SkGeometry.cpp revision b39d5617f60e8c26f76011cfcd984d7ad42d9fa9
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(SkScalar a, SkScalar b, SkScalar c) { 73 SkScalar ab = a - b; 74 SkScalar 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 return x > 0 && x < SK_Scalar1; 85} 86 87static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 88 SkASSERT(ratio); 89 90 if (numer < 0) { 91 numer = -numer; 92 denom = -denom; 93 } 94 95 if (denom == 0 || numer == 0 || numer >= denom) { 96 return 0; 97 } 98 99 SkScalar r = SkScalarDiv(numer, denom); 100 if (SkScalarIsNaN(r)) { 101 return 0; 102 } 103 SkASSERT(r >= 0 && r < SK_Scalar1); 104 if (r == 0) { // catch underflow if numer <<<< denom 105 return 0; 106 } 107 *ratio = r; 108 return 1; 109} 110 111/** From Numerical Recipes in C. 112 113 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 114 x1 = Q / A 115 x2 = C / Q 116*/ 117int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 118 SkASSERT(roots); 119 120 if (A == 0) { 121 return valid_unit_divide(-C, B, roots); 122 } 123 124 SkScalar* r = roots; 125 126 SkScalar R = B*B - 4*A*C; 127 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 128 return 0; 129 } 130 R = SkScalarSqrt(R); 131 132 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 133 r += valid_unit_divide(Q, A, r); 134 r += valid_unit_divide(C, Q, r); 135 if (r - roots == 2) { 136 if (roots[0] > roots[1]) 137 SkTSwap<SkScalar>(roots[0], roots[1]); 138 else if (roots[0] == roots[1]) // nearly-equal? 139 r -= 1; // skip the double root 140 } 141 return (int)(r - roots); 142} 143 144/////////////////////////////////////////////////////////////////////////////// 145/////////////////////////////////////////////////////////////////////////////// 146 147static SkScalar eval_quad(const SkScalar src[], SkScalar t) { 148 SkASSERT(src); 149 SkASSERT(t >= 0 && t <= SK_Scalar1); 150 151#ifdef DIRECT_EVAL_OF_POLYNOMIALS 152 SkScalar C = src[0]; 153 SkScalar A = src[4] - 2 * src[2] + C; 154 SkScalar B = 2 * (src[2] - C); 155 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 156#else 157 SkScalar ab = SkScalarInterp(src[0], src[2], t); 158 SkScalar bc = SkScalarInterp(src[2], src[4], t); 159 return SkScalarInterp(ab, bc, t); 160#endif 161} 162 163static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) { 164 SkScalar A = src[4] - 2 * src[2] + src[0]; 165 SkScalar B = src[2] - src[0]; 166 167 return 2 * SkScalarMulAdd(A, t, B); 168} 169 170static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) { 171 SkScalar A = src[4] - 2 * src[2] + src[0]; 172 SkScalar B = src[2] - src[0]; 173 return A + 2 * B; 174} 175 176void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, 177 SkVector* tangent) { 178 SkASSERT(src); 179 SkASSERT(t >= 0 && t <= SK_Scalar1); 180 181 if (pt) { 182 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 183 } 184 if (tangent) { 185 tangent->set(eval_quad_derivative(&src[0].fX, t), 186 eval_quad_derivative(&src[0].fY, t)); 187 } 188} 189 190void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) { 191 SkASSERT(src); 192 193 if (pt) { 194 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 195 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 196 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 197 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 198 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 199 } 200 if (tangent) { 201 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 202 eval_quad_derivative_at_half(&src[0].fY)); 203 } 204} 205 206static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) { 207 SkScalar ab = SkScalarInterp(src[0], src[2], t); 208 SkScalar bc = SkScalarInterp(src[2], src[4], t); 209 210 dst[0] = src[0]; 211 dst[2] = ab; 212 dst[4] = SkScalarInterp(ab, bc, t); 213 dst[6] = bc; 214 dst[8] = src[4]; 215} 216 217void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 218 SkASSERT(t > 0 && t < SK_Scalar1); 219 220 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 221 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 222} 223 224void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 225 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 226 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 227 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 228 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 229 230 dst[0] = src[0]; 231 dst[1].set(x01, y01); 232 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 233 dst[3].set(x12, y12); 234 dst[4] = src[2]; 235} 236 237/** Quad'(t) = At + B, where 238 A = 2(a - 2b + c) 239 B = 2(b - a) 240 Solve for t, only if it fits between 0 < t < 1 241*/ 242int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 243 /* At + B == 0 244 t = -B / A 245 */ 246 return valid_unit_divide(a - b, a - b - b + c, tValue); 247} 248 249static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 250 coords[2] = coords[6] = coords[4]; 251} 252 253/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 254 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 255 */ 256int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 257 SkASSERT(src); 258 SkASSERT(dst); 259 260 SkScalar a = src[0].fY; 261 SkScalar b = src[1].fY; 262 SkScalar c = src[2].fY; 263 264 if (is_not_monotonic(a, b, c)) { 265 SkScalar tValue; 266 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 267 SkChopQuadAt(src, dst, tValue); 268 flatten_double_quad_extrema(&dst[0].fY); 269 return 1; 270 } 271 // if we get here, we need to force dst to be monotonic, even though 272 // we couldn't compute a unit_divide value (probably underflow). 273 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 274 } 275 dst[0].set(src[0].fX, a); 276 dst[1].set(src[1].fX, b); 277 dst[2].set(src[2].fX, c); 278 return 0; 279} 280 281/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 282 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 283 */ 284int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 285 SkASSERT(src); 286 SkASSERT(dst); 287 288 SkScalar a = src[0].fX; 289 SkScalar b = src[1].fX; 290 SkScalar c = src[2].fX; 291 292 if (is_not_monotonic(a, b, c)) { 293 SkScalar tValue; 294 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 295 SkChopQuadAt(src, dst, tValue); 296 flatten_double_quad_extrema(&dst[0].fX); 297 return 1; 298 } 299 // if we get here, we need to force dst to be monotonic, even though 300 // we couldn't compute a unit_divide value (probably underflow). 301 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 302 } 303 dst[0].set(a, src[0].fY); 304 dst[1].set(b, src[1].fY); 305 dst[2].set(c, src[2].fY); 306 return 0; 307} 308 309// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 310// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 311// F''(t) = 2 (a - 2b + c) 312// 313// A = 2 (b - a) 314// B = 2 (a - 2b + c) 315// 316// Maximum curvature for a quadratic means solving 317// Fx' Fx'' + Fy' Fy'' = 0 318// 319// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 320// 321SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 322 SkScalar Ax = src[1].fX - src[0].fX; 323 SkScalar Ay = src[1].fY - src[0].fY; 324 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 325 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 326 SkScalar t = 0; // 0 means don't chop 327 328 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 329 return t; 330} 331 332int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 333 SkScalar t = SkFindQuadMaxCurvature(src); 334 if (t == 0) { 335 memcpy(dst, src, 3 * sizeof(SkPoint)); 336 return 1; 337 } else { 338 SkChopQuadAt(src, dst, t); 339 return 2; 340 } 341} 342 343#define SK_ScalarTwoThirds (0.666666666f) 344 345void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 346 const SkScalar scale = SK_ScalarTwoThirds; 347 dst[0] = src[0]; 348 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 349 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 350 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 351 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 352 dst[3] = src[2]; 353} 354 355////////////////////////////////////////////////////////////////////////////// 356///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 357////////////////////////////////////////////////////////////////////////////// 358 359static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) { 360 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 361 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 362 coeff[2] = 3*(pt[2] - pt[0]); 363 coeff[3] = pt[0]; 364} 365 366void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) { 367 SkASSERT(pts); 368 369 if (cx) { 370 get_cubic_coeff(&pts[0].fX, cx); 371 } 372 if (cy) { 373 get_cubic_coeff(&pts[0].fY, cy); 374 } 375} 376 377static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 378 SkASSERT(src); 379 SkASSERT(t >= 0 && t <= SK_Scalar1); 380 381 if (t == 0) { 382 return src[0]; 383 } 384 385#ifdef DIRECT_EVAL_OF_POLYNOMIALS 386 SkScalar D = src[0]; 387 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 388 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 389 SkScalar C = 3*(src[2] - D); 390 391 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 392#else 393 SkScalar ab = SkScalarInterp(src[0], src[2], t); 394 SkScalar bc = SkScalarInterp(src[2], src[4], t); 395 SkScalar cd = SkScalarInterp(src[4], src[6], t); 396 SkScalar abc = SkScalarInterp(ab, bc, t); 397 SkScalar bcd = SkScalarInterp(bc, cd, t); 398 return SkScalarInterp(abc, bcd, t); 399#endif 400} 401 402/** return At^2 + Bt + C 403*/ 404static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { 405 SkASSERT(t >= 0 && t <= SK_Scalar1); 406 407 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 408} 409 410static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { 411 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 412 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 413 SkScalar C = src[2] - src[0]; 414 415 return eval_quadratic(A, B, C, t); 416} 417 418static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { 419 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 420 SkScalar B = src[4] - 2 * src[2] + src[0]; 421 422 return SkScalarMulAdd(A, t, B); 423} 424 425void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 426 SkVector* tangent, SkVector* curvature) { 427 SkASSERT(src); 428 SkASSERT(t >= 0 && t <= SK_Scalar1); 429 430 if (loc) { 431 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 432 } 433 if (tangent) { 434 tangent->set(eval_cubic_derivative(&src[0].fX, t), 435 eval_cubic_derivative(&src[0].fY, t)); 436 } 437 if (curvature) { 438 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 439 eval_cubic_2ndDerivative(&src[0].fY, t)); 440 } 441} 442 443/** Cubic'(t) = At^2 + Bt + C, where 444 A = 3(-a + 3(b - c) + d) 445 B = 6(a - 2b + c) 446 C = 3(b - a) 447 Solve for t, keeping only those that fit betwee 0 < t < 1 448*/ 449int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 450 SkScalar tValues[2]) { 451 // we divide A,B,C by 3 to simplify 452 SkScalar A = d - a + 3*(b - c); 453 SkScalar B = 2*(a - b - b + c); 454 SkScalar C = b - a; 455 456 return SkFindUnitQuadRoots(A, B, C, tValues); 457} 458 459static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, 460 SkScalar t) { 461 SkScalar ab = SkScalarInterp(src[0], src[2], t); 462 SkScalar bc = SkScalarInterp(src[2], src[4], t); 463 SkScalar cd = SkScalarInterp(src[4], src[6], t); 464 SkScalar abc = SkScalarInterp(ab, bc, t); 465 SkScalar bcd = SkScalarInterp(bc, cd, t); 466 SkScalar abcd = SkScalarInterp(abc, bcd, t); 467 468 dst[0] = src[0]; 469 dst[2] = ab; 470 dst[4] = abc; 471 dst[6] = abcd; 472 dst[8] = bcd; 473 dst[10] = cd; 474 dst[12] = src[6]; 475} 476 477void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 478 SkASSERT(t > 0 && t < SK_Scalar1); 479 480 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 481 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 482} 483 484/* http://code.google.com/p/skia/issues/detail?id=32 485 486 This test code would fail when we didn't check the return result of 487 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 488 that after the first chop, the parameters to valid_unit_divide are equal 489 (thanks to finite float precision and rounding in the subtracts). Thus 490 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 491 up with 1.0, hence the need to check and just return the last cubic as 492 a degenerate clump of 4 points in the sampe place. 493 494 static void test_cubic() { 495 SkPoint src[4] = { 496 { 556.25000, 523.03003 }, 497 { 556.23999, 522.96002 }, 498 { 556.21997, 522.89001 }, 499 { 556.21997, 522.82001 } 500 }; 501 SkPoint dst[10]; 502 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 503 SkChopCubicAt(src, dst, tval, 2); 504 } 505 */ 506 507void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 508 const SkScalar tValues[], int roots) { 509#ifdef SK_DEBUG 510 { 511 for (int i = 0; i < roots - 1; i++) 512 { 513 SkASSERT(is_unit_interval(tValues[i])); 514 SkASSERT(is_unit_interval(tValues[i+1])); 515 SkASSERT(tValues[i] < tValues[i+1]); 516 } 517 } 518#endif 519 520 if (dst) { 521 if (roots == 0) { // nothing to chop 522 memcpy(dst, src, 4*sizeof(SkPoint)); 523 } else { 524 SkScalar t = tValues[0]; 525 SkPoint tmp[4]; 526 527 for (int i = 0; i < roots; i++) { 528 SkChopCubicAt(src, dst, t); 529 if (i == roots - 1) { 530 break; 531 } 532 533 dst += 3; 534 // have src point to the remaining cubic (after the chop) 535 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 536 src = tmp; 537 538 // watch out in case the renormalized t isn't in range 539 if (!valid_unit_divide(tValues[i+1] - tValues[i], 540 SK_Scalar1 - tValues[i], &t)) { 541 // if we can't, just create a degenerate cubic 542 dst[4] = dst[5] = dst[6] = src[3]; 543 break; 544 } 545 } 546 } 547 } 548} 549 550void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 551 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 552 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 553 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 554 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 555 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 556 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 557 558 SkScalar x012 = SkScalarAve(x01, x12); 559 SkScalar y012 = SkScalarAve(y01, y12); 560 SkScalar x123 = SkScalarAve(x12, x23); 561 SkScalar y123 = SkScalarAve(y12, y23); 562 563 dst[0] = src[0]; 564 dst[1].set(x01, y01); 565 dst[2].set(x012, y012); 566 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 567 dst[4].set(x123, y123); 568 dst[5].set(x23, y23); 569 dst[6] = src[3]; 570} 571 572static void flatten_double_cubic_extrema(SkScalar coords[14]) { 573 coords[4] = coords[8] = coords[6]; 574} 575 576/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 577 the resulting beziers are monotonic in Y. This is called by the scan converter. 578 Depending on what is returned, dst[] is treated as follows 579 0 dst[0..3] is the original cubic 580 1 dst[0..3] and dst[3..6] are the two new cubics 581 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 582 If dst == null, it is ignored and only the count is returned. 583*/ 584int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 585 SkScalar tValues[2]; 586 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 587 src[3].fY, tValues); 588 589 SkChopCubicAt(src, dst, tValues, roots); 590 if (dst && roots > 0) { 591 // we do some cleanup to ensure our Y extrema are flat 592 flatten_double_cubic_extrema(&dst[0].fY); 593 if (roots == 2) { 594 flatten_double_cubic_extrema(&dst[3].fY); 595 } 596 } 597 return roots; 598} 599 600int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 601 SkScalar tValues[2]; 602 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 603 src[3].fX, tValues); 604 605 SkChopCubicAt(src, dst, tValues, roots); 606 if (dst && roots > 0) { 607 // we do some cleanup to ensure our Y extrema are flat 608 flatten_double_cubic_extrema(&dst[0].fX); 609 if (roots == 2) { 610 flatten_double_cubic_extrema(&dst[3].fX); 611 } 612 } 613 return roots; 614} 615 616/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 617 618 Inflection means that curvature is zero. 619 Curvature is [F' x F''] / [F'^3] 620 So we solve F'x X F''y - F'y X F''y == 0 621 After some canceling of the cubic term, we get 622 A = b - a 623 B = c - 2b + a 624 C = d - 3c + 3b - a 625 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 626*/ 627int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 628 SkScalar Ax = src[1].fX - src[0].fX; 629 SkScalar Ay = src[1].fY - src[0].fY; 630 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 631 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 632 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 633 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 634 635 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 636} 637 638int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 639 SkScalar tValues[2]; 640 int count = SkFindCubicInflections(src, tValues); 641 642 if (dst) { 643 if (count == 0) { 644 memcpy(dst, src, 4 * sizeof(SkPoint)); 645 } else { 646 SkChopCubicAt(src, dst, tValues, count); 647 } 648 } 649 return count + 1; 650} 651 652template <typename T> void bubble_sort(T array[], int count) { 653 for (int i = count - 1; i > 0; --i) 654 for (int j = i; j > 0; --j) 655 if (array[j] < array[j-1]) 656 { 657 T tmp(array[j]); 658 array[j] = array[j-1]; 659 array[j-1] = tmp; 660 } 661} 662 663/** 664 * Given an array and count, remove all pair-wise duplicates from the array, 665 * keeping the existing sorting, and return the new count 666 */ 667static int collaps_duplicates(SkScalar array[], int count) { 668 for (int n = count; n > 1; --n) { 669 if (array[0] == array[1]) { 670 for (int i = 1; i < n; ++i) { 671 array[i - 1] = array[i]; 672 } 673 count -= 1; 674 } else { 675 array += 1; 676 } 677 } 678 return count; 679} 680 681#ifdef SK_DEBUG 682 683#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 684 685static void test_collaps_duplicates() { 686 static bool gOnce; 687 if (gOnce) { return; } 688 gOnce = true; 689 const SkScalar src0[] = { 0 }; 690 const SkScalar src1[] = { 0, 0 }; 691 const SkScalar src2[] = { 0, 1 }; 692 const SkScalar src3[] = { 0, 0, 0 }; 693 const SkScalar src4[] = { 0, 0, 1 }; 694 const SkScalar src5[] = { 0, 1, 1 }; 695 const SkScalar src6[] = { 0, 1, 2 }; 696 const struct { 697 const SkScalar* fData; 698 int fCount; 699 int fCollapsedCount; 700 } data[] = { 701 { TEST_COLLAPS_ENTRY(src0), 1 }, 702 { TEST_COLLAPS_ENTRY(src1), 1 }, 703 { TEST_COLLAPS_ENTRY(src2), 2 }, 704 { TEST_COLLAPS_ENTRY(src3), 1 }, 705 { TEST_COLLAPS_ENTRY(src4), 2 }, 706 { TEST_COLLAPS_ENTRY(src5), 2 }, 707 { TEST_COLLAPS_ENTRY(src6), 3 }, 708 }; 709 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 710 SkScalar dst[3]; 711 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 712 int count = collaps_duplicates(dst, data[i].fCount); 713 SkASSERT(data[i].fCollapsedCount == count); 714 for (int j = 1; j < count; ++j) { 715 SkASSERT(dst[j-1] < dst[j]); 716 } 717 } 718} 719#endif 720 721static SkScalar SkScalarCubeRoot(SkScalar x) { 722 return SkScalarPow(x, 0.3333333f); 723} 724 725/* Solve coeff(t) == 0, returning the number of roots that 726 lie withing 0 < t < 1. 727 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 728 729 Eliminates repeated roots (so that all tValues are distinct, and are always 730 in increasing order. 731*/ 732static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 733 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 734 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 735 } 736 737 SkScalar a, b, c, Q, R; 738 739 { 740 SkASSERT(coeff[0] != 0); 741 742 SkScalar inva = SkScalarInvert(coeff[0]); 743 a = coeff[1] * inva; 744 b = coeff[2] * inva; 745 c = coeff[3] * inva; 746 } 747 Q = (a*a - b*3) / 9; 748 R = (2*a*a*a - 9*a*b + 27*c) / 54; 749 750 SkScalar Q3 = Q * Q * Q; 751 SkScalar R2MinusQ3 = R * R - Q3; 752 SkScalar adiv3 = a / 3; 753 754 SkScalar* roots = tValues; 755 SkScalar r; 756 757 if (R2MinusQ3 < 0) { // we have 3 real roots 758 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 759 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 760 761 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 762 if (is_unit_interval(r)) { 763 *roots++ = r; 764 } 765 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 766 if (is_unit_interval(r)) { 767 *roots++ = r; 768 } 769 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 770 if (is_unit_interval(r)) { 771 *roots++ = r; 772 } 773 SkDEBUGCODE(test_collaps_duplicates();) 774 775 // now sort the roots 776 int count = (int)(roots - tValues); 777 SkASSERT((unsigned)count <= 3); 778 bubble_sort(tValues, count); 779 count = collaps_duplicates(tValues, count); 780 roots = tValues + count; // so we compute the proper count below 781 } else { // we have 1 real root 782 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 783 A = SkScalarCubeRoot(A); 784 if (R > 0) { 785 A = -A; 786 } 787 if (A != 0) { 788 A += Q / A; 789 } 790 r = A - adiv3; 791 if (is_unit_interval(r)) { 792 *roots++ = r; 793 } 794 } 795 796 return (int)(roots - tValues); 797} 798 799/* Looking for F' dot F'' == 0 800 801 A = b - a 802 B = c - 2b + a 803 C = d - 3c + 3b - a 804 805 F' = 3Ct^2 + 6Bt + 3A 806 F'' = 6Ct + 6B 807 808 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 809*/ 810static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 811 SkScalar a = src[2] - src[0]; 812 SkScalar b = src[4] - 2 * src[2] + src[0]; 813 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 814 815 coeff[0] = c * c; 816 coeff[1] = 3 * b * c; 817 coeff[2] = 2 * b * b + c * a; 818 coeff[3] = a * b; 819} 820 821/* Looking for F' dot F'' == 0 822 823 A = b - a 824 B = c - 2b + a 825 C = d - 3c + 3b - a 826 827 F' = 3Ct^2 + 6Bt + 3A 828 F'' = 6Ct + 6B 829 830 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 831*/ 832int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 833 SkScalar coeffX[4], coeffY[4]; 834 int i; 835 836 formulate_F1DotF2(&src[0].fX, coeffX); 837 formulate_F1DotF2(&src[0].fY, coeffY); 838 839 for (i = 0; i < 4; i++) { 840 coeffX[i] += coeffY[i]; 841 } 842 843 SkScalar t[3]; 844 int count = solve_cubic_poly(coeffX, t); 845 int maxCount = 0; 846 847 // now remove extrema where the curvature is zero (mins) 848 // !!!! need a test for this !!!! 849 for (i = 0; i < count; i++) { 850 // if (not_min_curvature()) 851 if (t[i] > 0 && t[i] < SK_Scalar1) { 852 tValues[maxCount++] = t[i]; 853 } 854 } 855 return maxCount; 856} 857 858int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 859 SkScalar tValues[3]) { 860 SkScalar t_storage[3]; 861 862 if (tValues == NULL) { 863 tValues = t_storage; 864 } 865 866 int count = SkFindCubicMaxCurvature(src, tValues); 867 868 if (dst) { 869 if (count == 0) { 870 memcpy(dst, src, 4 * sizeof(SkPoint)); 871 } else { 872 SkChopCubicAt(src, dst, tValues, count); 873 } 874 } 875 return count + 1; 876} 877 878bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], 879 bool* ambiguous) { 880 if (ambiguous) { 881 *ambiguous = false; 882 } 883 884 // Find the minimum and maximum y of the extrema, which are the 885 // first and last points since this cubic is monotonic 886 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 887 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 888 889 if (pt.fY == cubic[0].fY 890 || pt.fY < min_y 891 || pt.fY > max_y) { 892 // The query line definitely does not cross the curve 893 if (ambiguous) { 894 *ambiguous = (pt.fY == cubic[0].fY); 895 } 896 return false; 897 } 898 899 bool pt_at_extremum = (pt.fY == cubic[3].fY); 900 901 SkScalar min_x = 902 SkMinScalar( 903 SkMinScalar( 904 SkMinScalar(cubic[0].fX, cubic[1].fX), 905 cubic[2].fX), 906 cubic[3].fX); 907 if (pt.fX < min_x) { 908 // The query line definitely crosses the curve 909 if (ambiguous) { 910 *ambiguous = pt_at_extremum; 911 } 912 return true; 913 } 914 915 SkScalar max_x = 916 SkMaxScalar( 917 SkMaxScalar( 918 SkMaxScalar(cubic[0].fX, cubic[1].fX), 919 cubic[2].fX), 920 cubic[3].fX); 921 if (pt.fX > max_x) { 922 // The query line definitely does not cross the curve 923 return false; 924 } 925 926 // Do a binary search to find the parameter value which makes y as 927 // close as possible to the query point. See whether the query 928 // line's origin is to the left of the associated x coordinate. 929 930 // kMaxIter is chosen as the number of mantissa bits for a float, 931 // since there's no way we are going to get more precision by 932 // iterating more times than that. 933 const int kMaxIter = 23; 934 SkPoint eval; 935 int iter = 0; 936 SkScalar upper_t; 937 SkScalar lower_t; 938 // Need to invert direction of t parameter if cubic goes up 939 // instead of down 940 if (cubic[3].fY > cubic[0].fY) { 941 upper_t = SK_Scalar1; 942 lower_t = 0; 943 } else { 944 upper_t = 0; 945 lower_t = SK_Scalar1; 946 } 947 do { 948 SkScalar t = SkScalarAve(upper_t, lower_t); 949 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 950 if (pt.fY > eval.fY) { 951 lower_t = t; 952 } else { 953 upper_t = t; 954 } 955 } while (++iter < kMaxIter 956 && !SkScalarNearlyZero(eval.fY - pt.fY)); 957 if (pt.fX <= eval.fX) { 958 if (ambiguous) { 959 *ambiguous = pt_at_extremum; 960 } 961 return true; 962 } 963 return false; 964} 965 966int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 967 int num_crossings = 0; 968 SkPoint monotonic_cubics[10]; 969 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 970 if (ambiguous) { 971 *ambiguous = false; 972 } 973 bool locally_ambiguous; 974 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous)) 975 ++num_crossings; 976 if (ambiguous) { 977 *ambiguous |= locally_ambiguous; 978 } 979 if (num_monotonic_cubics > 0) 980 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous)) 981 ++num_crossings; 982 if (ambiguous) { 983 *ambiguous |= locally_ambiguous; 984 } 985 if (num_monotonic_cubics > 1) 986 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous)) 987 ++num_crossings; 988 if (ambiguous) { 989 *ambiguous |= locally_ambiguous; 990 } 991 return num_crossings; 992} 993 994/////////////////////////////////////////////////////////////////////////////// 995 996/* Find t value for quadratic [a, b, c] = d. 997 Return 0 if there is no solution within [0, 1) 998*/ 999static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { 1000 // At^2 + Bt + C = d 1001 SkScalar A = a - 2 * b + c; 1002 SkScalar B = 2 * (b - a); 1003 SkScalar C = a - d; 1004 1005 SkScalar roots[2]; 1006 int count = SkFindUnitQuadRoots(A, B, C, roots); 1007 1008 SkASSERT(count <= 1); 1009 return count == 1 ? roots[0] : 0; 1010} 1011 1012/* given a quad-curve and a point (x,y), chop the quad at that point and place 1013 the new off-curve point and endpoint into 'dest'. 1014 Should only return false if the computed pos is the start of the curve 1015 (i.e. root == 0) 1016*/ 1017static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, 1018 SkPoint* dest) { 1019 const SkScalar* base; 1020 SkScalar value; 1021 1022 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1023 base = &quad[0].fX; 1024 value = x; 1025 } else { 1026 base = &quad[0].fY; 1027 value = y; 1028 } 1029 1030 // note: this returns 0 if it thinks value is out of range, meaning the 1031 // root might return something outside of [0, 1) 1032 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1033 1034 if (t > 0) { 1035 SkPoint tmp[5]; 1036 SkChopQuadAt(quad, tmp, t); 1037 dest[0] = tmp[1]; 1038 dest[1].set(x, y); 1039 return true; 1040 } else { 1041 /* t == 0 means either the value triggered a root outside of [0, 1) 1042 For our purposes, we can ignore the <= 0 roots, but we want to 1043 catch the >= 1 roots (which given our caller, will basically mean 1044 a root of 1, give-or-take numerical instability). If we are in the 1045 >= 1 case, return the existing offCurve point. 1046 1047 The test below checks to see if we are close to the "end" of the 1048 curve (near base[4]). Rather than specifying a tolerance, I just 1049 check to see if value is on to the right/left of the middle point 1050 (depending on the direction/sign of the end points). 1051 */ 1052 if ((base[0] < base[4] && value > base[2]) || 1053 (base[0] > base[4] && value < base[2])) // should root have been 1 1054 { 1055 dest[0] = quad[1]; 1056 dest[1].set(x, y); 1057 return true; 1058 } 1059 } 1060 return false; 1061} 1062 1063static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1064// The mid point of the quadratic arc approximation is half way between the two 1065// control points. The float epsilon adjustment moves the on curve point out by 1066// two bits, distributing the convex test error between the round rect approximation 1067// and the convex cross product sign equality test. 1068#define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1069 { SK_Scalar1, 0 }, 1070 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1071 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1072 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1073 1074 { 0, SK_Scalar1 }, 1075 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1076 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1077 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1078 1079 { -SK_Scalar1, 0 }, 1080 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1081 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1082 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1083 1084 { 0, -SK_Scalar1 }, 1085 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1086 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1087 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1088 1089 { SK_Scalar1, 0 } 1090#undef SK_MID_RRECT_OFFSET 1091}; 1092 1093int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1094 SkRotationDirection dir, const SkMatrix* userMatrix, 1095 SkPoint quadPoints[]) { 1096 // rotate by x,y so that uStart is (1.0) 1097 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1098 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1099 1100 SkScalar absX = SkScalarAbs(x); 1101 SkScalar absY = SkScalarAbs(y); 1102 1103 int pointCount; 1104 1105 // check for (effectively) coincident vectors 1106 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1107 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1108 if (absY <= SK_ScalarNearlyZero && x > 0 && 1109 ((y >= 0 && kCW_SkRotationDirection == dir) || 1110 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1111 1112 // just return the start-point 1113 quadPoints[0].set(SK_Scalar1, 0); 1114 pointCount = 1; 1115 } else { 1116 if (dir == kCCW_SkRotationDirection) { 1117 y = -y; 1118 } 1119 // what octant (quadratic curve) is [xy] in? 1120 int oct = 0; 1121 bool sameSign = true; 1122 1123 if (0 == y) { 1124 oct = 4; // 180 1125 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1126 } else if (0 == x) { 1127 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1128 oct = y > 0 ? 2 : 6; // 90 : 270 1129 } else { 1130 if (y < 0) { 1131 oct += 4; 1132 } 1133 if ((x < 0) != (y < 0)) { 1134 oct += 2; 1135 sameSign = false; 1136 } 1137 if ((absX < absY) == sameSign) { 1138 oct += 1; 1139 } 1140 } 1141 1142 int wholeCount = oct << 1; 1143 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1144 1145 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1146 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { 1147 wholeCount += 2; 1148 } 1149 pointCount = wholeCount + 1; 1150 } 1151 1152 // now handle counter-clockwise and the initial unitStart rotation 1153 SkMatrix matrix; 1154 matrix.setSinCos(uStart.fY, uStart.fX); 1155 if (dir == kCCW_SkRotationDirection) { 1156 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1157 } 1158 if (userMatrix) { 1159 matrix.postConcat(*userMatrix); 1160 } 1161 matrix.mapPoints(quadPoints, pointCount); 1162 return pointCount; 1163} 1164 1165/////////////////////////////////////////////////////////////////////////////// 1166 1167// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1168// ------------------------------------------ 1169// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1170// 1171// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1172// ------------------------------------------------ 1173// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1174// 1175 1176// Take the parametric specification for the conic (either X or Y) and return 1177// in coeff[] the coefficients for the simple quadratic polynomial 1178// coeff[0] for t^2 1179// coeff[1] for t 1180// coeff[2] for constant term 1181// 1182static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1183 SkASSERT(src); 1184 SkASSERT(t >= 0 && t <= SK_Scalar1); 1185 1186 SkScalar src2w = SkScalarMul(src[2], w); 1187 SkScalar C = src[0]; 1188 SkScalar A = src[4] - 2 * src2w + C; 1189 SkScalar B = 2 * (src2w - C); 1190 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1191 1192 B = 2 * (w - SK_Scalar1); 1193 C = SK_Scalar1; 1194 A = -B; 1195 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1196 1197 return SkScalarDiv(numer, denom); 1198} 1199 1200// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1201// 1202// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1203// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1204// t^0 : -2 P0 w + 2 P1 w 1205// 1206// We disregard magnitude, so we can freely ignore the denominator of F', and 1207// divide the numerator by 2 1208// 1209// coeff[0] for t^2 1210// coeff[1] for t^1 1211// coeff[2] for t^0 1212// 1213static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1214 const SkScalar P20 = src[4] - src[0]; 1215 const SkScalar P10 = src[2] - src[0]; 1216 const SkScalar wP10 = w * P10; 1217 coeff[0] = w * P20 - P20; 1218 coeff[1] = P20 - 2 * wP10; 1219 coeff[2] = wP10; 1220} 1221 1222static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1223 SkScalar coeff[3]; 1224 conic_deriv_coeff(coord, w, coeff); 1225 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1226} 1227 1228static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1229 SkScalar coeff[3]; 1230 conic_deriv_coeff(src, w, coeff); 1231 1232 SkScalar tValues[2]; 1233 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1234 SkASSERT(0 == roots || 1 == roots); 1235 1236 if (1 == roots) { 1237 *t = tValues[0]; 1238 return true; 1239 } 1240 return false; 1241} 1242 1243struct SkP3D { 1244 SkScalar fX, fY, fZ; 1245 1246 void set(SkScalar x, SkScalar y, SkScalar z) { 1247 fX = x; fY = y; fZ = z; 1248 } 1249 1250 void projectDown(SkPoint* dst) const { 1251 dst->set(fX / fZ, fY / fZ); 1252 } 1253}; 1254 1255// We only interpolate one dimension at a time (the first, at +0, +3, +6). 1256static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 1257 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1258 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1259 dst[0] = ab; 1260 dst[3] = SkScalarInterp(ab, bc, t); 1261 dst[6] = bc; 1262} 1263 1264static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1265 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1266 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1267 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1268} 1269 1270void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1271 SkASSERT(t >= 0 && t <= SK_Scalar1); 1272 1273 if (pt) { 1274 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1275 conic_eval_pos(&fPts[0].fY, fW, t)); 1276 } 1277 if (tangent) { 1278 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1279 conic_eval_tan(&fPts[0].fY, fW, t)); 1280 } 1281} 1282 1283void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1284 SkP3D tmp[3], tmp2[3]; 1285 1286 ratquad_mapTo3D(fPts, fW, tmp); 1287 1288 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1289 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1290 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1291 1292 dst[0].fPts[0] = fPts[0]; 1293 tmp2[0].projectDown(&dst[0].fPts[1]); 1294 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1295 tmp2[2].projectDown(&dst[1].fPts[1]); 1296 dst[1].fPts[2] = fPts[2]; 1297 1298 // to put in "standard form", where w0 and w2 are both 1, we compute the 1299 // new w1 as sqrt(w1*w1/w0*w2) 1300 // or 1301 // w1 /= sqrt(w0*w2) 1302 // 1303 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1304 // 1305 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1306 dst[0].fW = tmp2[0].fZ / root; 1307 dst[1].fW = tmp2[2].fZ / root; 1308} 1309 1310static SkScalar subdivide_w_value(SkScalar w) { 1311 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1312} 1313 1314void SkConic::chop(SkConic dst[2]) const { 1315 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1316 SkScalar p1x = fW * fPts[1].fX; 1317 SkScalar p1y = fW * fPts[1].fY; 1318 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1319 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1320 1321 dst[0].fPts[0] = fPts[0]; 1322 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1323 (fPts[0].fY + p1y) * scale); 1324 dst[0].fPts[2].set(mx, my); 1325 1326 dst[1].fPts[0].set(mx, my); 1327 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1328 (p1y + fPts[2].fY) * scale); 1329 dst[1].fPts[2] = fPts[2]; 1330 1331 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1332} 1333 1334/* 1335 * "High order approximation of conic sections by quadratic splines" 1336 * by Michael Floater, 1993 1337 */ 1338#define AS_QUAD_ERROR_SETUP \ 1339 SkScalar a = fW - 1; \ 1340 SkScalar k = a / (4 * (2 + a)); \ 1341 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1342 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1343 1344void SkConic::computeAsQuadError(SkVector* err) const { 1345 AS_QUAD_ERROR_SETUP 1346 err->set(x, y); 1347} 1348 1349bool SkConic::asQuadTol(SkScalar tol) const { 1350 AS_QUAD_ERROR_SETUP 1351 return (x * x + y * y) <= tol * tol; 1352} 1353 1354int SkConic::computeQuadPOW2(SkScalar tol) const { 1355 AS_QUAD_ERROR_SETUP 1356 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1357 1358 if (error <= 0) { 1359 return 0; 1360 } 1361 uint32_t ierr = (uint32_t)error; 1362 return (34 - SkCLZ(ierr)) >> 1; 1363} 1364 1365static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1366 SkASSERT(level >= 0); 1367 1368 if (0 == level) { 1369 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1370 return pts + 2; 1371 } else { 1372 SkConic dst[2]; 1373 src.chop(dst); 1374 --level; 1375 pts = subdivide(dst[0], pts, level); 1376 return subdivide(dst[1], pts, level); 1377 } 1378} 1379 1380int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1381 SkASSERT(pow2 >= 0); 1382 *pts = fPts[0]; 1383 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1384 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1385 return 1 << pow2; 1386} 1387 1388bool SkConic::findXExtrema(SkScalar* t) const { 1389 return conic_find_extrema(&fPts[0].fX, fW, t); 1390} 1391 1392bool SkConic::findYExtrema(SkScalar* t) const { 1393 return conic_find_extrema(&fPts[0].fY, fW, t); 1394} 1395 1396bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1397 SkScalar t; 1398 if (this->findXExtrema(&t)) { 1399 this->chopAt(t, dst); 1400 // now clean-up the middle, since we know t was meant to be at 1401 // an X-extrema 1402 SkScalar value = dst[0].fPts[2].fX; 1403 dst[0].fPts[1].fX = value; 1404 dst[1].fPts[0].fX = value; 1405 dst[1].fPts[1].fX = value; 1406 return true; 1407 } 1408 return false; 1409} 1410 1411bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1412 SkScalar t; 1413 if (this->findYExtrema(&t)) { 1414 this->chopAt(t, dst); 1415 // now clean-up the middle, since we know t was meant to be at 1416 // an Y-extrema 1417 SkScalar value = dst[0].fPts[2].fY; 1418 dst[0].fPts[1].fY = value; 1419 dst[1].fPts[0].fY = value; 1420 dst[1].fPts[1].fY = value; 1421 return true; 1422 } 1423 return false; 1424} 1425 1426void SkConic::computeTightBounds(SkRect* bounds) const { 1427 SkPoint pts[4]; 1428 pts[0] = fPts[0]; 1429 pts[1] = fPts[2]; 1430 int count = 2; 1431 1432 SkScalar t; 1433 if (this->findXExtrema(&t)) { 1434 this->evalAt(t, &pts[count++]); 1435 } 1436 if (this->findYExtrema(&t)) { 1437 this->evalAt(t, &pts[count++]); 1438 } 1439 bounds->set(pts, count); 1440} 1441 1442void SkConic::computeFastBounds(SkRect* bounds) const { 1443 bounds->set(fPts, 3); 1444} 1445