SkGeometry.cpp revision 8f4d2306fa866a26f9448048ff63f692b2ba43aa
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// #define DIRECT_EVAL_OF_POLYNOMIALS 72 73//////////////////////////////////////////////////////////////////////// 74 75static int is_not_monotonic(float a, float b, float c) { 76 float ab = a - b; 77 float bc = b - c; 78 if (ab < 0) { 79 bc = -bc; 80 } 81 return ab == 0 || bc < 0; 82} 83 84//////////////////////////////////////////////////////////////////////// 85 86static bool is_unit_interval(SkScalar x) 87{ 88 return x > 0 && x < SK_Scalar1; 89} 90 91static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) 92{ 93 SkASSERT(ratio); 94 95 if (numer < 0) 96 { 97 numer = -numer; 98 denom = -denom; 99 } 100 101 if (denom == 0 || numer == 0 || numer >= denom) 102 return 0; 103 104 SkScalar r = SkScalarDiv(numer, denom); 105 if (SkScalarIsNaN(r)) { 106 return 0; 107 } 108 SkASSERT(r >= 0 && r < SK_Scalar1); 109 if (r == 0) // catch underflow if numer <<<< denom 110 return 0; 111 *ratio = r; 112 return 1; 113} 114 115/** From Numerical Recipes in C. 116 117 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 118 x1 = Q / A 119 x2 = C / Q 120*/ 121int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) 122{ 123 SkASSERT(roots); 124 125 if (A == 0) 126 return valid_unit_divide(-C, B, roots); 127 128 SkScalar* r = roots; 129 130 float R = B*B - 4*A*C; 131 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 132 return 0; 133 } 134 R = sk_float_sqrt(R); 135 136 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 137 r += valid_unit_divide(Q, A, r); 138 r += valid_unit_divide(C, Q, r); 139 if (r - roots == 2) 140 { 141 if (roots[0] > roots[1]) 142 SkTSwap<SkScalar>(roots[0], roots[1]); 143 else if (roots[0] == roots[1]) // nearly-equal? 144 r -= 1; // skip the double root 145 } 146 return (int)(r - roots); 147} 148 149/////////////////////////////////////////////////////////////////////////////// 150/////////////////////////////////////////////////////////////////////////////// 151 152static SkScalar eval_quad(const SkScalar src[], SkScalar t) 153{ 154 SkASSERT(src); 155 SkASSERT(t >= 0 && t <= SK_Scalar1); 156 157#ifdef DIRECT_EVAL_OF_POLYNOMIALS 158 SkScalar C = src[0]; 159 SkScalar A = src[4] - 2 * src[2] + C; 160 SkScalar B = 2 * (src[2] - C); 161 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 162#else 163 SkScalar ab = SkScalarInterp(src[0], src[2], t); 164 SkScalar bc = SkScalarInterp(src[2], src[4], t); 165 return SkScalarInterp(ab, bc, t); 166#endif 167} 168 169static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) 170{ 171 SkScalar A = src[4] - 2 * src[2] + src[0]; 172 SkScalar B = src[2] - src[0]; 173 174 return 2 * SkScalarMulAdd(A, t, B); 175} 176 177static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) 178{ 179 SkScalar A = src[4] - 2 * src[2] + src[0]; 180 SkScalar B = src[2] - src[0]; 181 return A + 2 * B; 182} 183 184void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) 185{ 186 SkASSERT(src); 187 SkASSERT(t >= 0 && t <= SK_Scalar1); 188 189 if (pt) 190 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 191 if (tangent) 192 tangent->set(eval_quad_derivative(&src[0].fX, t), 193 eval_quad_derivative(&src[0].fY, t)); 194} 195 196void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) 197{ 198 SkASSERT(src); 199 200 if (pt) 201 { 202 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 203 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 204 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 205 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 206 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 207 } 208 if (tangent) 209 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 210 eval_quad_derivative_at_half(&src[0].fY)); 211} 212 213static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 214{ 215 SkScalar ab = SkScalarInterp(src[0], src[2], t); 216 SkScalar bc = SkScalarInterp(src[2], src[4], t); 217 218 dst[0] = src[0]; 219 dst[2] = ab; 220 dst[4] = SkScalarInterp(ab, bc, t); 221 dst[6] = bc; 222 dst[8] = src[4]; 223} 224 225void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) 226{ 227 SkASSERT(t > 0 && t < SK_Scalar1); 228 229 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 230 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 231} 232 233void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) 234{ 235 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 236 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 237 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 238 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 239 240 dst[0] = src[0]; 241 dst[1].set(x01, y01); 242 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 243 dst[3].set(x12, y12); 244 dst[4] = src[2]; 245} 246 247/** Quad'(t) = At + B, where 248 A = 2(a - 2b + c) 249 B = 2(b - a) 250 Solve for t, only if it fits between 0 < t < 1 251*/ 252int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) 253{ 254 /* At + B == 0 255 t = -B / A 256 */ 257 return valid_unit_divide(a - b, a - b - b + c, tValue); 258} 259 260static inline void flatten_double_quad_extrema(SkScalar coords[14]) 261{ 262 coords[2] = coords[6] = coords[4]; 263} 264 265/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 266 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 267 */ 268int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) 269{ 270 SkASSERT(src); 271 SkASSERT(dst); 272 273#if 0 274 static bool once = true; 275 if (once) 276 { 277 once = false; 278 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; 279 SkPoint d[6]; 280 281 int n = SkChopQuadAtYExtrema(s, d); 282 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); 283 } 284#endif 285 286 SkScalar a = src[0].fY; 287 SkScalar b = src[1].fY; 288 SkScalar c = src[2].fY; 289 290 if (is_not_monotonic(a, b, c)) 291 { 292 SkScalar tValue; 293 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) 294 { 295 SkChopQuadAt(src, dst, tValue); 296 flatten_double_quad_extrema(&dst[0].fY); 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(src[0].fX, a); 304 dst[1].set(src[1].fX, b); 305 dst[2].set(src[2].fX, c); 306 return 0; 307} 308 309/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 310 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 311 */ 312int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) 313{ 314 SkASSERT(src); 315 SkASSERT(dst); 316 317 SkScalar a = src[0].fX; 318 SkScalar b = src[1].fX; 319 SkScalar c = src[2].fX; 320 321 if (is_not_monotonic(a, b, c)) { 322 SkScalar tValue; 323 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 324 SkChopQuadAt(src, dst, tValue); 325 flatten_double_quad_extrema(&dst[0].fX); 326 return 1; 327 } 328 // if we get here, we need to force dst to be monotonic, even though 329 // we couldn't compute a unit_divide value (probably underflow). 330 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 331 } 332 dst[0].set(a, src[0].fY); 333 dst[1].set(b, src[1].fY); 334 dst[2].set(c, src[2].fY); 335 return 0; 336} 337 338// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 339// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 340// F''(t) = 2 (a - 2b + c) 341// 342// A = 2 (b - a) 343// B = 2 (a - 2b + c) 344// 345// Maximum curvature for a quadratic means solving 346// Fx' Fx'' + Fy' Fy'' = 0 347// 348// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 349// 350float SkFindQuadMaxCurvature(const SkPoint src[3]) { 351 SkScalar Ax = src[1].fX - src[0].fX; 352 SkScalar Ay = src[1].fY - src[0].fY; 353 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 354 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 355 SkScalar t = 0; // 0 means don't chop 356 357 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 358 return t; 359} 360 361int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) 362{ 363 SkScalar t = SkFindQuadMaxCurvature(src); 364 if (t == 0) { 365 memcpy(dst, src, 3 * sizeof(SkPoint)); 366 return 1; 367 } else { 368 SkChopQuadAt(src, dst, t); 369 return 2; 370 } 371} 372 373#define SK_ScalarTwoThirds (0.666666666f) 374 375void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 376 const SkScalar scale = SK_ScalarTwoThirds; 377 dst[0] = src[0]; 378 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 379 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 380 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 381 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 382 dst[3] = src[2]; 383} 384 385//////////////////////////////////////////////////////////////////////////////////////// 386///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 387//////////////////////////////////////////////////////////////////////////////////////// 388 389static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) 390{ 391 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 392 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 393 coeff[2] = 3*(pt[2] - pt[0]); 394 coeff[3] = pt[0]; 395} 396 397void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) 398{ 399 SkASSERT(pts); 400 401 if (cx) 402 get_cubic_coeff(&pts[0].fX, cx); 403 if (cy) 404 get_cubic_coeff(&pts[0].fY, cy); 405} 406 407static SkScalar eval_cubic(const SkScalar src[], SkScalar t) 408{ 409 SkASSERT(src); 410 SkASSERT(t >= 0 && t <= SK_Scalar1); 411 412 if (t == 0) 413 return src[0]; 414 415#ifdef DIRECT_EVAL_OF_POLYNOMIALS 416 SkScalar D = src[0]; 417 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 418 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 419 SkScalar C = 3*(src[2] - D); 420 421 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 422#else 423 SkScalar ab = SkScalarInterp(src[0], src[2], t); 424 SkScalar bc = SkScalarInterp(src[2], src[4], t); 425 SkScalar cd = SkScalarInterp(src[4], src[6], t); 426 SkScalar abc = SkScalarInterp(ab, bc, t); 427 SkScalar bcd = SkScalarInterp(bc, cd, t); 428 return SkScalarInterp(abc, bcd, t); 429#endif 430} 431 432/** return At^2 + Bt + C 433*/ 434static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) 435{ 436 SkASSERT(t >= 0 && t <= SK_Scalar1); 437 438 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 439} 440 441static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) 442{ 443 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 444 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 445 SkScalar C = src[2] - src[0]; 446 447 return eval_quadratic(A, B, C, t); 448} 449 450static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) 451{ 452 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 453 SkScalar B = src[4] - 2 * src[2] + src[0]; 454 455 return SkScalarMulAdd(A, t, B); 456} 457 458void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature) 459{ 460 SkASSERT(src); 461 SkASSERT(t >= 0 && t <= SK_Scalar1); 462 463 if (loc) 464 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 465 if (tangent) 466 tangent->set(eval_cubic_derivative(&src[0].fX, t), 467 eval_cubic_derivative(&src[0].fY, t)); 468 if (curvature) 469 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 470 eval_cubic_2ndDerivative(&src[0].fY, t)); 471} 472 473/** Cubic'(t) = At^2 + Bt + C, where 474 A = 3(-a + 3(b - c) + d) 475 B = 6(a - 2b + c) 476 C = 3(b - a) 477 Solve for t, keeping only those that fit betwee 0 < t < 1 478*/ 479int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2]) 480{ 481 // we divide A,B,C by 3 to simplify 482 SkScalar A = d - a + 3*(b - c); 483 SkScalar B = 2*(a - b - b + c); 484 SkScalar C = b - a; 485 486 return SkFindUnitQuadRoots(A, B, C, tValues); 487} 488 489static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 490{ 491 SkScalar ab = SkScalarInterp(src[0], src[2], t); 492 SkScalar bc = SkScalarInterp(src[2], src[4], t); 493 SkScalar cd = SkScalarInterp(src[4], src[6], t); 494 SkScalar abc = SkScalarInterp(ab, bc, t); 495 SkScalar bcd = SkScalarInterp(bc, cd, t); 496 SkScalar abcd = SkScalarInterp(abc, bcd, t); 497 498 dst[0] = src[0]; 499 dst[2] = ab; 500 dst[4] = abc; 501 dst[6] = abcd; 502 dst[8] = bcd; 503 dst[10] = cd; 504 dst[12] = src[6]; 505} 506 507void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) 508{ 509 SkASSERT(t > 0 && t < SK_Scalar1); 510 511 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 512 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 513} 514 515/* http://code.google.com/p/skia/issues/detail?id=32 516 517 This test code would fail when we didn't check the return result of 518 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 519 that after the first chop, the parameters to valid_unit_divide are equal 520 (thanks to finite float precision and rounding in the subtracts). Thus 521 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 522 up with 1.0, hence the need to check and just return the last cubic as 523 a degenerate clump of 4 points in the sampe place. 524 525 static void test_cubic() { 526 SkPoint src[4] = { 527 { 556.25000, 523.03003 }, 528 { 556.23999, 522.96002 }, 529 { 556.21997, 522.89001 }, 530 { 556.21997, 522.82001 } 531 }; 532 SkPoint dst[10]; 533 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 534 SkChopCubicAt(src, dst, tval, 2); 535 } 536 */ 537 538void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots) 539{ 540#ifdef SK_DEBUG 541 { 542 for (int i = 0; i < roots - 1; i++) 543 { 544 SkASSERT(is_unit_interval(tValues[i])); 545 SkASSERT(is_unit_interval(tValues[i+1])); 546 SkASSERT(tValues[i] < tValues[i+1]); 547 } 548 } 549#endif 550 551 if (dst) 552 { 553 if (roots == 0) // nothing to chop 554 memcpy(dst, src, 4*sizeof(SkPoint)); 555 else 556 { 557 SkScalar t = tValues[0]; 558 SkPoint tmp[4]; 559 560 for (int i = 0; i < roots; i++) 561 { 562 SkChopCubicAt(src, dst, t); 563 if (i == roots - 1) 564 break; 565 566 dst += 3; 567 // have src point to the remaining cubic (after the chop) 568 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 569 src = tmp; 570 571 // watch out in case the renormalized t isn't in range 572 if (!valid_unit_divide(tValues[i+1] - tValues[i], 573 SK_Scalar1 - tValues[i], &t)) { 574 // if we can't, just create a degenerate cubic 575 dst[4] = dst[5] = dst[6] = src[3]; 576 break; 577 } 578 } 579 } 580 } 581} 582 583void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) 584{ 585 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 586 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 587 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 588 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 589 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 590 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 591 592 SkScalar x012 = SkScalarAve(x01, x12); 593 SkScalar y012 = SkScalarAve(y01, y12); 594 SkScalar x123 = SkScalarAve(x12, x23); 595 SkScalar y123 = SkScalarAve(y12, y23); 596 597 dst[0] = src[0]; 598 dst[1].set(x01, y01); 599 dst[2].set(x012, y012); 600 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 601 dst[4].set(x123, y123); 602 dst[5].set(x23, y23); 603 dst[6] = src[3]; 604} 605 606static void flatten_double_cubic_extrema(SkScalar coords[14]) 607{ 608 coords[4] = coords[8] = coords[6]; 609} 610 611/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 612 the resulting beziers are monotonic in Y. This is called by the scan converter. 613 Depending on what is returned, dst[] is treated as follows 614 0 dst[0..3] is the original cubic 615 1 dst[0..3] and dst[3..6] are the two new cubics 616 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 617 If dst == null, it is ignored and only the count is returned. 618*/ 619int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 620 SkScalar tValues[2]; 621 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 622 src[3].fY, tValues); 623 624 SkChopCubicAt(src, dst, tValues, roots); 625 if (dst && roots > 0) { 626 // we do some cleanup to ensure our Y extrema are flat 627 flatten_double_cubic_extrema(&dst[0].fY); 628 if (roots == 2) { 629 flatten_double_cubic_extrema(&dst[3].fY); 630 } 631 } 632 return roots; 633} 634 635int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 636 SkScalar tValues[2]; 637 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 638 src[3].fX, tValues); 639 640 SkChopCubicAt(src, dst, tValues, roots); 641 if (dst && roots > 0) { 642 // we do some cleanup to ensure our Y extrema are flat 643 flatten_double_cubic_extrema(&dst[0].fX); 644 if (roots == 2) { 645 flatten_double_cubic_extrema(&dst[3].fX); 646 } 647 } 648 return roots; 649} 650 651/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 652 653 Inflection means that curvature is zero. 654 Curvature is [F' x F''] / [F'^3] 655 So we solve F'x X F''y - F'y X F''y == 0 656 After some canceling of the cubic term, we get 657 A = b - a 658 B = c - 2b + a 659 C = d - 3c + 3b - a 660 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 661*/ 662int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) 663{ 664 SkScalar Ax = src[1].fX - src[0].fX; 665 SkScalar Ay = src[1].fY - src[0].fY; 666 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 667 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 668 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 669 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 670 671 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 672} 673 674int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) 675{ 676 SkScalar tValues[2]; 677 int count = SkFindCubicInflections(src, tValues); 678 679 if (dst) 680 { 681 if (count == 0) 682 memcpy(dst, src, 4 * sizeof(SkPoint)); 683 else 684 SkChopCubicAt(src, dst, tValues, count); 685 } 686 return count + 1; 687} 688 689template <typename T> void bubble_sort(T array[], int count) 690{ 691 for (int i = count - 1; i > 0; --i) 692 for (int j = i; j > 0; --j) 693 if (array[j] < array[j-1]) 694 { 695 T tmp(array[j]); 696 array[j] = array[j-1]; 697 array[j-1] = tmp; 698 } 699} 700 701// newton refinement 702#if 0 703static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) 704{ 705 // x1 = x0 - f(t) / f'(t) 706 707 SkFP T = SkScalarToFloat(root); 708 SkFP N, D; 709 710 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] 711 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); 712 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); 713 D = SkFPAdd(D, coeff[2]); 714 715 if (D == 0) 716 return root; 717 718 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] 719 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); 720 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); 721 N = SkFPAdd(N, SkFPMul(T, coeff[2])); 722 N = SkFPAdd(N, coeff[3]); 723 724 if (N) 725 { 726 SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); 727 728 if (delta) 729 root -= delta; 730 } 731 return root; 732} 733#endif 734 735/** 736 * Given an array and count, remove all pair-wise duplicates from the array, 737 * keeping the existing sorting, and return the new count 738 */ 739static int collaps_duplicates(float array[], int count) { 740 for (int n = count; n > 1; --n) { 741 if (array[0] == array[1]) { 742 for (int i = 1; i < n; ++i) { 743 array[i - 1] = array[i]; 744 } 745 count -= 1; 746 } else { 747 array += 1; 748 } 749 } 750 return count; 751} 752 753#ifdef SK_DEBUG 754 755#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 756 757static void test_collaps_duplicates() { 758 static bool gOnce; 759 if (gOnce) { return; } 760 gOnce = true; 761 const float src0[] = { 0 }; 762 const float src1[] = { 0, 0 }; 763 const float src2[] = { 0, 1 }; 764 const float src3[] = { 0, 0, 0 }; 765 const float src4[] = { 0, 0, 1 }; 766 const float src5[] = { 0, 1, 1 }; 767 const float src6[] = { 0, 1, 2 }; 768 const struct { 769 const float* fData; 770 int fCount; 771 int fCollapsedCount; 772 } data[] = { 773 { TEST_COLLAPS_ENTRY(src0), 1 }, 774 { TEST_COLLAPS_ENTRY(src1), 1 }, 775 { TEST_COLLAPS_ENTRY(src2), 2 }, 776 { TEST_COLLAPS_ENTRY(src3), 1 }, 777 { TEST_COLLAPS_ENTRY(src4), 2 }, 778 { TEST_COLLAPS_ENTRY(src5), 2 }, 779 { TEST_COLLAPS_ENTRY(src6), 3 }, 780 }; 781 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 782 float dst[3]; 783 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 784 int count = collaps_duplicates(dst, data[i].fCount); 785 SkASSERT(data[i].fCollapsedCount == count); 786 for (int j = 1; j < count; ++j) { 787 SkASSERT(dst[j-1] < dst[j]); 788 } 789 } 790} 791#endif 792 793static SkScalar SkScalarCubeRoot(SkScalar x) { 794 return sk_float_pow(x, 0.3333333f); 795} 796 797/* Solve coeff(t) == 0, returning the number of roots that 798 lie withing 0 < t < 1. 799 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 800 801 Eliminates repeated roots (so that all tValues are distinct, and are always 802 in increasing order. 803*/ 804static int solve_cubic_polynomial(const SkScalar coeff[4], SkScalar tValues[3]) 805{ 806 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic 807 { 808 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 809 } 810 811 SkScalar a, b, c, Q, R; 812 813 { 814 SkASSERT(coeff[0] != 0); 815 816 SkScalar inva = SkScalarInvert(coeff[0]); 817 a = coeff[1] * inva; 818 b = coeff[2] * inva; 819 c = coeff[3] * inva; 820 } 821 Q = (a*a - b*3) / 9; 822 R = (2*a*a*a - 9*a*b + 27*c) / 54; 823 824 SkScalar Q3 = Q * Q * Q; 825 SkScalar R2MinusQ3 = R * R - Q3; 826 SkScalar adiv3 = a / 3; 827 828 SkScalar* roots = tValues; 829 SkScalar r; 830 831 if (R2MinusQ3 < 0) // we have 3 real roots 832 { 833 float theta = sk_float_acos(R / sk_float_sqrt(Q3)); 834 float neg2RootQ = -2 * sk_float_sqrt(Q); 835 836 r = neg2RootQ * sk_float_cos(theta/3) - adiv3; 837 if (is_unit_interval(r)) 838 *roots++ = r; 839 840 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; 841 if (is_unit_interval(r)) 842 *roots++ = r; 843 844 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; 845 if (is_unit_interval(r)) 846 *roots++ = r; 847 848 SkDEBUGCODE(test_collaps_duplicates();) 849 850 // now sort the roots 851 int count = (int)(roots - tValues); 852 SkASSERT((unsigned)count <= 3); 853 bubble_sort(tValues, count); 854 count = collaps_duplicates(tValues, count); 855 roots = tValues + count; // so we compute the proper count below 856 } 857 else // we have 1 real root 858 { 859 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 860 A = SkScalarCubeRoot(A); 861 if (R > 0) 862 A = -A; 863 864 if (A != 0) 865 A += Q / A; 866 r = A - adiv3; 867 if (is_unit_interval(r)) 868 *roots++ = r; 869 } 870 871 return (int)(roots - tValues); 872} 873 874/* Looking for F' dot F'' == 0 875 876 A = b - a 877 B = c - 2b + a 878 C = d - 3c + 3b - a 879 880 F' = 3Ct^2 + 6Bt + 3A 881 F'' = 6Ct + 6B 882 883 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 884*/ 885static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) 886{ 887 SkScalar a = src[2] - src[0]; 888 SkScalar b = src[4] - 2 * src[2] + src[0]; 889 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 890 891 coeff[0] = c * c; 892 coeff[1] = 3 * b * c; 893 coeff[2] = 2 * b * b + c * a; 894 coeff[3] = a * b; 895} 896 897// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 898//#define kMinTValueForChopping (SK_Scalar1 / 256) 899#define kMinTValueForChopping 0 900 901/* Looking for F' dot F'' == 0 902 903 A = b - a 904 B = c - 2b + a 905 C = d - 3c + 3b - a 906 907 F' = 3Ct^2 + 6Bt + 3A 908 F'' = 6Ct + 6B 909 910 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 911*/ 912int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) 913{ 914 SkScalar coeffX[4], coeffY[4]; 915 int i; 916 917 formulate_F1DotF2(&src[0].fX, coeffX); 918 formulate_F1DotF2(&src[0].fY, coeffY); 919 920 for (i = 0; i < 4; i++) 921 coeffX[i] += coeffY[i]; 922 923 SkScalar t[3]; 924 int count = solve_cubic_polynomial(coeffX, t); 925 int maxCount = 0; 926 927 // now remove extrema where the curvature is zero (mins) 928 // !!!! need a test for this !!!! 929 for (i = 0; i < count; i++) 930 { 931 // if (not_min_curvature()) 932 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) 933 tValues[maxCount++] = t[i]; 934 } 935 return maxCount; 936} 937 938int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3]) 939{ 940 SkScalar t_storage[3]; 941 942 if (tValues == NULL) 943 tValues = t_storage; 944 945 int count = SkFindCubicMaxCurvature(src, tValues); 946 947 if (dst) { 948 if (count == 0) 949 memcpy(dst, src, 4 * sizeof(SkPoint)); 950 else 951 SkChopCubicAt(src, dst, tValues, count); 952 } 953 return count + 1; 954} 955 956bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 957 if (ambiguous) { 958 *ambiguous = false; 959 } 960 961 // Find the minimum and maximum y of the extrema, which are the 962 // first and last points since this cubic is monotonic 963 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 964 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 965 966 if (pt.fY == cubic[0].fY 967 || pt.fY < min_y 968 || pt.fY > max_y) { 969 // The query line definitely does not cross the curve 970 if (ambiguous) { 971 *ambiguous = (pt.fY == cubic[0].fY); 972 } 973 return false; 974 } 975 976 bool pt_at_extremum = (pt.fY == cubic[3].fY); 977 978 SkScalar min_x = 979 SkMinScalar( 980 SkMinScalar( 981 SkMinScalar(cubic[0].fX, cubic[1].fX), 982 cubic[2].fX), 983 cubic[3].fX); 984 if (pt.fX < min_x) { 985 // The query line definitely crosses the curve 986 if (ambiguous) { 987 *ambiguous = pt_at_extremum; 988 } 989 return true; 990 } 991 992 SkScalar max_x = 993 SkMaxScalar( 994 SkMaxScalar( 995 SkMaxScalar(cubic[0].fX, cubic[1].fX), 996 cubic[2].fX), 997 cubic[3].fX); 998 if (pt.fX > max_x) { 999 // The query line definitely does not cross the curve 1000 return false; 1001 } 1002 1003 // Do a binary search to find the parameter value which makes y as 1004 // close as possible to the query point. See whether the query 1005 // line's origin is to the left of the associated x coordinate. 1006 1007 // kMaxIter is chosen as the number of mantissa bits for a float, 1008 // since there's no way we are going to get more precision by 1009 // iterating more times than that. 1010 const int kMaxIter = 23; 1011 SkPoint eval; 1012 int iter = 0; 1013 SkScalar upper_t; 1014 SkScalar lower_t; 1015 // Need to invert direction of t parameter if cubic goes up 1016 // instead of down 1017 if (cubic[3].fY > cubic[0].fY) { 1018 upper_t = SK_Scalar1; 1019 lower_t = 0; 1020 } else { 1021 upper_t = 0; 1022 lower_t = SK_Scalar1; 1023 } 1024 do { 1025 SkScalar t = SkScalarAve(upper_t, lower_t); 1026 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 1027 if (pt.fY > eval.fY) { 1028 lower_t = t; 1029 } else { 1030 upper_t = t; 1031 } 1032 } while (++iter < kMaxIter 1033 && !SkScalarNearlyZero(eval.fY - pt.fY)); 1034 if (pt.fX <= eval.fX) { 1035 if (ambiguous) { 1036 *ambiguous = pt_at_extremum; 1037 } 1038 return true; 1039 } 1040 return false; 1041} 1042 1043int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1044 int num_crossings = 0; 1045 SkPoint monotonic_cubics[10]; 1046 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 1047 if (ambiguous) { 1048 *ambiguous = false; 1049 } 1050 bool locally_ambiguous; 1051 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous)) 1052 ++num_crossings; 1053 if (ambiguous) { 1054 *ambiguous |= locally_ambiguous; 1055 } 1056 if (num_monotonic_cubics > 0) 1057 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous)) 1058 ++num_crossings; 1059 if (ambiguous) { 1060 *ambiguous |= locally_ambiguous; 1061 } 1062 if (num_monotonic_cubics > 1) 1063 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous)) 1064 ++num_crossings; 1065 if (ambiguous) { 1066 *ambiguous |= locally_ambiguous; 1067 } 1068 return num_crossings; 1069} 1070//////////////////////////////////////////////////////////////////////////////// 1071 1072/* Find t value for quadratic [a, b, c] = d. 1073 Return 0 if there is no solution within [0, 1) 1074*/ 1075static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) 1076{ 1077 // At^2 + Bt + C = d 1078 SkScalar A = a - 2 * b + c; 1079 SkScalar B = 2 * (b - a); 1080 SkScalar C = a - d; 1081 1082 SkScalar roots[2]; 1083 int count = SkFindUnitQuadRoots(A, B, C, roots); 1084 1085 SkASSERT(count <= 1); 1086 return count == 1 ? roots[0] : 0; 1087} 1088 1089/* given a quad-curve and a point (x,y), chop the quad at that point and place 1090 the new off-curve point and endpoint into 'dest'. 1091 Should only return false if the computed pos is the start of the curve 1092 (i.e. root == 0) 1093*/ 1094static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest) 1095{ 1096 const SkScalar* base; 1097 SkScalar value; 1098 1099 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1100 base = &quad[0].fX; 1101 value = x; 1102 } else { 1103 base = &quad[0].fY; 1104 value = y; 1105 } 1106 1107 // note: this returns 0 if it thinks value is out of range, meaning the 1108 // root might return something outside of [0, 1) 1109 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1110 1111 if (t > 0) 1112 { 1113 SkPoint tmp[5]; 1114 SkChopQuadAt(quad, tmp, t); 1115 dest[0] = tmp[1]; 1116 dest[1].set(x, y); 1117 return true; 1118 } else { 1119 /* t == 0 means either the value triggered a root outside of [0, 1) 1120 For our purposes, we can ignore the <= 0 roots, but we want to 1121 catch the >= 1 roots (which given our caller, will basically mean 1122 a root of 1, give-or-take numerical instability). If we are in the 1123 >= 1 case, return the existing offCurve point. 1124 1125 The test below checks to see if we are close to the "end" of the 1126 curve (near base[4]). Rather than specifying a tolerance, I just 1127 check to see if value is on to the right/left of the middle point 1128 (depending on the direction/sign of the end points). 1129 */ 1130 if ((base[0] < base[4] && value > base[2]) || 1131 (base[0] > base[4] && value < base[2])) // should root have been 1 1132 { 1133 dest[0] = quad[1]; 1134 dest[1].set(x, y); 1135 return true; 1136 } 1137 } 1138 return false; 1139} 1140 1141static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1142// The mid point of the quadratic arc approximation is half way between the two 1143// control points. The float epsilon adjustment moves the on curve point out by 1144// two bits, distributing the convex test error between the round rect approximation 1145// and the convex cross product sign equality test. 1146#define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1147 { SK_Scalar1, 0 }, 1148 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1149 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1150 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1151 1152 { 0, SK_Scalar1 }, 1153 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1154 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1155 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1156 1157 { -SK_Scalar1, 0 }, 1158 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1159 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1160 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1161 1162 { 0, -SK_Scalar1 }, 1163 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1164 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1165 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1166 1167 { SK_Scalar1, 0 } 1168#undef SK_MID_RRECT_OFFSET 1169}; 1170 1171int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1172 SkRotationDirection dir, const SkMatrix* userMatrix, 1173 SkPoint quadPoints[]) 1174{ 1175 // rotate by x,y so that uStart is (1.0) 1176 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1177 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1178 1179 SkScalar absX = SkScalarAbs(x); 1180 SkScalar absY = SkScalarAbs(y); 1181 1182 int pointCount; 1183 1184 // check for (effectively) coincident vectors 1185 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1186 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1187 if (absY <= SK_ScalarNearlyZero && x > 0 && 1188 ((y >= 0 && kCW_SkRotationDirection == dir) || 1189 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1190 1191 // just return the start-point 1192 quadPoints[0].set(SK_Scalar1, 0); 1193 pointCount = 1; 1194 } else { 1195 if (dir == kCCW_SkRotationDirection) 1196 y = -y; 1197 1198 // what octant (quadratic curve) is [xy] in? 1199 int oct = 0; 1200 bool sameSign = true; 1201 1202 if (0 == y) 1203 { 1204 oct = 4; // 180 1205 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1206 } 1207 else if (0 == x) 1208 { 1209 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1210 if (y > 0) 1211 oct = 2; // 90 1212 else 1213 oct = 6; // 270 1214 } 1215 else 1216 { 1217 if (y < 0) 1218 oct += 4; 1219 if ((x < 0) != (y < 0)) 1220 { 1221 oct += 2; 1222 sameSign = false; 1223 } 1224 if ((absX < absY) == sameSign) 1225 oct += 1; 1226 } 1227 1228 int wholeCount = oct << 1; 1229 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1230 1231 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1232 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) 1233 { 1234 wholeCount += 2; 1235 } 1236 pointCount = wholeCount + 1; 1237 } 1238 1239 // now handle counter-clockwise and the initial unitStart rotation 1240 SkMatrix matrix; 1241 matrix.setSinCos(uStart.fY, uStart.fX); 1242 if (dir == kCCW_SkRotationDirection) { 1243 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1244 } 1245 if (userMatrix) { 1246 matrix.postConcat(*userMatrix); 1247 } 1248 matrix.mapPoints(quadPoints, pointCount); 1249 return pointCount; 1250} 1251 1252/////////////////////////////////////////////////////////////////////////////// 1253 1254// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1255// ------------------------------------------ 1256// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1257// 1258// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1259// ------------------------------------------------ 1260// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1261// 1262 1263// Take the parametric specification for the conic (either X or Y) and return 1264// in coeff[] the coefficients for the simple quadratic polynomial 1265// coeff[0] for t^2 1266// coeff[1] for t 1267// coeff[2] for constant term 1268// 1269static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1270 SkASSERT(src); 1271 SkASSERT(t >= 0 && t <= SK_Scalar1); 1272 1273 SkScalar src2w = SkScalarMul(src[2], w); 1274 SkScalar C = src[0]; 1275 SkScalar A = src[4] - 2 * src2w + C; 1276 SkScalar B = 2 * (src2w - C); 1277 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1278 1279 B = 2 * (w - SK_Scalar1); 1280 C = SK_Scalar1; 1281 A = -B; 1282 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1283 1284 return SkScalarDiv(numer, denom); 1285} 1286 1287// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1288// 1289// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1290// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1291// t^0 : -2 P0 w + 2 P1 w 1292// 1293// We disregard magnitude, so we can freely ignore the denominator of F', and 1294// divide the numerator by 2 1295// 1296// coeff[0] for t^2 1297// coeff[1] for t^1 1298// coeff[2] for t^0 1299// 1300static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1301 const SkScalar P20 = src[4] - src[0]; 1302 const SkScalar P10 = src[2] - src[0]; 1303 const SkScalar wP10 = w * P10; 1304 coeff[0] = w * P20 - P20; 1305 coeff[1] = P20 - 2 * wP10; 1306 coeff[2] = wP10; 1307} 1308 1309static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1310 SkScalar coeff[3]; 1311 conic_deriv_coeff(coord, w, coeff); 1312 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1313} 1314 1315static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1316 SkScalar coeff[3]; 1317 conic_deriv_coeff(src, w, coeff); 1318 1319 SkScalar tValues[2]; 1320 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1321 SkASSERT(0 == roots || 1 == roots); 1322 1323 if (1 == roots) { 1324 *t = tValues[0]; 1325 return true; 1326 } 1327 return false; 1328} 1329 1330struct SkP3D { 1331 SkScalar fX, fY, fZ; 1332 1333 void set(SkScalar x, SkScalar y, SkScalar z) { 1334 fX = x; fY = y; fZ = z; 1335 } 1336 1337 void projectDown(SkPoint* dst) const { 1338 dst->set(fX / fZ, fY / fZ); 1339 } 1340}; 1341 1342// we just return the middle 3 points, since the first and last are dups of src 1343// 1344static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) { 1345 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1346 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1347 dst[0] = ab; 1348 dst[3] = SkScalarInterp(ab, bc, t); 1349 dst[6] = bc; 1350} 1351 1352static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1353 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1354 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1355 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1356} 1357 1358void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1359 SkASSERT(t >= 0 && t <= SK_Scalar1); 1360 1361 if (pt) { 1362 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1363 conic_eval_pos(&fPts[0].fY, fW, t)); 1364 } 1365 if (tangent) { 1366 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1367 conic_eval_tan(&fPts[0].fY, fW, t)); 1368 } 1369} 1370 1371void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1372 SkP3D tmp[3], tmp2[3]; 1373 1374 ratquad_mapTo3D(fPts, fW, tmp); 1375 1376 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1377 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1378 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1379 1380 dst[0].fPts[0] = fPts[0]; 1381 tmp2[0].projectDown(&dst[0].fPts[1]); 1382 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1383 tmp2[2].projectDown(&dst[1].fPts[1]); 1384 dst[1].fPts[2] = fPts[2]; 1385 1386 // to put in "standard form", where w0 and w2 are both 1, we compute the 1387 // new w1 as sqrt(w1*w1/w0*w2) 1388 // or 1389 // w1 /= sqrt(w0*w2) 1390 // 1391 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1392 // 1393 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1394 dst[0].fW = tmp2[0].fZ / root; 1395 dst[1].fW = tmp2[2].fZ / root; 1396} 1397 1398static SkScalar subdivide_w_value(SkScalar w) { 1399 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1400} 1401 1402void SkConic::chop(SkConic dst[2]) const { 1403 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1404 SkScalar p1x = fW * fPts[1].fX; 1405 SkScalar p1y = fW * fPts[1].fY; 1406 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1407 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1408 1409 dst[0].fPts[0] = fPts[0]; 1410 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1411 (fPts[0].fY + p1y) * scale); 1412 dst[0].fPts[2].set(mx, my); 1413 1414 dst[1].fPts[0].set(mx, my); 1415 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1416 (p1y + fPts[2].fY) * scale); 1417 dst[1].fPts[2] = fPts[2]; 1418 1419 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1420} 1421 1422/* 1423 * "High order approximation of conic sections by quadratic splines" 1424 * by Michael Floater, 1993 1425 */ 1426#define AS_QUAD_ERROR_SETUP \ 1427 SkScalar a = fW - 1; \ 1428 SkScalar k = a / (4 * (2 + a)); \ 1429 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1430 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1431 1432void SkConic::computeAsQuadError(SkVector* err) const { 1433 AS_QUAD_ERROR_SETUP 1434 err->set(x, y); 1435} 1436 1437bool SkConic::asQuadTol(SkScalar tol) const { 1438 AS_QUAD_ERROR_SETUP 1439 return (x * x + y * y) <= tol * tol; 1440} 1441 1442int SkConic::computeQuadPOW2(SkScalar tol) const { 1443 AS_QUAD_ERROR_SETUP 1444 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1445 1446 if (error <= 0) { 1447 return 0; 1448 } 1449 uint32_t ierr = (uint32_t)error; 1450 return (34 - SkCLZ(ierr)) >> 1; 1451} 1452 1453static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1454 SkASSERT(level >= 0); 1455 1456 if (0 == level) { 1457 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1458 return pts + 2; 1459 } else { 1460 SkConic dst[2]; 1461 src.chop(dst); 1462 --level; 1463 pts = subdivide(dst[0], pts, level); 1464 return subdivide(dst[1], pts, level); 1465 } 1466} 1467 1468int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1469 SkASSERT(pow2 >= 0); 1470 *pts = fPts[0]; 1471 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1472 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1473 return 1 << pow2; 1474} 1475 1476bool SkConic::findXExtrema(SkScalar* t) const { 1477 return conic_find_extrema(&fPts[0].fX, fW, t); 1478} 1479 1480bool SkConic::findYExtrema(SkScalar* t) const { 1481 return conic_find_extrema(&fPts[0].fY, fW, t); 1482} 1483 1484bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1485 SkScalar t; 1486 if (this->findXExtrema(&t)) { 1487 this->chopAt(t, dst); 1488 // now clean-up the middle, since we know t was meant to be at 1489 // an X-extrema 1490 SkScalar value = dst[0].fPts[2].fX; 1491 dst[0].fPts[1].fX = value; 1492 dst[1].fPts[0].fX = value; 1493 dst[1].fPts[1].fX = value; 1494 return true; 1495 } 1496 return false; 1497} 1498 1499bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1500 SkScalar t; 1501 if (this->findYExtrema(&t)) { 1502 this->chopAt(t, dst); 1503 // now clean-up the middle, since we know t was meant to be at 1504 // an Y-extrema 1505 SkScalar value = dst[0].fPts[2].fY; 1506 dst[0].fPts[1].fY = value; 1507 dst[1].fPts[0].fY = value; 1508 dst[1].fPts[1].fY = value; 1509 return true; 1510 } 1511 return false; 1512} 1513 1514void SkConic::computeTightBounds(SkRect* bounds) const { 1515 SkPoint pts[4]; 1516 pts[0] = fPts[0]; 1517 pts[1] = fPts[2]; 1518 int count = 2; 1519 1520 SkScalar t; 1521 if (this->findXExtrema(&t)) { 1522 this->evalAt(t, &pts[count++]); 1523 } 1524 if (this->findYExtrema(&t)) { 1525 this->evalAt(t, &pts[count++]); 1526 } 1527 bounds->set(pts, count); 1528} 1529 1530void SkConic::computeFastBounds(SkRect* bounds) const { 1531 bounds->set(fPts, 3); 1532} 1533