, nc)); 186 SkASSERT(shift >= 0); 187 188 return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots); 189} 190#endif 191 192///////////////////////////////////////////////////////////////////////////////////// 193///////////////////////////////////////////////////////////////////////////////////// 194 195static SkScalar eval_quad(const SkScalar src[], SkScalar t) 196{ 197 SkASSERT(src); 198 SkASSERT(t >= 0 && t <= SK_Scalar1); 199 200#ifdef DIRECT_EVAL_OF_POLYNOMIALS 201 SkScalar C = src[0]; 202 SkScalar A = src[4] - 2 * src[2] + C; 203 SkScalar B = 2 * (src[2] - C); 204 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 205#else 206 SkScalar ab = SkScalarInterp(src[0], src[2], t); 207 SkScalar bc = SkScalarInterp(src[2], src[4], t); 208 return SkScalarInterp(ab, bc, t); 209#endif 210} 211 212static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) 213{ 214 SkScalar A = src[4] - 2 * src[2] + src[0]; 215 SkScalar B = src[2] - src[0]; 216 217 return 2 * SkScalarMulAdd(A, t, B); 218} 219 220static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) 221{ 222 SkScalar A = src[4] - 2 * src[2] + src[0]; 223 SkScalar B = src[2] - src[0]; 224 return A + 2 * B; 225} 226 227void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) 228{ 229 SkASSERT(src); 230 SkASSERT(t >= 0 && t <= SK_Scalar1); 231 232 if (pt) 233 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 234 if (tangent) 235 tangent->set(eval_quad_derivative(&src[0].fX, t), 236 eval_quad_derivative(&src[0].fY, t)); 237} 238 239void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) 240{ 241 SkASSERT(src); 242 243 if (pt) 244 { 245 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 246 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 247 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 248 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 249 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 250 } 251 if (tangent) 252 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 253 eval_quad_derivative_at_half(&src[0].fY)); 254} 255 256static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 257{ 258 SkScalar ab = SkScalarInterp(src[0], src[2], t); 259 SkScalar bc = SkScalarInterp(src[2], src[4], t); 260 261 dst[0] = src[0]; 262 dst[2] = ab; 263 dst[4] = SkScalarInterp(ab, bc, t); 264 dst[6] = bc; 265 dst[8] = src[4]; 266} 267 268void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) 269{ 270 SkASSERT(t > 0 && t < SK_Scalar1); 271 272 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 273 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 274} 275 276void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) 277{ 278 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 279 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 280 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 281 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 282 283 dst[0] = src[0]; 284 dst[1].set(x01, y01); 285 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 286 dst[3].set(x12, y12); 287 dst[4] = src[2]; 288} 289 290/** Quad'(t) = At + B, where 291 A = 2(a - 2b + c) 292 B = 2(b - a) 293 Solve for t, only if it fits between 0 < t < 1 294*/ 295int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) 296{ 297 /* At + B == 0 298 t = -B / A 299 */ 300#ifdef SK_SCALAR_IS_FIXED 301 return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue); 302#else 303 return valid_unit_divide(a - b, a - b - b + c, tValue); 304#endif 305} 306 307static inline void flatten_double_quad_extrema(SkScalar coords[14]) 308{ 309 coords[2] = coords[6] = coords[4]; 310} 311 312/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 313 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 314 */ 315int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) 316{ 317 SkASSERT(src); 318 SkASSERT(dst); 319 320#if 0 321 static bool once = true; 322 if (once) 323 { 324 once = false; 325 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; 326 SkPoint d[6]; 327 328 int n = SkChopQuadAtYExtrema(s, d); 329 SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY); 330 } 331#endif 332 333 SkScalar a = src[0].fY; 334 SkScalar b = src[1].fY; 335 SkScalar c = src[2].fY; 336 337 if (is_not_monotonic(a, b, c)) 338 { 339 SkScalar tValue; 340 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) 341 { 342 SkChopQuadAt(src, dst, tValue); 343 flatten_double_quad_extrema(&dst[0].fY); 344 return 1; 345 } 346 // if we get here, we need to force dst to be monotonic, even though 347 // we couldn't compute a unit_divide value (probably underflow). 348 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 349 } 350 dst[0].set(src[0].fX, a); 351 dst[1].set(src[1].fX, b); 352 dst[2].set(src[2].fX, c); 353 return 0; 354} 355 356/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 357 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 358 */ 359int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) 360{ 361 SkASSERT(src); 362 SkASSERT(dst); 363 364 SkScalar a = src[0].fX; 365 SkScalar b = src[1].fX; 366 SkScalar c = src[2].fX; 367 368 if (is_not_monotonic(a, b, c)) { 369 SkScalar tValue; 370 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 371 SkChopQuadAt(src, dst, tValue); 372 flatten_double_quad_extrema(&dst[0].fX); 373 return 1; 374 } 375 // if we get here, we need to force dst to be monotonic, even though 376 // we couldn't compute a unit_divide value (probably underflow). 377 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 378 } 379 dst[0].set(a, src[0].fY); 380 dst[1].set(b, src[1].fY); 381 dst[2].set(c, src[2].fY); 382 return 0; 383} 384 385// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 386// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 387// F''(t) = 2 (a - 2b + c) 388// 389// A = 2 (b - a) 390// B = 2 (a - 2b + c) 391// 392// Maximum curvature for a quadratic means solving 393// Fx' Fx'' + Fy' Fy'' = 0 394// 395// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 396// 397int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) 398{ 399 SkScalar Ax = src[1].fX - src[0].fX; 400 SkScalar Ay = src[1].fY - src[0].fY; 401 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 402 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 403 SkScalar t = 0; // 0 means don't chop 404 405#ifdef SK_SCALAR_IS_FLOAT 406 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 407#else 408 // !!! should I use SkFloat here? seems like it 409 Sk64 numer, denom, tmp; 410 411 numer.setMul(Ax, -Bx); 412 tmp.setMul(Ay, -By); 413 numer.add(tmp); 414 415 if (numer.isPos()) // do nothing if numer <= 0 416 { 417 denom.setMul(Bx, Bx); 418 tmp.setMul(By, By); 419 denom.add(tmp); 420 SkASSERT(!denom.isNeg()); 421 if (numer < denom) 422 { 423 t = numer.getFixedDiv(denom); 424 SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!) 425 if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability 426 t = 0; // ignore the chop 427 } 428 } 429#endif 430 431 if (t == 0) 432 { 433 memcpy(dst, src, 3 * sizeof(SkPoint)); 434 return 1; 435 } 436 else 437 { 438 SkChopQuadAt(src, dst, t); 439 return 2; 440 } 441} 442 443#ifdef SK_SCALAR_IS_FLOAT 444 #define SK_ScalarTwoThirds (0.666666666f) 445#else 446 #define SK_ScalarTwoThirds ((SkFixed)(43691)) 447#endif 448 449void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 450 const SkScalar scale = SK_ScalarTwoThirds; 451 dst[0] = src[0]; 452 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 453 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 454 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 455 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 456 dst[3] = src[2]; 457} 458 459//////////////////////////////////////////////////////////////////////////////////////// 460///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 461//////////////////////////////////////////////////////////////////////////////////////// 462 463static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) 464{ 465 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 466 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 467 coeff[2] = 3*(pt[2] - pt[0]); 468 coeff[3] = pt[0]; 469} 470 471void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) 472{ 473 SkASSERT(pts); 474 475 if (cx) 476 get_cubic_coeff(&pts[0].fX, cx); 477 if (cy) 478 get_cubic_coeff(&pts[0].fY, cy); 479} 480 481static SkScalar eval_cubic(const SkScalar src[], SkScalar t) 482{ 483 SkASSERT(src); 484 SkASSERT(t >= 0 && t <= SK_Scalar1); 485 486 if (t == 0) 487 return src[0]; 488 489#ifdef DIRECT_EVAL_OF_POLYNOMIALS 490 SkScalar D = src[0]; 491 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 492 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 493 SkScalar C = 3*(src[2] - D); 494 495 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 496#else 497 SkScalar ab = SkScalarInterp(src[0], src[2], t); 498 SkScalar bc = SkScalarInterp(src[2], src[4], t); 499 SkScalar cd = SkScalarInterp(src[4], src[6], t); 500 SkScalar abc = SkScalarInterp(ab, bc, t); 501 SkScalar bcd = SkScalarInterp(bc, cd, t); 502 return SkScalarInterp(abc, bcd, t); 503#endif 504} 505 506/** return At^2 + Bt + C 507*/ 508static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) 509{ 510 SkASSERT(t >= 0 && t <= SK_Scalar1); 511 512 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 513} 514 515static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) 516{ 517 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 518 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 519 SkScalar C = src[2] - src[0]; 520 521 return eval_quadratic(A, B, C, t); 522} 523 524static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) 525{ 526 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 527 SkScalar B = src[4] - 2 * src[2] + src[0]; 528 529 return SkScalarMulAdd(A, t, B); 530} 531 532void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature) 533{ 534 SkASSERT(src); 535 SkASSERT(t >= 0 && t <= SK_Scalar1); 536 537 if (loc) 538 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 539 if (tangent) 540 tangent->set(eval_cubic_derivative(&src[0].fX, t), 541 eval_cubic_derivative(&src[0].fY, t)); 542 if (curvature) 543 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 544 eval_cubic_2ndDerivative(&src[0].fY, t)); 545} 546 547/** Cubic'(t) = At^2 + Bt + C, where 548 A = 3(-a + 3(b - c) + d) 549 B = 6(a - 2b + c) 550 C = 3(b - a) 551 Solve for t, keeping only those that fit betwee 0 < t < 1 552*/ 553int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2]) 554{ 555#ifdef SK_SCALAR_IS_FIXED 556 if (!is_not_monotonic(a, b, c, d)) 557 return 0; 558#endif 559 560 // we divide A,B,C by 3 to simplify 561 SkScalar A = d - a + 3*(b - c); 562 SkScalar B = 2*(a - b - b + c); 563 SkScalar C = b - a; 564 565 return SkFindUnitQuadRoots(A, B, C, tValues); 566} 567 568static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 569{ 570 SkScalar ab = SkScalarInterp(src[0], src[2], t); 571 SkScalar bc = SkScalarInterp(src[2], src[4], t); 572 SkScalar cd = SkScalarInterp(src[4], src[6], t); 573 SkScalar abc = SkScalarInterp(ab, bc, t); 574 SkScalar bcd = SkScalarInterp(bc, cd, t); 575 SkScalar abcd = SkScalarInterp(abc, bcd, t); 576 577 dst[0] = src[0]; 578 dst[2] = ab; 579 dst[4] = abc; 580 dst[6] = abcd; 581 dst[8] = bcd; 582 dst[10] = cd; 583 dst[12] = src[6]; 584} 585 586void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) 587{ 588 SkASSERT(t > 0 && t < SK_Scalar1); 589 590 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 591 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 592} 593 594/* http://code.google.com/p/skia/issues/detail?id=32 595 596 This test code would fail when we didn't check the return result of 597 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 598 that after the first chop, the parameters to valid_unit_divide are equal 599 (thanks to finite float precision and rounding in the subtracts). Thus 600 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 601 up with 1.0, hence the need to check and just return the last cubic as 602 a degenerate clump of 4 points in the sampe place. 603 604 static void test_cubic() { 605 SkPoint src[4] = { 606 { 556.25000, 523.03003 }, 607 { 556.23999, 522.96002 }, 608 { 556.21997, 522.89001 }, 609 { 556.21997, 522.82001 } 610 }; 611 SkPoint dst[10]; 612 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 613 SkChopCubicAt(src, dst, tval, 2); 614 } 615 */ 616 617void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots) 618{ 619#ifdef SK_DEBUG 620 { 621 for (int i = 0; i < roots - 1; i++) 622 { 623 SkASSERT(is_unit_interval(tValues[i])); 624 SkASSERT(is_unit_interval(tValues[i+1])); 625 SkASSERT(tValues[i] < tValues[i+1]); 626 } 627 } 628#endif 629 630 if (dst) 631 { 632 if (roots == 0) // nothing to chop 633 memcpy(dst, src, 4*sizeof(SkPoint)); 634 else 635 { 636 SkScalar t = tValues[0]; 637 SkPoint tmp[4]; 638 639 for (int i = 0; i < roots; i++) 640 { 641 SkChopCubicAt(src, dst, t); 642 if (i == roots - 1) 643 break; 644 645 dst += 3; 646 // have src point to the remaining cubic (after the chop) 647 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 648 src = tmp; 649 650 // watch out in case the renormalized t isn't in range 651 if (!valid_unit_divide(tValues[i+1] - tValues[i], 652 SK_Scalar1 - tValues[i], &t)) { 653 // if we can't, just create a degenerate cubic 654 dst[4] = dst[5] = dst[6] = src[3]; 655 break; 656 } 657 } 658 } 659 } 660} 661 662void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) 663{ 664 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 665 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 666 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 667 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 668 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 669 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 670 671 SkScalar x012 = SkScalarAve(x01, x12); 672 SkScalar y012 = SkScalarAve(y01, y12); 673 SkScalar x123 = SkScalarAve(x12, x23); 674 SkScalar y123 = SkScalarAve(y12, y23); 675 676 dst[0] = src[0]; 677 dst[1].set(x01, y01); 678 dst[2].set(x012, y012); 679 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 680 dst[4].set(x123, y123); 681 dst[5].set(x23, y23); 682 dst[6] = src[3]; 683} 684 685static void flatten_double_cubic_extrema(SkScalar coords[14]) 686{ 687 coords[4] = coords[8] = coords[6]; 688} 689 690/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 691 the resulting beziers are monotonic in Y. This is called by the scan converter. 692 Depending on what is returned, dst[] is treated as follows 693 0 dst[0..3] is the original cubic 694 1 dst[0..3] and dst[3..6] are the two new cubics 695 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 696 If dst == null, it is ignored and only the count is returned. 697*/ 698int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 699 SkScalar tValues[2]; 700 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 701 src[3].fY, tValues); 702 703 SkChopCubicAt(src, dst, tValues, roots); 704 if (dst && roots > 0) { 705 // we do some cleanup to ensure our Y extrema are flat 706 flatten_double_cubic_extrema(&dst[0].fY); 707 if (roots == 2) { 708 flatten_double_cubic_extrema(&dst[3].fY); 709 } 710 } 711 return roots; 712} 713 714int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 715 SkScalar tValues[2]; 716 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 717 src[3].fX, tValues); 718 719 SkChopCubicAt(src, dst, tValues, roots); 720 if (dst && roots > 0) { 721 // we do some cleanup to ensure our Y extrema are flat 722 flatten_double_cubic_extrema(&dst[0].fX); 723 if (roots == 2) { 724 flatten_double_cubic_extrema(&dst[3].fX); 725 } 726 } 727 return roots; 728} 729 730/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 731 732 Inflection means that curvature is zero. 733 Curvature is [F' x F''] / [F'^3] 734 So we solve F'x X F''y - F'y X F''y == 0 735 After some canceling of the cubic term, we get 736 A = b - a 737 B = c - 2b + a 738 C = d - 3c + 3b - a 739 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 740*/ 741int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) 742{ 743 SkScalar Ax = src[1].fX - src[0].fX; 744 SkScalar Ay = src[1].fY - src[0].fY; 745 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 746 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 747 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 748 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 749 int count; 750 751#ifdef SK_SCALAR_IS_FLOAT 752 count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 753#else 754 Sk64 A, B, C, tmp; 755 756 A.setMul(Bx, Cy); 757 tmp.setMul(By, Cx); 758 A.sub(tmp); 759 760 B.setMul(Ax, Cy); 761 tmp.setMul(Ay, Cx); 762 B.sub(tmp); 763 764 C.setMul(Ax, By); 765 tmp.setMul(Ay, Bx); 766 C.sub(tmp); 767 768 count = Sk64FindFixedQuadRoots(A, B, C, tValues); 769#endif 770 771 return count; 772} 773 774int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) 775{ 776 SkScalar tValues[2]; 777 int count = SkFindCubicInflections(src, tValues); 778 779 if (dst) 780 { 781 if (count == 0) 782 memcpy(dst, src, 4 * sizeof(SkPoint)); 783 else 784 SkChopCubicAt(src, dst, tValues, count); 785 } 786 return count + 1; 787} 788 789template <typename T> void bubble_sort(T array[], int count) 790{ 791 for (int i = count - 1; i > 0; --i) 792 for (int j = i; j > 0; --j) 793 if (array[j] < array[j-1]) 794 { 795 T tmp(array[j]); 796 array[j] = array[j-1]; 797 array[j-1] = tmp; 798 } 799} 800 801#include "SkFP.h" 802 803// newton refinement 804#if 0 805static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) 806{ 807 // x1 = x0 - f(t) / f'(t) 808 809 SkFP T = SkScalarToFloat(root); 810 SkFP N, D; 811 812 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] 813 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); 814 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); 815 D = SkFPAdd(D, coeff[2]); 816 817 if (D == 0) 818 return root; 819 820 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] 821 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); 822 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); 823 N = SkFPAdd(N, SkFPMul(T, coeff[2])); 824 N = SkFPAdd(N, coeff[3]); 825 826 if (N) 827 { 828 SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); 829 830 if (delta) 831 root -= delta; 832 } 833 return root; 834} 835#endif 836 837#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop 838#pragma warning ( disable : 4702 ) 839#endif 840 841/* Solve coeff(t) == 0, returning the number of roots that 842 lie withing 0 < t < 1. 843 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 844*/ 845static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3]) 846{ 847#ifndef SK_SCALAR_IS_FLOAT 848 return 0; // this is not yet implemented for software float 849#endif 850 851 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic 852 { 853 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 854 } 855 856 SkFP a, b, c, Q, R; 857 858 { 859 SkASSERT(coeff[0] != 0); 860 861 SkFP inva = SkFPInvert(coeff[0]); 862 a = SkFPMul(coeff[1], inva); 863 b = SkFPMul(coeff[2], inva); 864 c = SkFPMul(coeff[3], inva); 865 } 866 Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); 867// R = (2*a*a*a - 9*a*b + 27*c) / 54; 868 R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); 869 R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); 870 R = SkFPAdd(R, SkFPMulInt(c, 27)); 871 R = SkFPDivInt(R, 54); 872 873 SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); 874 SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); 875 SkFP adiv3 = SkFPDivInt(a, 3); 876 877 SkScalar* roots = tValues; 878 SkScalar r; 879 880 if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots 881 { 882#ifdef SK_SCALAR_IS_FLOAT 883 float theta = sk_float_acos(R / sk_float_sqrt(Q3)); 884 float neg2RootQ = -2 * sk_float_sqrt(Q); 885 886 r = neg2RootQ * sk_float_cos(theta/3) - adiv3; 887 if (is_unit_interval(r)) 888 *roots++ = r; 889 890 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; 891 if (is_unit_interval(r)) 892 *roots++ = r; 893 894 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; 895 if (is_unit_interval(r)) 896 *roots++ = r; 897 898 // now sort the roots 899 bubble_sort(tValues, (int)(roots - tValues)); 900#endif 901 } 902 else // we have 1 real root 903 { 904 SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3)); 905 A = SkFPCubeRoot(A); 906 if (SkFPGT(R, 0)) 907 A = SkFPNeg(A); 908 909 if (A != 0) 910 A = SkFPAdd(A, SkFPDiv(Q, A)); 911 r = SkFPToScalar(SkFPSub(A, adiv3)); 912 if (is_unit_interval(r)) 913 *roots++ = r; 914 } 915 916 return (int)(roots - tValues); 917} 918 919/* Looking for F' dot F'' == 0 920 921 A = b - a 922 B = c - 2b + a 923 C = d - 3c + 3b - a 924 925 F' = 3Ct^2 + 6Bt + 3A 926 F'' = 6Ct + 6B 927 928 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 929*/ 930static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4]) 931{ 932 SkScalar a = src[2] - src[0]; 933 SkScalar b = src[4] - 2 * src[2] + src[0]; 934 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 935 936 SkFP A = SkScalarToFP(a); 937 SkFP B = SkScalarToFP(b); 938 SkFP C = SkScalarToFP(c); 939 940 coeff[0] = SkFPMul(C, C); 941 coeff[1] = SkFPMulInt(SkFPMul(B, C), 3); 942 coeff[2] = SkFPMulInt(SkFPMul(B, B), 2); 943 coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A)); 944 coeff[3] = SkFPMul(A, B); 945} 946 947// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 948//#define kMinTValueForChopping (SK_Scalar1 / 256) 949#define kMinTValueForChopping 0 950 951/* Looking for F' dot F'' == 0 952 953 A = b - a 954 B = c - 2b + a 955 C = d - 3c + 3b - a 956 957 F' = 3Ct^2 + 6Bt + 3A 958 F'' = 6Ct + 6B 959 960 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 961*/ 962int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) 963{ 964 SkFP coeffX[4], coeffY[4]; 965 int i; 966 967 formulate_F1DotF2(&src[0].fX, coeffX); 968 formulate_F1DotF2(&src[0].fY, coeffY); 969 970 for (i = 0; i < 4; i++) 971 coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]); 972 973 SkScalar t[3]; 974 int count = solve_cubic_polynomial(coeffX, t); 975 int maxCount = 0; 976 977 // now remove extrema where the curvature is zero (mins) 978 // !!!! need a test for this !!!! 979 for (i = 0; i < count; i++) 980 { 981 // if (not_min_curvature()) 982 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) 983 tValues[maxCount++] = t[i]; 984 } 985 return maxCount; 986} 987 988int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3]) 989{ 990 SkScalar t_storage[3]; 991 992 if (tValues == NULL) 993 tValues = t_storage; 994 995 int count = SkFindCubicMaxCurvature(src, tValues); 996 997 if (dst) 998 { 999 if (count == 0) 1000 memcpy(dst, src, 4 * sizeof(SkPoint)); 1001 else 1002 SkChopCubicAt(src, dst, tValues, count); 1003 } 1004 return count + 1; 1005} 1006 1007bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1008 if (ambiguous) { 1009 *ambiguous = false; 1010 } 1011 1012 // Find the minimum and maximum y of the extrema, which are the 1013 // first and last points since this cubic is monotonic 1014 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 1015 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 1016 1017 if (pt.fY == cubic[0].fY 1018 || pt.fY < min_y 1019 || pt.fY > max_y) { 1020 // The query line definitely does not cross the curve 1021 if (ambiguous) { 1022 *ambiguous = (pt.fY == cubic[0].fY); 1023 } 1024 return false; 1025 } 1026 1027 bool pt_at_extremum = (pt.fY == cubic[3].fY); 1028 1029 SkScalar min_x = 1030 SkMinScalar( 1031 SkMinScalar( 1032 SkMinScalar(cubic[0].fX, cubic[1].fX), 1033 cubic[2].fX), 1034 cubic[3].fX); 1035 if (pt.fX < min_x) { 1036 // The query line definitely crosses the curve 1037 if (ambiguous) { 1038 *ambiguous = pt_at_extremum; 1039 } 1040 return true; 1041 } 1042 1043 SkScalar max_x = 1044 SkMaxScalar( 1045 SkMaxScalar( 1046 SkMaxScalar(cubic[0].fX, cubic[1].fX), 1047 cubic[2].fX), 1048 cubic[3].fX); 1049 if (pt.fX > max_x) { 1050 // The query line definitely does not cross the curve 1051 return false; 1052 } 1053 1054 // Do a binary search to find the parameter value which makes y as 1055 // close as possible to the query point. See whether the query 1056 // line's origin is to the left of the associated x coordinate. 1057 1058 // kMaxIter is chosen as the number of mantissa bits for a float, 1059 // since there's no way we are going to get more precision by 1060 // iterating more times than that. 1061 const int kMaxIter = 23; 1062 SkPoint eval; 1063 int iter = 0; 1064 SkScalar upper_t; 1065 SkScalar lower_t; 1066 // Need to invert direction of t parameter if cubic goes up 1067 // instead of down 1068 if (cubic[3].fY > cubic[0].fY) { 1069 upper_t = SK_Scalar1; 1070 lower_t = SkFloatToScalar(0); 1071 } else { 1072 upper_t = SkFloatToScalar(0); 1073 lower_t = SK_Scalar1; 1074 } 1075 do { 1076 SkScalar t = SkScalarAve(upper_t, lower_t); 1077 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 1078 if (pt.fY > eval.fY) { 1079 lower_t = t; 1080 } else { 1081 upper_t = t; 1082 } 1083 } while (++iter < kMaxIter 1084 && !SkScalarNearlyZero(eval.fY - pt.fY)); 1085 if (pt.fX <= eval.fX) { 1086 if (ambiguous) { 1087 *ambiguous = pt_at_extremum; 1088 } 1089 return true; 1090 } 1091 return false; 1092} 1093 1094int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1095 int num_crossings = 0; 1096 SkPoint monotonic_cubics[10]; 1097 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 1098 if (ambiguous) { 1099 *ambiguous = false; 1100 } 1101 bool locally_ambiguous; 1102 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous)) 1103 ++num_crossings; 1104 if (ambiguous) { 1105 *ambiguous |= locally_ambiguous; 1106 } 1107 if (num_monotonic_cubics > 0) 1108 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous)) 1109 ++num_crossings; 1110 if (ambiguous) { 1111 *ambiguous |= locally_ambiguous; 1112 } 1113 if (num_monotonic_cubics > 1) 1114 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous)) 1115 ++num_crossings; 1116 if (ambiguous) { 1117 *ambiguous |= locally_ambiguous; 1118 } 1119 return num_crossings; 1120} 1121 1122//////////////////////////////////////////////////////////////////////////////// 1123 1124/* Find t value for quadratic [a, b, c] = d. 1125 Return 0 if there is no solution within [0, 1) 1126*/ 1127static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) 1128{ 1129 // At^2 + Bt + C = d 1130 SkScalar A = a - 2 * b + c; 1131 SkScalar B = 2 * (b - a); 1132 SkScalar C = a - d; 1133 1134 SkScalar roots[2]; 1135 int count = SkFindUnitQuadRoots(A, B, C, roots); 1136 1137 SkASSERT(count <= 1); 1138 return count == 1 ? roots[0] : 0; 1139} 1140 1141/* given a quad-curve and a point (x,y), chop the quad at that point and return 1142 the new quad's offCurve point. Should only return false if the computed pos 1143 is the start of the curve (i.e. root == 0) 1144*/ 1145static bool quad_pt2OffCurve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* offCurve) 1146{ 1147 const SkScalar* base; 1148 SkScalar value; 1149 1150 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1151 base = &quad[0].fX; 1152 value = x; 1153 } else { 1154 base = &quad[0].fY; 1155 value = y; 1156 } 1157 1158 // note: this returns 0 if it thinks value is out of range, meaning the 1159 // root might return something outside of [0, 1) 1160 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1161 1162 if (t > 0) 1163 { 1164 SkPoint tmp[5]; 1165 SkChopQuadAt(quad, tmp, t); 1166 *offCurve = tmp[1]; 1167 return true; 1168 } else { 1169 /* t == 0 means either the value triggered a root outside of [0, 1) 1170 For our purposes, we can ignore the <= 0 roots, but we want to 1171 catch the >= 1 roots (which given our caller, will basically mean 1172 a root of 1, give-or-take numerical instability). If we are in the 1173 >= 1 case, return the existing offCurve point. 1174 1175 The test below checks to see if we are close to the "end" of the 1176 curve (near base[4]). Rather than specifying a tolerance, I just 1177 check to see if value is on to the right/left of the middle point 1178 (depending on the direction/sign of the end points). 1179 */ 1180 if ((base[0] < base[4] && value > base[2]) || 1181 (base[0] > base[4] && value < base[2])) // should root have been 1 1182 { 1183 *offCurve = quad[1]; 1184 return true; 1185 } 1186 } 1187 return false; 1188} 1189 1190static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1191 { SK_Scalar1, 0 }, 1192 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1193 { SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, 1194 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1195 1196 { 0, SK_Scalar1 }, 1197 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1198 { -SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, 1199 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1200 1201 { -SK_Scalar1, 0 }, 1202 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1203 { -SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, 1204 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1205 1206 { 0, -SK_Scalar1 }, 1207 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1208 { SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, 1209 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1210 1211 { SK_Scalar1, 0 } 1212}; 1213 1214int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1215 SkRotationDirection dir, const SkMatrix* userMatrix, 1216 SkPoint quadPoints[]) 1217{ 1218 // rotate by x,y so that uStart is (1.0) 1219 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1220 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1221 1222 SkScalar absX = SkScalarAbs(x); 1223 SkScalar absY = SkScalarAbs(y); 1224 1225 int pointCount; 1226 1227 // check for (effectively) coincident vectors 1228 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1229 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1230 if (absY <= SK_ScalarNearlyZero && x > 0 && 1231 ((y >= 0 && kCW_SkRotationDirection == dir) || 1232 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1233 1234 // just return the start-point 1235 quadPoints[0].set(SK_Scalar1, 0); 1236 pointCount = 1; 1237 } else { 1238 if (dir == kCCW_SkRotationDirection) 1239 y = -y; 1240 1241 // what octant (quadratic curve) is [xy] in? 1242 int oct = 0; 1243 bool sameSign = true; 1244 1245 if (0 == y) 1246 { 1247 oct = 4; // 180 1248 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1249 } 1250 else if (0 == x) 1251 { 1252 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1253 if (y > 0) 1254 oct = 2; // 90 1255 else 1256 oct = 6; // 270 1257 } 1258 else 1259 { 1260 if (y < 0) 1261 oct += 4; 1262 if ((x < 0) != (y < 0)) 1263 { 1264 oct += 2; 1265 sameSign = false; 1266 } 1267 if ((absX < absY) == sameSign) 1268 oct += 1; 1269 } 1270 1271 int wholeCount = oct << 1; 1272 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1273 1274 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1275 if (quad_pt2OffCurve(arc, x, y, &quadPoints[wholeCount + 1])) 1276 { 1277 quadPoints[wholeCount + 2].set(x, y); 1278 wholeCount += 2; 1279 } 1280 pointCount = wholeCount + 1; 1281 } 1282 1283 // now handle counter-clockwise and the initial unitStart rotation 1284 SkMatrix matrix; 1285 matrix.setSinCos(uStart.fY, uStart.fX); 1286 if (dir == kCCW_SkRotationDirection) { 1287 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1288 } 1289 if (userMatrix) { 1290 matrix.postConcat(*userMatrix); 1291 } 1292 matrix.mapPoints(quadPoints, pointCount); 1293 return pointCount; 1294} 1295 1296