SkGeometry.cpp revision bf0001d0472d727266762c5967ec0d919a6df083
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 11bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) { 12 if (ambiguous) { 13 *ambiguous = false; 14 } 15 // Determine quick discards. 16 // Consider query line going exactly through point 0 to not 17 // intersect, for symmetry with SkXRayCrossesMonotonicCubic. 18 if (pt.fY == pts[0].fY) { 19 if (ambiguous) { 20 *ambiguous = true; 21 } 22 return false; 23 } 24 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) 25 return false; 26 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) 27 return false; 28 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) 29 return false; 30 // Determine degenerate cases 31 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) 32 return false; 33 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { 34 // We've already determined the query point lies within the 35 // vertical range of the line segment. 36 if (pt.fX <= pts[0].fX) { 37 if (ambiguous) { 38 *ambiguous = (pt.fY == pts[1].fY); 39 } 40 return true; 41 } 42 return false; 43 } 44 // Ambiguity check 45 if (pt.fY == pts[1].fY) { 46 if (pt.fX <= pts[1].fX) { 47 if (ambiguous) { 48 *ambiguous = true; 49 } 50 return true; 51 } 52 return false; 53 } 54 // Full line segment evaluation 55 SkScalar delta_y = pts[1].fY - pts[0].fY; 56 SkScalar delta_x = pts[1].fX - pts[0].fX; 57 SkScalar slope = SkScalarDiv(delta_y, delta_x); 58 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); 59 // Solve for x coordinate at y = pt.fY 60 SkScalar x = SkScalarDiv(pt.fY - b, slope); 61 return pt.fX <= x; 62} 63 64/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 65 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 66 May also introduce overflow of fixed when we compute our setup. 67*/ 68// #define DIRECT_EVAL_OF_POLYNOMIALS 69 70//////////////////////////////////////////////////////////////////////// 71 72static int is_not_monotonic(float a, float b, float c) { 73 float ab = a - b; 74 float bc = b - c; 75 if (ab < 0) { 76 bc = -bc; 77 } 78 return ab == 0 || bc < 0; 79} 80 81//////////////////////////////////////////////////////////////////////// 82 83static bool is_unit_interval(SkScalar x) 84{ 85 return x > 0 && x < SK_Scalar1; 86} 87 88static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) 89{ 90 SkASSERT(ratio); 91 92 if (numer < 0) 93 { 94 numer = -numer; 95 denom = -denom; 96 } 97 98 if (denom == 0 || numer == 0 || numer >= denom) 99 return 0; 100 101 SkScalar r = SkScalarDiv(numer, denom); 102 if (SkScalarIsNaN(r)) { 103 return 0; 104 } 105 SkASSERT(r >= 0 && r < SK_Scalar1); 106 if (r == 0) // catch underflow if numer <<<< denom 107 return 0; 108 *ratio = r; 109 return 1; 110} 111 112/** From Numerical Recipes in C. 113 114 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 115 x1 = Q / A 116 x2 = C / Q 117*/ 118int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) 119{ 120 SkASSERT(roots); 121 122 if (A == 0) 123 return valid_unit_divide(-C, B, roots); 124 125 SkScalar* r = roots; 126 127 float R = B*B - 4*A*C; 128 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 129 return 0; 130 } 131 R = sk_float_sqrt(R); 132 133 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 134 r += valid_unit_divide(Q, A, r); 135 r += valid_unit_divide(C, Q, r); 136 if (r - roots == 2) 137 { 138 if (roots[0] > roots[1]) 139 SkTSwap<SkScalar>(roots[0], roots[1]); 140 else if (roots[0] == roots[1]) // nearly-equal? 141 r -= 1; // skip the double root 142 } 143 return (int)(r - roots); 144} 145 146/////////////////////////////////////////////////////////////////////////////// 147/////////////////////////////////////////////////////////////////////////////// 148 149static SkScalar eval_quad(const SkScalar src[], SkScalar t) 150{ 151 SkASSERT(src); 152 SkASSERT(t >= 0 && t <= SK_Scalar1); 153 154#ifdef DIRECT_EVAL_OF_POLYNOMIALS 155 SkScalar C = src[0]; 156 SkScalar A = src[4] - 2 * src[2] + C; 157 SkScalar B = 2 * (src[2] - C); 158 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 159#else 160 SkScalar ab = SkScalarInterp(src[0], src[2], t); 161 SkScalar bc = SkScalarInterp(src[2], src[4], t); 162 return SkScalarInterp(ab, bc, t); 163#endif 164} 165 166static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) 167{ 168 SkScalar A = src[4] - 2 * src[2] + src[0]; 169 SkScalar B = src[2] - src[0]; 170 171 return 2 * SkScalarMulAdd(A, t, B); 172} 173 174static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) 175{ 176 SkScalar A = src[4] - 2 * src[2] + src[0]; 177 SkScalar B = src[2] - src[0]; 178 return A + 2 * B; 179} 180 181void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) 182{ 183 SkASSERT(src); 184 SkASSERT(t >= 0 && t <= SK_Scalar1); 185 186 if (pt) 187 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 188 if (tangent) 189 tangent->set(eval_quad_derivative(&src[0].fX, t), 190 eval_quad_derivative(&src[0].fY, t)); 191} 192 193void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) 194{ 195 SkASSERT(src); 196 197 if (pt) 198 { 199 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 200 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 201 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 202 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 203 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 204 } 205 if (tangent) 206 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 207 eval_quad_derivative_at_half(&src[0].fY)); 208} 209 210static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 211{ 212 SkScalar ab = SkScalarInterp(src[0], src[2], t); 213 SkScalar bc = SkScalarInterp(src[2], src[4], t); 214 215 dst[0] = src[0]; 216 dst[2] = ab; 217 dst[4] = SkScalarInterp(ab, bc, t); 218 dst[6] = bc; 219 dst[8] = src[4]; 220} 221 222void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) 223{ 224 SkASSERT(t > 0 && t < SK_Scalar1); 225 226 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 227 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 228} 229 230void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) 231{ 232 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 233 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 234 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 235 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 236 237 dst[0] = src[0]; 238 dst[1].set(x01, y01); 239 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 240 dst[3].set(x12, y12); 241 dst[4] = src[2]; 242} 243 244/** Quad'(t) = At + B, where 245 A = 2(a - 2b + c) 246 B = 2(b - a) 247 Solve for t, only if it fits between 0 < t < 1 248*/ 249int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) 250{ 251 /* At + B == 0 252 t = -B / A 253 */ 254 return valid_unit_divide(a - b, a - b - b + c, tValue); 255} 256 257static inline void flatten_double_quad_extrema(SkScalar coords[14]) 258{ 259 coords[2] = coords[6] = coords[4]; 260} 261 262/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 263 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 264 */ 265int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) 266{ 267 SkASSERT(src); 268 SkASSERT(dst); 269 270#if 0 271 static bool once = true; 272 if (once) 273 { 274 once = false; 275 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; 276 SkPoint d[6]; 277 278 int n = SkChopQuadAtYExtrema(s, d); 279 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); 280 } 281#endif 282 283 SkScalar a = src[0].fY; 284 SkScalar b = src[1].fY; 285 SkScalar c = src[2].fY; 286 287 if (is_not_monotonic(a, b, c)) 288 { 289 SkScalar tValue; 290 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) 291 { 292 SkChopQuadAt(src, dst, tValue); 293 flatten_double_quad_extrema(&dst[0].fY); 294 return 1; 295 } 296 // if we get here, we need to force dst to be monotonic, even though 297 // we couldn't compute a unit_divide value (probably underflow). 298 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 299 } 300 dst[0].set(src[0].fX, a); 301 dst[1].set(src[1].fX, b); 302 dst[2].set(src[2].fX, c); 303 return 0; 304} 305 306/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 307 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 308 */ 309int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) 310{ 311 SkASSERT(src); 312 SkASSERT(dst); 313 314 SkScalar a = src[0].fX; 315 SkScalar b = src[1].fX; 316 SkScalar c = src[2].fX; 317 318 if (is_not_monotonic(a, b, c)) { 319 SkScalar tValue; 320 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 321 SkChopQuadAt(src, dst, tValue); 322 flatten_double_quad_extrema(&dst[0].fX); 323 return 1; 324 } 325 // if we get here, we need to force dst to be monotonic, even though 326 // we couldn't compute a unit_divide value (probably underflow). 327 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 328 } 329 dst[0].set(a, src[0].fY); 330 dst[1].set(b, src[1].fY); 331 dst[2].set(c, src[2].fY); 332 return 0; 333} 334 335// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 336// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 337// F''(t) = 2 (a - 2b + c) 338// 339// A = 2 (b - a) 340// B = 2 (a - 2b + c) 341// 342// Maximum curvature for a quadratic means solving 343// Fx' Fx'' + Fy' Fy'' = 0 344// 345// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 346// 347float SkFindQuadMaxCurvature(const SkPoint src[3]) { 348 SkScalar Ax = src[1].fX - src[0].fX; 349 SkScalar Ay = src[1].fY - src[0].fY; 350 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 351 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 352 SkScalar t = 0; // 0 means don't chop 353 354 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 355 return t; 356} 357 358int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) 359{ 360 SkScalar t = SkFindQuadMaxCurvature(src); 361 if (t == 0) { 362 memcpy(dst, src, 3 * sizeof(SkPoint)); 363 return 1; 364 } else { 365 SkChopQuadAt(src, dst, t); 366 return 2; 367 } 368} 369 370#define SK_ScalarTwoThirds (0.666666666f) 371 372void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 373 const SkScalar scale = SK_ScalarTwoThirds; 374 dst[0] = src[0]; 375 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 376 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 377 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 378 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 379 dst[3] = src[2]; 380} 381 382//////////////////////////////////////////////////////////////////////////////////////// 383///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 384//////////////////////////////////////////////////////////////////////////////////////// 385 386static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) 387{ 388 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 389 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 390 coeff[2] = 3*(pt[2] - pt[0]); 391 coeff[3] = pt[0]; 392} 393 394void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) 395{ 396 SkASSERT(pts); 397 398 if (cx) 399 get_cubic_coeff(&pts[0].fX, cx); 400 if (cy) 401 get_cubic_coeff(&pts[0].fY, cy); 402} 403 404static SkScalar eval_cubic(const SkScalar src[], SkScalar t) 405{ 406 SkASSERT(src); 407 SkASSERT(t >= 0 && t <= SK_Scalar1); 408 409 if (t == 0) 410 return src[0]; 411 412#ifdef DIRECT_EVAL_OF_POLYNOMIALS 413 SkScalar D = src[0]; 414 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 415 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 416 SkScalar C = 3*(src[2] - D); 417 418 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 419#else 420 SkScalar ab = SkScalarInterp(src[0], src[2], t); 421 SkScalar bc = SkScalarInterp(src[2], src[4], t); 422 SkScalar cd = SkScalarInterp(src[4], src[6], t); 423 SkScalar abc = SkScalarInterp(ab, bc, t); 424 SkScalar bcd = SkScalarInterp(bc, cd, t); 425 return SkScalarInterp(abc, bcd, t); 426#endif 427} 428 429/** return At^2 + Bt + C 430*/ 431static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) 432{ 433 SkASSERT(t >= 0 && t <= SK_Scalar1); 434 435 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 436} 437 438static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) 439{ 440 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 441 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 442 SkScalar C = src[2] - src[0]; 443 444 return eval_quadratic(A, B, C, t); 445} 446 447static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) 448{ 449 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 450 SkScalar B = src[4] - 2 * src[2] + src[0]; 451 452 return SkScalarMulAdd(A, t, B); 453} 454 455void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature) 456{ 457 SkASSERT(src); 458 SkASSERT(t >= 0 && t <= SK_Scalar1); 459 460 if (loc) 461 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 462 if (tangent) 463 tangent->set(eval_cubic_derivative(&src[0].fX, t), 464 eval_cubic_derivative(&src[0].fY, t)); 465 if (curvature) 466 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 467 eval_cubic_2ndDerivative(&src[0].fY, t)); 468} 469 470/** Cubic'(t) = At^2 + Bt + C, where 471 A = 3(-a + 3(b - c) + d) 472 B = 6(a - 2b + c) 473 C = 3(b - a) 474 Solve for t, keeping only those that fit betwee 0 < t < 1 475*/ 476int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2]) 477{ 478 // we divide A,B,C by 3 to simplify 479 SkScalar A = d - a + 3*(b - c); 480 SkScalar B = 2*(a - b - b + c); 481 SkScalar C = b - a; 482 483 return SkFindUnitQuadRoots(A, B, C, tValues); 484} 485 486static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 487{ 488 SkScalar ab = SkScalarInterp(src[0], src[2], t); 489 SkScalar bc = SkScalarInterp(src[2], src[4], t); 490 SkScalar cd = SkScalarInterp(src[4], src[6], t); 491 SkScalar abc = SkScalarInterp(ab, bc, t); 492 SkScalar bcd = SkScalarInterp(bc, cd, t); 493 SkScalar abcd = SkScalarInterp(abc, bcd, t); 494 495 dst[0] = src[0]; 496 dst[2] = ab; 497 dst[4] = abc; 498 dst[6] = abcd; 499 dst[8] = bcd; 500 dst[10] = cd; 501 dst[12] = src[6]; 502} 503 504void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) 505{ 506 SkASSERT(t > 0 && t < SK_Scalar1); 507 508 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 509 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 510} 511 512/* http://code.google.com/p/skia/issues/detail?id=32 513 514 This test code would fail when we didn't check the return result of 515 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 516 that after the first chop, the parameters to valid_unit_divide are equal 517 (thanks to finite float precision and rounding in the subtracts). Thus 518 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 519 up with 1.0, hence the need to check and just return the last cubic as 520 a degenerate clump of 4 points in the sampe place. 521 522 static void test_cubic() { 523 SkPoint src[4] = { 524 { 556.25000, 523.03003 }, 525 { 556.23999, 522.96002 }, 526 { 556.21997, 522.89001 }, 527 { 556.21997, 522.82001 } 528 }; 529 SkPoint dst[10]; 530 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 531 SkChopCubicAt(src, dst, tval, 2); 532 } 533 */ 534 535void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots) 536{ 537#ifdef SK_DEBUG 538 { 539 for (int i = 0; i < roots - 1; i++) 540 { 541 SkASSERT(is_unit_interval(tValues[i])); 542 SkASSERT(is_unit_interval(tValues[i+1])); 543 SkASSERT(tValues[i] < tValues[i+1]); 544 } 545 } 546#endif 547 548 if (dst) 549 { 550 if (roots == 0) // nothing to chop 551 memcpy(dst, src, 4*sizeof(SkPoint)); 552 else 553 { 554 SkScalar t = tValues[0]; 555 SkPoint tmp[4]; 556 557 for (int i = 0; i < roots; i++) 558 { 559 SkChopCubicAt(src, dst, t); 560 if (i == roots - 1) 561 break; 562 563 dst += 3; 564 // have src point to the remaining cubic (after the chop) 565 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 566 src = tmp; 567 568 // watch out in case the renormalized t isn't in range 569 if (!valid_unit_divide(tValues[i+1] - tValues[i], 570 SK_Scalar1 - tValues[i], &t)) { 571 // if we can't, just create a degenerate cubic 572 dst[4] = dst[5] = dst[6] = src[3]; 573 break; 574 } 575 } 576 } 577 } 578} 579 580void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) 581{ 582 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 583 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 584 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 585 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 586 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 587 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 588 589 SkScalar x012 = SkScalarAve(x01, x12); 590 SkScalar y012 = SkScalarAve(y01, y12); 591 SkScalar x123 = SkScalarAve(x12, x23); 592 SkScalar y123 = SkScalarAve(y12, y23); 593 594 dst[0] = src[0]; 595 dst[1].set(x01, y01); 596 dst[2].set(x012, y012); 597 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 598 dst[4].set(x123, y123); 599 dst[5].set(x23, y23); 600 dst[6] = src[3]; 601} 602 603static void flatten_double_cubic_extrema(SkScalar coords[14]) 604{ 605 coords[4] = coords[8] = coords[6]; 606} 607 608/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 609 the resulting beziers are monotonic in Y. This is called by the scan converter. 610 Depending on what is returned, dst[] is treated as follows 611 0 dst[0..3] is the original cubic 612 1 dst[0..3] and dst[3..6] are the two new cubics 613 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 614 If dst == null, it is ignored and only the count is returned. 615*/ 616int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 617 SkScalar tValues[2]; 618 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 619 src[3].fY, tValues); 620 621 SkChopCubicAt(src, dst, tValues, roots); 622 if (dst && roots > 0) { 623 // we do some cleanup to ensure our Y extrema are flat 624 flatten_double_cubic_extrema(&dst[0].fY); 625 if (roots == 2) { 626 flatten_double_cubic_extrema(&dst[3].fY); 627 } 628 } 629 return roots; 630} 631 632int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 633 SkScalar tValues[2]; 634 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 635 src[3].fX, tValues); 636 637 SkChopCubicAt(src, dst, tValues, roots); 638 if (dst && roots > 0) { 639 // we do some cleanup to ensure our Y extrema are flat 640 flatten_double_cubic_extrema(&dst[0].fX); 641 if (roots == 2) { 642 flatten_double_cubic_extrema(&dst[3].fX); 643 } 644 } 645 return roots; 646} 647 648/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 649 650 Inflection means that curvature is zero. 651 Curvature is [F' x F''] / [F'^3] 652 So we solve F'x X F''y - F'y X F''y == 0 653 After some canceling of the cubic term, we get 654 A = b - a 655 B = c - 2b + a 656 C = d - 3c + 3b - a 657 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 658*/ 659int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) 660{ 661 SkScalar Ax = src[1].fX - src[0].fX; 662 SkScalar Ay = src[1].fY - src[0].fY; 663 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 664 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 665 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 666 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 667 668 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 669} 670 671int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) 672{ 673 SkScalar tValues[2]; 674 int count = SkFindCubicInflections(src, tValues); 675 676 if (dst) 677 { 678 if (count == 0) 679 memcpy(dst, src, 4 * sizeof(SkPoint)); 680 else 681 SkChopCubicAt(src, dst, tValues, count); 682 } 683 return count + 1; 684} 685 686template <typename T> void bubble_sort(T array[], int count) 687{ 688 for (int i = count - 1; i > 0; --i) 689 for (int j = i; j > 0; --j) 690 if (array[j] < array[j-1]) 691 { 692 T tmp(array[j]); 693 array[j] = array[j-1]; 694 array[j-1] = tmp; 695 } 696} 697 698// newton refinement 699#if 0 700static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) 701{ 702 // x1 = x0 - f(t) / f'(t) 703 704 SkFP T = SkScalarToFloat(root); 705 SkFP N, D; 706 707 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] 708 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); 709 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); 710 D = SkFPAdd(D, coeff[2]); 711 712 if (D == 0) 713 return root; 714 715 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] 716 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); 717 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); 718 N = SkFPAdd(N, SkFPMul(T, coeff[2])); 719 N = SkFPAdd(N, coeff[3]); 720 721 if (N) 722 { 723 SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); 724 725 if (delta) 726 root -= delta; 727 } 728 return root; 729} 730#endif 731 732/** 733 * Given an array and count, remove all pair-wise duplicates from the array, 734 * keeping the existing sorting, and return the new count 735 */ 736static int collaps_duplicates(float array[], int count) { 737 for (int n = count; n > 1; --n) { 738 if (array[0] == array[1]) { 739 for (int i = 1; i < n; ++i) { 740 array[i - 1] = array[i]; 741 } 742 count -= 1; 743 } else { 744 array += 1; 745 } 746 } 747 return count; 748} 749 750#ifdef SK_DEBUG 751 752#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 753 754static void test_collaps_duplicates() { 755 static bool gOnce; 756 if (gOnce) { return; } 757 gOnce = true; 758 const float src0[] = { 0 }; 759 const float src1[] = { 0, 0 }; 760 const float src2[] = { 0, 1 }; 761 const float src3[] = { 0, 0, 0 }; 762 const float src4[] = { 0, 0, 1 }; 763 const float src5[] = { 0, 1, 1 }; 764 const float src6[] = { 0, 1, 2 }; 765 const struct { 766 const float* fData; 767 int fCount; 768 int fCollapsedCount; 769 } data[] = { 770 { TEST_COLLAPS_ENTRY(src0), 1 }, 771 { TEST_COLLAPS_ENTRY(src1), 1 }, 772 { TEST_COLLAPS_ENTRY(src2), 2 }, 773 { TEST_COLLAPS_ENTRY(src3), 1 }, 774 { TEST_COLLAPS_ENTRY(src4), 2 }, 775 { TEST_COLLAPS_ENTRY(src5), 2 }, 776 { TEST_COLLAPS_ENTRY(src6), 3 }, 777 }; 778 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 779 float dst[3]; 780 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 781 int count = collaps_duplicates(dst, data[i].fCount); 782 SkASSERT(data[i].fCollapsedCount == count); 783 for (int j = 1; j < count; ++j) { 784 SkASSERT(dst[j-1] < dst[j]); 785 } 786 } 787} 788#endif 789 790static SkScalar SkScalarCubeRoot(SkScalar x) { 791 return sk_float_pow(x, 0.3333333f); 792} 793 794/* Solve coeff(t) == 0, returning the number of roots that 795 lie withing 0 < t < 1. 796 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 797 798 Eliminates repeated roots (so that all tValues are distinct, and are always 799 in increasing order. 800*/ 801static int solve_cubic_polynomial(const SkScalar coeff[4], SkScalar tValues[3]) 802{ 803 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic 804 { 805 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 806 } 807 808 SkScalar a, b, c, Q, R; 809 810 { 811 SkASSERT(coeff[0] != 0); 812 813 SkScalar inva = SkScalarInvert(coeff[0]); 814 a = coeff[1] * inva; 815 b = coeff[2] * inva; 816 c = coeff[3] * inva; 817 } 818 Q = (a*a - b*3) / 9; 819 R = (2*a*a*a - 9*a*b + 27*c) / 54; 820 821 SkScalar Q3 = Q * Q * Q; 822 SkScalar R2MinusQ3 = R * R - Q3; 823 SkScalar adiv3 = a / 3; 824 825 SkScalar* roots = tValues; 826 SkScalar r; 827 828 if (R2MinusQ3 < 0) // we have 3 real roots 829 { 830 float theta = sk_float_acos(R / sk_float_sqrt(Q3)); 831 float neg2RootQ = -2 * sk_float_sqrt(Q); 832 833 r = neg2RootQ * sk_float_cos(theta/3) - adiv3; 834 if (is_unit_interval(r)) 835 *roots++ = r; 836 837 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; 838 if (is_unit_interval(r)) 839 *roots++ = r; 840 841 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; 842 if (is_unit_interval(r)) 843 *roots++ = r; 844 845 SkDEBUGCODE(test_collaps_duplicates();) 846 847 // now sort the roots 848 int count = (int)(roots - tValues); 849 SkASSERT((unsigned)count <= 3); 850 bubble_sort(tValues, count); 851 count = collaps_duplicates(tValues, count); 852 roots = tValues + count; // so we compute the proper count below 853 } 854 else // we have 1 real root 855 { 856 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 857 A = SkScalarCubeRoot(A); 858 if (R > 0) 859 A = -A; 860 861 if (A != 0) 862 A += Q / A; 863 r = A - adiv3; 864 if (is_unit_interval(r)) 865 *roots++ = r; 866 } 867 868 return (int)(roots - tValues); 869} 870 871/* Looking for F' dot F'' == 0 872 873 A = b - a 874 B = c - 2b + a 875 C = d - 3c + 3b - a 876 877 F' = 3Ct^2 + 6Bt + 3A 878 F'' = 6Ct + 6B 879 880 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 881*/ 882static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) 883{ 884 SkScalar a = src[2] - src[0]; 885 SkScalar b = src[4] - 2 * src[2] + src[0]; 886 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 887 888 coeff[0] = c * c; 889 coeff[1] = 3 * b * c; 890 coeff[2] = 2 * b * b + c * a; 891 coeff[3] = a * b; 892} 893 894// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 895//#define kMinTValueForChopping (SK_Scalar1 / 256) 896#define kMinTValueForChopping 0 897 898/* Looking for F' dot F'' == 0 899 900 A = b - a 901 B = c - 2b + a 902 C = d - 3c + 3b - a 903 904 F' = 3Ct^2 + 6Bt + 3A 905 F'' = 6Ct + 6B 906 907 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 908*/ 909int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) 910{ 911 SkScalar coeffX[4], coeffY[4]; 912 int i; 913 914 formulate_F1DotF2(&src[0].fX, coeffX); 915 formulate_F1DotF2(&src[0].fY, coeffY); 916 917 for (i = 0; i < 4; i++) 918 coeffX[i] += coeffY[i]; 919 920 SkScalar t[3]; 921 int count = solve_cubic_polynomial(coeffX, t); 922 int maxCount = 0; 923 924 // now remove extrema where the curvature is zero (mins) 925 // !!!! need a test for this !!!! 926 for (i = 0; i < count; i++) 927 { 928 // if (not_min_curvature()) 929 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) 930 tValues[maxCount++] = t[i]; 931 } 932 return maxCount; 933} 934 935int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3]) 936{ 937 SkScalar t_storage[3]; 938 939 if (tValues == NULL) 940 tValues = t_storage; 941 942 int count = SkFindCubicMaxCurvature(src, tValues); 943 944 if (dst) { 945 if (count == 0) 946 memcpy(dst, src, 4 * sizeof(SkPoint)); 947 else 948 SkChopCubicAt(src, dst, tValues, count); 949 } 950 return count + 1; 951} 952 953bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 954 if (ambiguous) { 955 *ambiguous = false; 956 } 957 958 // Find the minimum and maximum y of the extrema, which are the 959 // first and last points since this cubic is monotonic 960 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 961 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 962 963 if (pt.fY == cubic[0].fY 964 || pt.fY < min_y 965 || pt.fY > max_y) { 966 // The query line definitely does not cross the curve 967 if (ambiguous) { 968 *ambiguous = (pt.fY == cubic[0].fY); 969 } 970 return false; 971 } 972 973 bool pt_at_extremum = (pt.fY == cubic[3].fY); 974 975 SkScalar min_x = 976 SkMinScalar( 977 SkMinScalar( 978 SkMinScalar(cubic[0].fX, cubic[1].fX), 979 cubic[2].fX), 980 cubic[3].fX); 981 if (pt.fX < min_x) { 982 // The query line definitely crosses the curve 983 if (ambiguous) { 984 *ambiguous = pt_at_extremum; 985 } 986 return true; 987 } 988 989 SkScalar max_x = 990 SkMaxScalar( 991 SkMaxScalar( 992 SkMaxScalar(cubic[0].fX, cubic[1].fX), 993 cubic[2].fX), 994 cubic[3].fX); 995 if (pt.fX > max_x) { 996 // The query line definitely does not cross the curve 997 return false; 998 } 999 1000 // Do a binary search to find the parameter value which makes y as 1001 // close as possible to the query point. See whether the query 1002 // line's origin is to the left of the associated x coordinate. 1003 1004 // kMaxIter is chosen as the number of mantissa bits for a float, 1005 // since there's no way we are going to get more precision by 1006 // iterating more times than that. 1007 const int kMaxIter = 23; 1008 SkPoint eval; 1009 int iter = 0; 1010 SkScalar upper_t; 1011 SkScalar lower_t; 1012 // Need to invert direction of t parameter if cubic goes up 1013 // instead of down 1014 if (cubic[3].fY > cubic[0].fY) { 1015 upper_t = SK_Scalar1; 1016 lower_t = 0; 1017 } else { 1018 upper_t = 0; 1019 lower_t = SK_Scalar1; 1020 } 1021 do { 1022 SkScalar t = SkScalarAve(upper_t, lower_t); 1023 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 1024 if (pt.fY > eval.fY) { 1025 lower_t = t; 1026 } else { 1027 upper_t = t; 1028 } 1029 } while (++iter < kMaxIter 1030 && !SkScalarNearlyZero(eval.fY - pt.fY)); 1031 if (pt.fX <= eval.fX) { 1032 if (ambiguous) { 1033 *ambiguous = pt_at_extremum; 1034 } 1035 return true; 1036 } 1037 return false; 1038} 1039 1040int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1041 int num_crossings = 0; 1042 SkPoint monotonic_cubics[10]; 1043 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 1044 if (ambiguous) { 1045 *ambiguous = false; 1046 } 1047 bool locally_ambiguous; 1048 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous)) 1049 ++num_crossings; 1050 if (ambiguous) { 1051 *ambiguous |= locally_ambiguous; 1052 } 1053 if (num_monotonic_cubics > 0) 1054 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous)) 1055 ++num_crossings; 1056 if (ambiguous) { 1057 *ambiguous |= locally_ambiguous; 1058 } 1059 if (num_monotonic_cubics > 1) 1060 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous)) 1061 ++num_crossings; 1062 if (ambiguous) { 1063 *ambiguous |= locally_ambiguous; 1064 } 1065 return num_crossings; 1066} 1067//////////////////////////////////////////////////////////////////////////////// 1068 1069/* Find t value for quadratic [a, b, c] = d. 1070 Return 0 if there is no solution within [0, 1) 1071*/ 1072static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) 1073{ 1074 // At^2 + Bt + C = d 1075 SkScalar A = a - 2 * b + c; 1076 SkScalar B = 2 * (b - a); 1077 SkScalar C = a - d; 1078 1079 SkScalar roots[2]; 1080 int count = SkFindUnitQuadRoots(A, B, C, roots); 1081 1082 SkASSERT(count <= 1); 1083 return count == 1 ? roots[0] : 0; 1084} 1085 1086/* given a quad-curve and a point (x,y), chop the quad at that point and place 1087 the new off-curve point and endpoint into 'dest'. 1088 Should only return false if the computed pos is the start of the curve 1089 (i.e. root == 0) 1090*/ 1091static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest) 1092{ 1093 const SkScalar* base; 1094 SkScalar value; 1095 1096 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1097 base = &quad[0].fX; 1098 value = x; 1099 } else { 1100 base = &quad[0].fY; 1101 value = y; 1102 } 1103 1104 // note: this returns 0 if it thinks value is out of range, meaning the 1105 // root might return something outside of [0, 1) 1106 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1107 1108 if (t > 0) 1109 { 1110 SkPoint tmp[5]; 1111 SkChopQuadAt(quad, tmp, t); 1112 dest[0] = tmp[1]; 1113 dest[1].set(x, y); 1114 return true; 1115 } else { 1116 /* t == 0 means either the value triggered a root outside of [0, 1) 1117 For our purposes, we can ignore the <= 0 roots, but we want to 1118 catch the >= 1 roots (which given our caller, will basically mean 1119 a root of 1, give-or-take numerical instability). If we are in the 1120 >= 1 case, return the existing offCurve point. 1121 1122 The test below checks to see if we are close to the "end" of the 1123 curve (near base[4]). Rather than specifying a tolerance, I just 1124 check to see if value is on to the right/left of the middle point 1125 (depending on the direction/sign of the end points). 1126 */ 1127 if ((base[0] < base[4] && value > base[2]) || 1128 (base[0] > base[4] && value < base[2])) // should root have been 1 1129 { 1130 dest[0] = quad[1]; 1131 dest[1].set(x, y); 1132 return true; 1133 } 1134 } 1135 return false; 1136} 1137 1138static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1139// The mid point of the quadratic arc approximation is half way between the two 1140// control points. The float epsilon adjustment moves the on curve point out by 1141// two bits, distributing the convex test error between the round rect approximation 1142// and the convex cross product sign equality test. 1143#define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1144 { SK_Scalar1, 0 }, 1145 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1146 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1147 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1148 1149 { 0, SK_Scalar1 }, 1150 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1151 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1152 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1153 1154 { -SK_Scalar1, 0 }, 1155 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1156 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1157 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1158 1159 { 0, -SK_Scalar1 }, 1160 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1161 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1162 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1163 1164 { SK_Scalar1, 0 } 1165#undef SK_MID_RRECT_OFFSET 1166}; 1167 1168int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1169 SkRotationDirection dir, const SkMatrix* userMatrix, 1170 SkPoint quadPoints[]) 1171{ 1172 // rotate by x,y so that uStart is (1.0) 1173 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1174 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1175 1176 SkScalar absX = SkScalarAbs(x); 1177 SkScalar absY = SkScalarAbs(y); 1178 1179 int pointCount; 1180 1181 // check for (effectively) coincident vectors 1182 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1183 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1184 if (absY <= SK_ScalarNearlyZero && x > 0 && 1185 ((y >= 0 && kCW_SkRotationDirection == dir) || 1186 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1187 1188 // just return the start-point 1189 quadPoints[0].set(SK_Scalar1, 0); 1190 pointCount = 1; 1191 } else { 1192 if (dir == kCCW_SkRotationDirection) 1193 y = -y; 1194 1195 // what octant (quadratic curve) is [xy] in? 1196 int oct = 0; 1197 bool sameSign = true; 1198 1199 if (0 == y) 1200 { 1201 oct = 4; // 180 1202 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1203 } 1204 else if (0 == x) 1205 { 1206 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1207 if (y > 0) 1208 oct = 2; // 90 1209 else 1210 oct = 6; // 270 1211 } 1212 else 1213 { 1214 if (y < 0) 1215 oct += 4; 1216 if ((x < 0) != (y < 0)) 1217 { 1218 oct += 2; 1219 sameSign = false; 1220 } 1221 if ((absX < absY) == sameSign) 1222 oct += 1; 1223 } 1224 1225 int wholeCount = oct << 1; 1226 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1227 1228 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1229 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) 1230 { 1231 wholeCount += 2; 1232 } 1233 pointCount = wholeCount + 1; 1234 } 1235 1236 // now handle counter-clockwise and the initial unitStart rotation 1237 SkMatrix matrix; 1238 matrix.setSinCos(uStart.fY, uStart.fX); 1239 if (dir == kCCW_SkRotationDirection) { 1240 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1241 } 1242 if (userMatrix) { 1243 matrix.postConcat(*userMatrix); 1244 } 1245 matrix.mapPoints(quadPoints, pointCount); 1246 return pointCount; 1247} 1248 1249/////////////////////////////////////////////////////////////////////////////// 1250 1251// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1252// ------------------------------------------ 1253// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1254// 1255// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1256// ------------------------------------------------ 1257// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1258// 1259 1260// Take the parametric specification for the conic (either X or Y) and return 1261// in coeff[] the coefficients for the simple quadratic polynomial 1262// coeff[0] for t^2 1263// coeff[1] for t 1264// coeff[2] for constant term 1265// 1266static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1267 SkASSERT(src); 1268 SkASSERT(t >= 0 && t <= SK_Scalar1); 1269 1270 SkScalar src2w = SkScalarMul(src[2], w); 1271 SkScalar C = src[0]; 1272 SkScalar A = src[4] - 2 * src2w + C; 1273 SkScalar B = 2 * (src2w - C); 1274 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1275 1276 B = 2 * (w - SK_Scalar1); 1277 C = SK_Scalar1; 1278 A = -B; 1279 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1280 1281 return SkScalarDiv(numer, denom); 1282} 1283 1284// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1285// 1286// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1287// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1288// t^0 : -2 P0 w + 2 P1 w 1289// 1290// We disregard magnitude, so we can freely ignore the denominator of F', and 1291// divide the numerator by 2 1292// 1293// coeff[0] for t^2 1294// coeff[1] for t^1 1295// coeff[2] for t^0 1296// 1297static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1298 const SkScalar P20 = src[4] - src[0]; 1299 const SkScalar P10 = src[2] - src[0]; 1300 const SkScalar wP10 = w * P10; 1301 coeff[0] = w * P20 - P20; 1302 coeff[1] = P20 - 2 * wP10; 1303 coeff[2] = wP10; 1304} 1305 1306static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1307 SkScalar coeff[3]; 1308 conic_deriv_coeff(coord, w, coeff); 1309 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1310} 1311 1312static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1313 SkScalar coeff[3]; 1314 conic_deriv_coeff(src, w, coeff); 1315 1316 SkScalar tValues[2]; 1317 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1318 SkASSERT(0 == roots || 1 == roots); 1319 1320 if (1 == roots) { 1321 *t = tValues[0]; 1322 return true; 1323 } 1324 return false; 1325} 1326 1327struct SkP3D { 1328 SkScalar fX, fY, fZ; 1329 1330 void set(SkScalar x, SkScalar y, SkScalar z) { 1331 fX = x; fY = y; fZ = z; 1332 } 1333 1334 void projectDown(SkPoint* dst) const { 1335 dst->set(fX / fZ, fY / fZ); 1336 } 1337}; 1338 1339// we just return the middle 3 points, since the first and last are dups of src 1340// 1341static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) { 1342 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1343 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1344 dst[0] = ab; 1345 dst[3] = SkScalarInterp(ab, bc, t); 1346 dst[6] = bc; 1347} 1348 1349static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1350 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1351 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1352 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1353} 1354 1355void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1356 SkASSERT(t >= 0 && t <= SK_Scalar1); 1357 1358 if (pt) { 1359 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1360 conic_eval_pos(&fPts[0].fY, fW, t)); 1361 } 1362 if (tangent) { 1363 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1364 conic_eval_tan(&fPts[0].fY, fW, t)); 1365 } 1366} 1367 1368void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1369 SkP3D tmp[3], tmp2[3]; 1370 1371 ratquad_mapTo3D(fPts, fW, tmp); 1372 1373 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1374 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1375 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1376 1377 dst[0].fPts[0] = fPts[0]; 1378 tmp2[0].projectDown(&dst[0].fPts[1]); 1379 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1380 tmp2[2].projectDown(&dst[1].fPts[1]); 1381 dst[1].fPts[2] = fPts[2]; 1382 1383 // to put in "standard form", where w0 and w2 are both 1, we compute the 1384 // new w1 as sqrt(w1*w1/w0*w2) 1385 // or 1386 // w1 /= sqrt(w0*w2) 1387 // 1388 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1389 // 1390 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1391 dst[0].fW = tmp2[0].fZ / root; 1392 dst[1].fW = tmp2[2].fZ / root; 1393} 1394 1395static SkScalar subdivide_w_value(SkScalar w) { 1396 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1397} 1398 1399void SkConic::chop(SkConic dst[2]) const { 1400 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1401 SkScalar p1x = fW * fPts[1].fX; 1402 SkScalar p1y = fW * fPts[1].fY; 1403 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1404 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1405 1406 dst[0].fPts[0] = fPts[0]; 1407 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1408 (fPts[0].fY + p1y) * scale); 1409 dst[0].fPts[2].set(mx, my); 1410 1411 dst[1].fPts[0].set(mx, my); 1412 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1413 (p1y + fPts[2].fY) * scale); 1414 dst[1].fPts[2] = fPts[2]; 1415 1416 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1417} 1418 1419/* 1420 * "High order approximation of conic sections by quadratic splines" 1421 * by Michael Floater, 1993 1422 */ 1423#define AS_QUAD_ERROR_SETUP \ 1424 SkScalar a = fW - 1; \ 1425 SkScalar k = a / (4 * (2 + a)); \ 1426 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1427 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1428 1429void SkConic::computeAsQuadError(SkVector* err) const { 1430 AS_QUAD_ERROR_SETUP 1431 err->set(x, y); 1432} 1433 1434bool SkConic::asQuadTol(SkScalar tol) const { 1435 AS_QUAD_ERROR_SETUP 1436 return (x * x + y * y) <= tol * tol; 1437} 1438 1439int SkConic::computeQuadPOW2(SkScalar tol) const { 1440 AS_QUAD_ERROR_SETUP 1441 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1442 1443 if (error <= 0) { 1444 return 0; 1445 } 1446 uint32_t ierr = (uint32_t)error; 1447 return (34 - SkCLZ(ierr)) >> 1; 1448} 1449 1450static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1451 SkASSERT(level >= 0); 1452 1453 if (0 == level) { 1454 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1455 return pts + 2; 1456 } else { 1457 SkConic dst[2]; 1458 src.chop(dst); 1459 --level; 1460 pts = subdivide(dst[0], pts, level); 1461 return subdivide(dst[1], pts, level); 1462 } 1463} 1464 1465int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1466 SkASSERT(pow2 >= 0); 1467 *pts = fPts[0]; 1468 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1469 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1470 return 1 << pow2; 1471} 1472 1473bool SkConic::findXExtrema(SkScalar* t) const { 1474 return conic_find_extrema(&fPts[0].fX, fW, t); 1475} 1476 1477bool SkConic::findYExtrema(SkScalar* t) const { 1478 return conic_find_extrema(&fPts[0].fY, fW, t); 1479} 1480 1481bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1482 SkScalar t; 1483 if (this->findXExtrema(&t)) { 1484 this->chopAt(t, dst); 1485 // now clean-up the middle, since we know t was meant to be at 1486 // an X-extrema 1487 SkScalar value = dst[0].fPts[2].fX; 1488 dst[0].fPts[1].fX = value; 1489 dst[1].fPts[0].fX = value; 1490 dst[1].fPts[1].fX = value; 1491 return true; 1492 } 1493 return false; 1494} 1495 1496bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1497 SkScalar t; 1498 if (this->findYExtrema(&t)) { 1499 this->chopAt(t, dst); 1500 // now clean-up the middle, since we know t was meant to be at 1501 // an Y-extrema 1502 SkScalar value = dst[0].fPts[2].fY; 1503 dst[0].fPts[1].fY = value; 1504 dst[1].fPts[0].fY = value; 1505 dst[1].fPts[1].fY = value; 1506 return true; 1507 } 1508 return false; 1509} 1510 1511void SkConic::computeTightBounds(SkRect* bounds) const { 1512 SkPoint pts[4]; 1513 pts[0] = fPts[0]; 1514 pts[1] = fPts[2]; 1515 int count = 2; 1516 1517 SkScalar t; 1518 if (this->findXExtrema(&t)) { 1519 this->evalAt(t, &pts[count++]); 1520 } 1521 if (this->findYExtrema(&t)) { 1522 this->evalAt(t, &pts[count++]); 1523 } 1524 bounds->set(pts, count); 1525} 1526 1527void SkConic::computeFastBounds(SkRect* bounds) const { 1528 bounds->set(fPts, 3); 1529} 1530