SkGeometry.cpp revision b0889a5aa610552bf306edc8d9a35d2d601acdb9
1 2/* 3 * Copyright 2006 The Android Open Source Project 4 * 5 * Use of this source code is governed by a BSD-style license that can be 6 * found in the LICENSE file. 7 */ 8 9 10#include "SkGeometry.h" 11#include "Sk64.h" 12#include "SkMatrix.h" 13 14bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) { 15 if (ambiguous) { 16 *ambiguous = false; 17 } 18 // Determine quick discards. 19 // Consider query line going exactly through point 0 to not 20 // intersect, for symmetry with SkXRayCrossesMonotonicCubic. 21 if (pt.fY == pts[0].fY) { 22 if (ambiguous) { 23 *ambiguous = true; 24 } 25 return false; 26 } 27 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) 28 return false; 29 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) 30 return false; 31 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) 32 return false; 33 // Determine degenerate cases 34 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) 35 return false; 36 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { 37 // We've already determined the query point lies within the 38 // vertical range of the line segment. 39 if (pt.fX <= pts[0].fX) { 40 if (ambiguous) { 41 *ambiguous = (pt.fY == pts[1].fY); 42 } 43 return true; 44 } 45 return false; 46 } 47 // Ambiguity check 48 if (pt.fY == pts[1].fY) { 49 if (pt.fX <= pts[1].fX) { 50 if (ambiguous) { 51 *ambiguous = true; 52 } 53 return true; 54 } 55 return false; 56 } 57 // Full line segment evaluation 58 SkScalar delta_y = pts[1].fY - pts[0].fY; 59 SkScalar delta_x = pts[1].fX - pts[0].fX; 60 SkScalar slope = SkScalarDiv(delta_y, delta_x); 61 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); 62 // Solve for x coordinate at y = pt.fY 63 SkScalar x = SkScalarDiv(pt.fY - b, slope); 64 return pt.fX <= x; 65} 66 67/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 68 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 69 May also introduce overflow of fixed when we compute our setup. 70*/ 71#ifdef SK_SCALAR_IS_FIXED 72 #define DIRECT_EVAL_OF_POLYNOMIALS 73#endif 74 75//////////////////////////////////////////////////////////////////////// 76 77#ifdef SK_SCALAR_IS_FIXED 78 static int is_not_monotonic(int a, int b, int c, int d) 79 { 80 return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31; 81 } 82 83 static int is_not_monotonic(int a, int b, int c) 84 { 85 return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31; 86 } 87#else 88 static int is_not_monotonic(float a, float b, float c) 89 { 90 float ab = a - b; 91 float bc = b - c; 92 if (ab < 0) 93 bc = -bc; 94 return ab == 0 || bc < 0; 95 } 96#endif 97 98//////////////////////////////////////////////////////////////////////// 99 100static bool is_unit_interval(SkScalar x) 101{ 102 return x > 0 && x < SK_Scalar1; 103} 104 105static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) 106{ 107 SkASSERT(ratio); 108 109 if (numer < 0) 110 { 111 numer = -numer; 112 denom = -denom; 113 } 114 115 if (denom == 0 || numer == 0 || numer >= denom) 116 return 0; 117 118 SkScalar r = SkScalarDiv(numer, denom); 119 if (SkScalarIsNaN(r)) { 120 return 0; 121 } 122 SkASSERT(r >= 0 && r < SK_Scalar1); 123 if (r == 0) // catch underflow if numer <<<< denom 124 return 0; 125 *ratio = r; 126 return 1; 127} 128 129/** From Numerical Recipes in C. 130 131 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 132 x1 = Q / A 133 x2 = C / Q 134*/ 135int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) 136{ 137 SkASSERT(roots); 138 139 if (A == 0) 140 return valid_unit_divide(-C, B, roots); 141 142 SkScalar* r = roots; 143 144#ifdef SK_SCALAR_IS_FLOAT 145 float R = B*B - 4*A*C; 146 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 147 return 0; 148 } 149 R = sk_float_sqrt(R); 150#else 151 Sk64 RR, tmp; 152 153 RR.setMul(B,B); 154 tmp.setMul(A,C); 155 tmp.shiftLeft(2); 156 RR.sub(tmp); 157 if (RR.isNeg()) 158 return 0; 159 SkFixed R = RR.getSqrt(); 160#endif 161 162 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 163 r += valid_unit_divide(Q, A, r); 164 r += valid_unit_divide(C, Q, r); 165 if (r - roots == 2) 166 { 167 if (roots[0] > roots[1]) 168 SkTSwap<SkScalar>(roots[0], roots[1]); 169 else if (roots[0] == roots[1]) // nearly-equal? 170 r -= 1; // skip the double root 171 } 172 return (int)(r - roots); 173} 174 175#ifdef SK_SCALAR_IS_FIXED 176/** Trim A/B/C down so that they are all <= 32bits 177 and then call SkFindUnitQuadRoots() 178*/ 179static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2]) 180{ 181 int na = A.shiftToMake32(); 182 int nb = B.shiftToMake32(); 183 int nc = C.shiftToMake32(); 184 185 int shift = SkMax32(na, SkMax32(nb, nc)); 186 SkASSERT(shift >= 0); 187 188 return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots); 189} 190#endif 191 192///////////////////////////////////////////////////////////////////////////////////// 193///////////////////////////////////////////////////////////////////////////////////// 194 195static SkScalar eval_quad(const SkScalar src[], SkScalar t) 196{ 197 SkASSERT(src); 198 SkASSERT(t >= 0 && t <= SK_Scalar1); 199 200#ifdef DIRECT_EVAL_OF_POLYNOMIALS 201 SkScalar C = src[0]; 202 SkScalar A = src[4] - 2 * src[2] + C; 203 SkScalar B = 2 * (src[2] - C); 204 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 205#else 206 SkScalar ab = SkScalarInterp(src[0], src[2], t); 207 SkScalar bc = SkScalarInterp(src[2], src[4], t); 208 return SkScalarInterp(ab, bc, t); 209#endif 210} 211 212static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) 213{ 214 SkScalar A = src[4] - 2 * src[2] + src[0]; 215 SkScalar B = src[2] - src[0]; 216 217 return 2 * SkScalarMulAdd(A, t, B); 218} 219 220static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) 221{ 222 SkScalar A = src[4] - 2 * src[2] + src[0]; 223 SkScalar B = src[2] - src[0]; 224 return A + 2 * B; 225} 226 227void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) 228{ 229 SkASSERT(src); 230 SkASSERT(t >= 0 && t <= SK_Scalar1); 231 232 if (pt) 233 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 234 if (tangent) 235 tangent->set(eval_quad_derivative(&src[0].fX, t), 236 eval_quad_derivative(&src[0].fY, t)); 237} 238 239void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) 240{ 241 SkASSERT(src); 242 243 if (pt) 244 { 245 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 246 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 247 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 248 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 249 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 250 } 251 if (tangent) 252 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 253 eval_quad_derivative_at_half(&src[0].fY)); 254} 255 256static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 257{ 258 SkScalar ab = SkScalarInterp(src[0], src[2], t); 259 SkScalar bc = SkScalarInterp(src[2], src[4], t); 260 261 dst[0] = src[0]; 262 dst[2] = ab; 263 dst[4] = SkScalarInterp(ab, bc, t); 264 dst[6] = bc; 265 dst[8] = src[4]; 266} 267 268void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) 269{ 270 SkASSERT(t > 0 && t < SK_Scalar1); 271 272 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 273 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 274} 275 276void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) 277{ 278 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 279 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 280 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 281 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 282 283 dst[0] = src[0]; 284 dst[1].set(x01, y01); 285 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 286 dst[3].set(x12, y12); 287 dst[4] = src[2]; 288} 289 290/** Quad'(t) = At + B, where 291 A = 2(a - 2b + c) 292 B = 2(b - a) 293 Solve for t, only if it fits between 0 < t < 1 294*/ 295int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) 296{ 297 /* At + B == 0 298 t = -B / A 299 */ 300#ifdef SK_SCALAR_IS_FIXED 301 return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue); 302#else 303 return valid_unit_divide(a - b, a - b - b + c, tValue); 304#endif 305} 306 307static inline void flatten_double_quad_extrema(SkScalar coords[14]) 308{ 309 coords[2] = coords[6] = coords[4]; 310} 311 312/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 313 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 314 */ 315int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) 316{ 317 SkASSERT(src); 318 SkASSERT(dst); 319 320#if 0 321 static bool once = true; 322 if (once) 323 { 324 once = false; 325 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; 326 SkPoint d[6]; 327 328 int n = SkChopQuadAtYExtrema(s, d); 329 SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY); 330 } 331#endif 332 333 SkScalar a = src[0].fY; 334 SkScalar b = src[1].fY; 335 SkScalar c = src[2].fY; 336 337 if (is_not_monotonic(a, b, c)) 338 { 339 SkScalar tValue; 340 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) 341 { 342 SkChopQuadAt(src, dst, tValue); 343 flatten_double_quad_extrema(&dst[0].fY); 344 return 1; 345 } 346 // if we get here, we need to force dst to be monotonic, even though 347 // we couldn't compute a unit_divide value (probably underflow). 348 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 349 } 350 dst[0].set(src[0].fX, a); 351 dst[1].set(src[1].fX, b); 352 dst[2].set(src[2].fX, c); 353 return 0; 354} 355 356/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 357 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 358 */ 359int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) 360{ 361 SkASSERT(src); 362 SkASSERT(dst); 363 364 SkScalar a = src[0].fX; 365 SkScalar b = src[1].fX; 366 SkScalar c = src[2].fX; 367 368 if (is_not_monotonic(a, b, c)) { 369 SkScalar tValue; 370 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 371 SkChopQuadAt(src, dst, tValue); 372 flatten_double_quad_extrema(&dst[0].fX); 373 return 1; 374 } 375 // if we get here, we need to force dst to be monotonic, even though 376 // we couldn't compute a unit_divide value (probably underflow). 377 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 378 } 379 dst[0].set(a, src[0].fY); 380 dst[1].set(b, src[1].fY); 381 dst[2].set(c, src[2].fY); 382 return 0; 383} 384 385// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 386// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 387// F''(t) = 2 (a - 2b + c) 388// 389// A = 2 (b - a) 390// B = 2 (a - 2b + c) 391// 392// Maximum curvature for a quadratic means solving 393// Fx' Fx'' + Fy' Fy'' = 0 394// 395// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 396// 397float SkFindQuadMaxCurvature(const SkPoint src[3]) { 398 SkScalar Ax = src[1].fX - src[0].fX; 399 SkScalar Ay = src[1].fY - src[0].fY; 400 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 401 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 402 SkScalar t = 0; // 0 means don't chop 403 404#ifdef SK_SCALAR_IS_FLOAT 405 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 406#else 407 // !!! should I use SkFloat here? seems like it 408 Sk64 numer, denom, tmp; 409 410 numer.setMul(Ax, -Bx); 411 tmp.setMul(Ay, -By); 412 numer.add(tmp); 413 414 if (numer.isPos()) // do nothing if numer <= 0 415 { 416 denom.setMul(Bx, Bx); 417 tmp.setMul(By, By); 418 denom.add(tmp); 419 SkASSERT(!denom.isNeg()); 420 if (numer < denom) 421 { 422 t = numer.getFixedDiv(denom); 423 SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!) 424 if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability 425 t = 0; // ignore the chop 426 } 427 } 428#endif 429 return t; 430} 431 432int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) 433{ 434 SkScalar t = SkFindQuadMaxCurvature(src); 435 if (t == 0) { 436 memcpy(dst, src, 3 * sizeof(SkPoint)); 437 return 1; 438 } else { 439 SkChopQuadAt(src, dst, t); 440 return 2; 441 } 442} 443 444#ifdef SK_SCALAR_IS_FLOAT 445 #define SK_ScalarTwoThirds (0.666666666f) 446#else 447 #define SK_ScalarTwoThirds ((SkFixed)(43691)) 448#endif 449 450void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 451 const SkScalar scale = SK_ScalarTwoThirds; 452 dst[0] = src[0]; 453 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 454 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 455 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 456 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 457 dst[3] = src[2]; 458} 459 460//////////////////////////////////////////////////////////////////////////////////////// 461///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 462//////////////////////////////////////////////////////////////////////////////////////// 463 464static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) 465{ 466 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 467 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 468 coeff[2] = 3*(pt[2] - pt[0]); 469 coeff[3] = pt[0]; 470} 471 472void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) 473{ 474 SkASSERT(pts); 475 476 if (cx) 477 get_cubic_coeff(&pts[0].fX, cx); 478 if (cy) 479 get_cubic_coeff(&pts[0].fY, cy); 480} 481 482static SkScalar eval_cubic(const SkScalar src[], SkScalar t) 483{ 484 SkASSERT(src); 485 SkASSERT(t >= 0 && t <= SK_Scalar1); 486 487 if (t == 0) 488 return src[0]; 489 490#ifdef DIRECT_EVAL_OF_POLYNOMIALS 491 SkScalar D = src[0]; 492 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 493 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 494 SkScalar C = 3*(src[2] - D); 495 496 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 497#else 498 SkScalar ab = SkScalarInterp(src[0], src[2], t); 499 SkScalar bc = SkScalarInterp(src[2], src[4], t); 500 SkScalar cd = SkScalarInterp(src[4], src[6], t); 501 SkScalar abc = SkScalarInterp(ab, bc, t); 502 SkScalar bcd = SkScalarInterp(bc, cd, t); 503 return SkScalarInterp(abc, bcd, t); 504#endif 505} 506 507/** return At^2 + Bt + C 508*/ 509static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) 510{ 511 SkASSERT(t >= 0 && t <= SK_Scalar1); 512 513 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 514} 515 516static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) 517{ 518 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 519 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 520 SkScalar C = src[2] - src[0]; 521 522 return eval_quadratic(A, B, C, t); 523} 524 525static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) 526{ 527 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 528 SkScalar B = src[4] - 2 * src[2] + src[0]; 529 530 return SkScalarMulAdd(A, t, B); 531} 532 533void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature) 534{ 535 SkASSERT(src); 536 SkASSERT(t >= 0 && t <= SK_Scalar1); 537 538 if (loc) 539 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 540 if (tangent) 541 tangent->set(eval_cubic_derivative(&src[0].fX, t), 542 eval_cubic_derivative(&src[0].fY, t)); 543 if (curvature) 544 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 545 eval_cubic_2ndDerivative(&src[0].fY, t)); 546} 547 548/** Cubic'(t) = At^2 + Bt + C, where 549 A = 3(-a + 3(b - c) + d) 550 B = 6(a - 2b + c) 551 C = 3(b - a) 552 Solve for t, keeping only those that fit betwee 0 < t < 1 553*/ 554int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2]) 555{ 556#ifdef SK_SCALAR_IS_FIXED 557 if (!is_not_monotonic(a, b, c, d)) 558 return 0; 559#endif 560 561 // we divide A,B,C by 3 to simplify 562 SkScalar A = d - a + 3*(b - c); 563 SkScalar B = 2*(a - b - b + c); 564 SkScalar C = b - a; 565 566 return SkFindUnitQuadRoots(A, B, C, tValues); 567} 568 569static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 570{ 571 SkScalar ab = SkScalarInterp(src[0], src[2], t); 572 SkScalar bc = SkScalarInterp(src[2], src[4], t); 573 SkScalar cd = SkScalarInterp(src[4], src[6], t); 574 SkScalar abc = SkScalarInterp(ab, bc, t); 575 SkScalar bcd = SkScalarInterp(bc, cd, t); 576 SkScalar abcd = SkScalarInterp(abc, bcd, t); 577 578 dst[0] = src[0]; 579 dst[2] = ab; 580 dst[4] = abc; 581 dst[6] = abcd; 582 dst[8] = bcd; 583 dst[10] = cd; 584 dst[12] = src[6]; 585} 586 587void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) 588{ 589 SkASSERT(t > 0 && t < SK_Scalar1); 590 591 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 592 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 593} 594 595/* http://code.google.com/p/skia/issues/detail?id=32 596 597 This test code would fail when we didn't check the return result of 598 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 599 that after the first chop, the parameters to valid_unit_divide are equal 600 (thanks to finite float precision and rounding in the subtracts). Thus 601 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 602 up with 1.0, hence the need to check and just return the last cubic as 603 a degenerate clump of 4 points in the sampe place. 604 605 static void test_cubic() { 606 SkPoint src[4] = { 607 { 556.25000, 523.03003 }, 608 { 556.23999, 522.96002 }, 609 { 556.21997, 522.89001 }, 610 { 556.21997, 522.82001 } 611 }; 612 SkPoint dst[10]; 613 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 614 SkChopCubicAt(src, dst, tval, 2); 615 } 616 */ 617 618void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots) 619{ 620#ifdef SK_DEBUG 621 { 622 for (int i = 0; i < roots - 1; i++) 623 { 624 SkASSERT(is_unit_interval(tValues[i])); 625 SkASSERT(is_unit_interval(tValues[i+1])); 626 SkASSERT(tValues[i] < tValues[i+1]); 627 } 628 } 629#endif 630 631 if (dst) 632 { 633 if (roots == 0) // nothing to chop 634 memcpy(dst, src, 4*sizeof(SkPoint)); 635 else 636 { 637 SkScalar t = tValues[0]; 638 SkPoint tmp[4]; 639 640 for (int i = 0; i < roots; i++) 641 { 642 SkChopCubicAt(src, dst, t); 643 if (i == roots - 1) 644 break; 645 646 dst += 3; 647 // have src point to the remaining cubic (after the chop) 648 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 649 src = tmp; 650 651 // watch out in case the renormalized t isn't in range 652 if (!valid_unit_divide(tValues[i+1] - tValues[i], 653 SK_Scalar1 - tValues[i], &t)) { 654 // if we can't, just create a degenerate cubic 655 dst[4] = dst[5] = dst[6] = src[3]; 656 break; 657 } 658 } 659 } 660 } 661} 662 663void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) 664{ 665 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 666 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 667 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 668 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 669 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 670 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 671 672 SkScalar x012 = SkScalarAve(x01, x12); 673 SkScalar y012 = SkScalarAve(y01, y12); 674 SkScalar x123 = SkScalarAve(x12, x23); 675 SkScalar y123 = SkScalarAve(y12, y23); 676 677 dst[0] = src[0]; 678 dst[1].set(x01, y01); 679 dst[2].set(x012, y012); 680 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 681 dst[4].set(x123, y123); 682 dst[5].set(x23, y23); 683 dst[6] = src[3]; 684} 685 686static void flatten_double_cubic_extrema(SkScalar coords[14]) 687{ 688 coords[4] = coords[8] = coords[6]; 689} 690 691/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 692 the resulting beziers are monotonic in Y. This is called by the scan converter. 693 Depending on what is returned, dst[] is treated as follows 694 0 dst[0..3] is the original cubic 695 1 dst[0..3] and dst[3..6] are the two new cubics 696 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 697 If dst == null, it is ignored and only the count is returned. 698*/ 699int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 700 SkScalar tValues[2]; 701 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 702 src[3].fY, tValues); 703 704 SkChopCubicAt(src, dst, tValues, roots); 705 if (dst && roots > 0) { 706 // we do some cleanup to ensure our Y extrema are flat 707 flatten_double_cubic_extrema(&dst[0].fY); 708 if (roots == 2) { 709 flatten_double_cubic_extrema(&dst[3].fY); 710 } 711 } 712 return roots; 713} 714 715int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 716 SkScalar tValues[2]; 717 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 718 src[3].fX, tValues); 719 720 SkChopCubicAt(src, dst, tValues, roots); 721 if (dst && roots > 0) { 722 // we do some cleanup to ensure our Y extrema are flat 723 flatten_double_cubic_extrema(&dst[0].fX); 724 if (roots == 2) { 725 flatten_double_cubic_extrema(&dst[3].fX); 726 } 727 } 728 return roots; 729} 730 731/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 732 733 Inflection means that curvature is zero. 734 Curvature is [F' x F''] / [F'^3] 735 So we solve F'x X F''y - F'y X F''y == 0 736 After some canceling of the cubic term, we get 737 A = b - a 738 B = c - 2b + a 739 C = d - 3c + 3b - a 740 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 741*/ 742int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) 743{ 744 SkScalar Ax = src[1].fX - src[0].fX; 745 SkScalar Ay = src[1].fY - src[0].fY; 746 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 747 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 748 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 749 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 750 int count; 751 752#ifdef SK_SCALAR_IS_FLOAT 753 count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 754#else 755 Sk64 A, B, C, tmp; 756 757 A.setMul(Bx, Cy); 758 tmp.setMul(By, Cx); 759 A.sub(tmp); 760 761 B.setMul(Ax, Cy); 762 tmp.setMul(Ay, Cx); 763 B.sub(tmp); 764 765 C.setMul(Ax, By); 766 tmp.setMul(Ay, Bx); 767 C.sub(tmp); 768 769 count = Sk64FindFixedQuadRoots(A, B, C, tValues); 770#endif 771 772 return count; 773} 774 775int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) 776{ 777 SkScalar tValues[2]; 778 int count = SkFindCubicInflections(src, tValues); 779 780 if (dst) 781 { 782 if (count == 0) 783 memcpy(dst, src, 4 * sizeof(SkPoint)); 784 else 785 SkChopCubicAt(src, dst, tValues, count); 786 } 787 return count + 1; 788} 789 790template <typename T> void bubble_sort(T array[], int count) 791{ 792 for (int i = count - 1; i > 0; --i) 793 for (int j = i; j > 0; --j) 794 if (array[j] < array[j-1]) 795 { 796 T tmp(array[j]); 797 array[j] = array[j-1]; 798 array[j-1] = tmp; 799 } 800} 801 802#include "SkFP.h" 803 804// newton refinement 805#if 0 806static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) 807{ 808 // x1 = x0 - f(t) / f'(t) 809 810 SkFP T = SkScalarToFloat(root); 811 SkFP N, D; 812 813 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] 814 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); 815 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); 816 D = SkFPAdd(D, coeff[2]); 817 818 if (D == 0) 819 return root; 820 821 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] 822 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); 823 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); 824 N = SkFPAdd(N, SkFPMul(T, coeff[2])); 825 N = SkFPAdd(N, coeff[3]); 826 827 if (N) 828 { 829 SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); 830 831 if (delta) 832 root -= delta; 833 } 834 return root; 835} 836#endif 837 838/** 839 * Given an array and count, remove all pair-wise duplicates from the array, 840 * keeping the existing sorting, and return the new count 841 */ 842static int collaps_duplicates(float array[], int count) { 843 for (int n = count; n > 1; --n) { 844 if (array[0] == array[1]) { 845 for (int i = 1; i < n; ++i) { 846 array[i - 1] = array[i]; 847 } 848 count -= 1; 849 } else { 850 array += 1; 851 } 852 } 853 return count; 854} 855 856#ifdef SK_DEBUG 857 858#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 859 860static void test_collaps_duplicates() { 861 static bool gOnce; 862 if (gOnce) { return; } 863 gOnce = true; 864 const float src0[] = { 0 }; 865 const float src1[] = { 0, 0 }; 866 const float src2[] = { 0, 1 }; 867 const float src3[] = { 0, 0, 0 }; 868 const float src4[] = { 0, 0, 1 }; 869 const float src5[] = { 0, 1, 1 }; 870 const float src6[] = { 0, 1, 2 }; 871 const struct { 872 const float* fData; 873 int fCount; 874 int fCollapsedCount; 875 } data[] = { 876 { TEST_COLLAPS_ENTRY(src0), 1 }, 877 { TEST_COLLAPS_ENTRY(src1), 1 }, 878 { TEST_COLLAPS_ENTRY(src2), 2 }, 879 { TEST_COLLAPS_ENTRY(src3), 1 }, 880 { TEST_COLLAPS_ENTRY(src4), 2 }, 881 { TEST_COLLAPS_ENTRY(src5), 2 }, 882 { TEST_COLLAPS_ENTRY(src6), 3 }, 883 }; 884 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 885 float dst[3]; 886 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 887 int count = collaps_duplicates(dst, data[i].fCount); 888 SkASSERT(data[i].fCollapsedCount == count); 889 for (int j = 1; j < count; ++j) { 890 SkASSERT(dst[j-1] < dst[j]); 891 } 892 } 893} 894#endif 895 896#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop 897#pragma warning ( disable : 4702 ) 898#endif 899 900/* Solve coeff(t) == 0, returning the number of roots that 901 lie withing 0 < t < 1. 902 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 903 904 Eliminates repeated roots (so that all tValues are distinct, and are always 905 in increasing order. 906*/ 907static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3]) 908{ 909#ifndef SK_SCALAR_IS_FLOAT 910 return 0; // this is not yet implemented for software float 911#endif 912 913 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic 914 { 915 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 916 } 917 918 SkFP a, b, c, Q, R; 919 920 { 921 SkASSERT(coeff[0] != 0); 922 923 SkFP inva = SkFPInvert(coeff[0]); 924 a = SkFPMul(coeff[1], inva); 925 b = SkFPMul(coeff[2], inva); 926 c = SkFPMul(coeff[3], inva); 927 } 928 Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); 929// R = (2*a*a*a - 9*a*b + 27*c) / 54; 930 R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); 931 R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); 932 R = SkFPAdd(R, SkFPMulInt(c, 27)); 933 R = SkFPDivInt(R, 54); 934 935 SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); 936 SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); 937 SkFP adiv3 = SkFPDivInt(a, 3); 938 939 SkScalar* roots = tValues; 940 SkScalar r; 941 942 if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots 943 { 944#ifdef SK_SCALAR_IS_FLOAT 945 float theta = sk_float_acos(R / sk_float_sqrt(Q3)); 946 float neg2RootQ = -2 * sk_float_sqrt(Q); 947 948 r = neg2RootQ * sk_float_cos(theta/3) - adiv3; 949 if (is_unit_interval(r)) 950 *roots++ = r; 951 952 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; 953 if (is_unit_interval(r)) 954 *roots++ = r; 955 956 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; 957 if (is_unit_interval(r)) 958 *roots++ = r; 959 960 SkDEBUGCODE(test_collaps_duplicates();) 961 962 // now sort the roots 963 int count = (int)(roots - tValues); 964 SkASSERT((unsigned)count <= 3); 965 bubble_sort(tValues, count); 966 count = collaps_duplicates(tValues, count); 967 roots = tValues + count; // so we compute the proper count below 968#endif 969 } 970 else // we have 1 real root 971 { 972 SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3)); 973 A = SkFPCubeRoot(A); 974 if (SkFPGT(R, 0)) 975 A = SkFPNeg(A); 976 977 if (A != 0) 978 A = SkFPAdd(A, SkFPDiv(Q, A)); 979 r = SkFPToScalar(SkFPSub(A, adiv3)); 980 if (is_unit_interval(r)) 981 *roots++ = r; 982 } 983 984 return (int)(roots - tValues); 985} 986 987/* Looking for F' dot F'' == 0 988 989 A = b - a 990 B = c - 2b + a 991 C = d - 3c + 3b - a 992 993 F' = 3Ct^2 + 6Bt + 3A 994 F'' = 6Ct + 6B 995 996 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 997*/ 998static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4]) 999{ 1000 SkScalar a = src[2] - src[0]; 1001 SkScalar b = src[4] - 2 * src[2] + src[0]; 1002 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 1003 1004 SkFP A = SkScalarToFP(a); 1005 SkFP B = SkScalarToFP(b); 1006 SkFP C = SkScalarToFP(c); 1007 1008 coeff[0] = SkFPMul(C, C); 1009 coeff[1] = SkFPMulInt(SkFPMul(B, C), 3); 1010 coeff[2] = SkFPMulInt(SkFPMul(B, B), 2); 1011 coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A)); 1012 coeff[3] = SkFPMul(A, B); 1013} 1014 1015// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 1016//#define kMinTValueForChopping (SK_Scalar1 / 256) 1017#define kMinTValueForChopping 0 1018 1019/* Looking for F' dot F'' == 0 1020 1021 A = b - a 1022 B = c - 2b + a 1023 C = d - 3c + 3b - a 1024 1025 F' = 3Ct^2 + 6Bt + 3A 1026 F'' = 6Ct + 6B 1027 1028 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 1029*/ 1030int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) 1031{ 1032 SkFP coeffX[4], coeffY[4]; 1033 int i; 1034 1035 formulate_F1DotF2(&src[0].fX, coeffX); 1036 formulate_F1DotF2(&src[0].fY, coeffY); 1037 1038 for (i = 0; i < 4; i++) 1039 coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]); 1040 1041 SkScalar t[3]; 1042 int count = solve_cubic_polynomial(coeffX, t); 1043 int maxCount = 0; 1044 1045 // now remove extrema where the curvature is zero (mins) 1046 // !!!! need a test for this !!!! 1047 for (i = 0; i < count; i++) 1048 { 1049 // if (not_min_curvature()) 1050 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) 1051 tValues[maxCount++] = t[i]; 1052 } 1053 return maxCount; 1054} 1055 1056int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3]) 1057{ 1058 SkScalar t_storage[3]; 1059 1060 if (tValues == NULL) 1061 tValues = t_storage; 1062 1063 int count = SkFindCubicMaxCurvature(src, tValues); 1064 1065 if (dst) { 1066 if (count == 0) 1067 memcpy(dst, src, 4 * sizeof(SkPoint)); 1068 else 1069 SkChopCubicAt(src, dst, tValues, count); 1070 } 1071 return count + 1; 1072} 1073 1074bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1075 if (ambiguous) { 1076 *ambiguous = false; 1077 } 1078 1079 // Find the minimum and maximum y of the extrema, which are the 1080 // first and last points since this cubic is monotonic 1081 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 1082 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 1083 1084 if (pt.fY == cubic[0].fY 1085 || pt.fY < min_y 1086 || pt.fY > max_y) { 1087 // The query line definitely does not cross the curve 1088 if (ambiguous) { 1089 *ambiguous = (pt.fY == cubic[0].fY); 1090 } 1091 return false; 1092 } 1093 1094 bool pt_at_extremum = (pt.fY == cubic[3].fY); 1095 1096 SkScalar min_x = 1097 SkMinScalar( 1098 SkMinScalar( 1099 SkMinScalar(cubic[0].fX, cubic[1].fX), 1100 cubic[2].fX), 1101 cubic[3].fX); 1102 if (pt.fX < min_x) { 1103 // The query line definitely crosses the curve 1104 if (ambiguous) { 1105 *ambiguous = pt_at_extremum; 1106 } 1107 return true; 1108 } 1109 1110 SkScalar max_x = 1111 SkMaxScalar( 1112 SkMaxScalar( 1113 SkMaxScalar(cubic[0].fX, cubic[1].fX), 1114 cubic[2].fX), 1115 cubic[3].fX); 1116 if (pt.fX > max_x) { 1117 // The query line definitely does not cross the curve 1118 return false; 1119 } 1120 1121 // Do a binary search to find the parameter value which makes y as 1122 // close as possible to the query point. See whether the query 1123 // line's origin is to the left of the associated x coordinate. 1124 1125 // kMaxIter is chosen as the number of mantissa bits for a float, 1126 // since there's no way we are going to get more precision by 1127 // iterating more times than that. 1128 const int kMaxIter = 23; 1129 SkPoint eval; 1130 int iter = 0; 1131 SkScalar upper_t; 1132 SkScalar lower_t; 1133 // Need to invert direction of t parameter if cubic goes up 1134 // instead of down 1135 if (cubic[3].fY > cubic[0].fY) { 1136 upper_t = SK_Scalar1; 1137 lower_t = SkFloatToScalar(0); 1138 } else { 1139 upper_t = SkFloatToScalar(0); 1140 lower_t = SK_Scalar1; 1141 } 1142 do { 1143 SkScalar t = SkScalarAve(upper_t, lower_t); 1144 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 1145 if (pt.fY > eval.fY) { 1146 lower_t = t; 1147 } else { 1148 upper_t = t; 1149 } 1150 } while (++iter < kMaxIter 1151 && !SkScalarNearlyZero(eval.fY - pt.fY)); 1152 if (pt.fX <= eval.fX) { 1153 if (ambiguous) { 1154 *ambiguous = pt_at_extremum; 1155 } 1156 return true; 1157 } 1158 return false; 1159} 1160 1161int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) { 1162 int num_crossings = 0; 1163 SkPoint monotonic_cubics[10]; 1164 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 1165 if (ambiguous) { 1166 *ambiguous = false; 1167 } 1168 bool locally_ambiguous; 1169 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous)) 1170 ++num_crossings; 1171 if (ambiguous) { 1172 *ambiguous |= locally_ambiguous; 1173 } 1174 if (num_monotonic_cubics > 0) 1175 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous)) 1176 ++num_crossings; 1177 if (ambiguous) { 1178 *ambiguous |= locally_ambiguous; 1179 } 1180 if (num_monotonic_cubics > 1) 1181 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous)) 1182 ++num_crossings; 1183 if (ambiguous) { 1184 *ambiguous |= locally_ambiguous; 1185 } 1186 return num_crossings; 1187} 1188//////////////////////////////////////////////////////////////////////////////// 1189 1190/* Find t value for quadratic [a, b, c] = d. 1191 Return 0 if there is no solution within [0, 1) 1192*/ 1193static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) 1194{ 1195 // At^2 + Bt + C = d 1196 SkScalar A = a - 2 * b + c; 1197 SkScalar B = 2 * (b - a); 1198 SkScalar C = a - d; 1199 1200 SkScalar roots[2]; 1201 int count = SkFindUnitQuadRoots(A, B, C, roots); 1202 1203 SkASSERT(count <= 1); 1204 return count == 1 ? roots[0] : 0; 1205} 1206 1207/* given a quad-curve and a point (x,y), chop the quad at that point and place 1208 the new off-curve point and endpoint into 'dest'. 1209 Should only return false if the computed pos is the start of the curve 1210 (i.e. root == 0) 1211*/ 1212static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest) 1213{ 1214 const SkScalar* base; 1215 SkScalar value; 1216 1217 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1218 base = &quad[0].fX; 1219 value = x; 1220 } else { 1221 base = &quad[0].fY; 1222 value = y; 1223 } 1224 1225 // note: this returns 0 if it thinks value is out of range, meaning the 1226 // root might return something outside of [0, 1) 1227 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1228 1229 if (t > 0) 1230 { 1231 SkPoint tmp[5]; 1232 SkChopQuadAt(quad, tmp, t); 1233 dest[0] = tmp[1]; 1234 dest[1].set(x, y); 1235 return true; 1236 } else { 1237 /* t == 0 means either the value triggered a root outside of [0, 1) 1238 For our purposes, we can ignore the <= 0 roots, but we want to 1239 catch the >= 1 roots (which given our caller, will basically mean 1240 a root of 1, give-or-take numerical instability). If we are in the 1241 >= 1 case, return the existing offCurve point. 1242 1243 The test below checks to see if we are close to the "end" of the 1244 curve (near base[4]). Rather than specifying a tolerance, I just 1245 check to see if value is on to the right/left of the middle point 1246 (depending on the direction/sign of the end points). 1247 */ 1248 if ((base[0] < base[4] && value > base[2]) || 1249 (base[0] > base[4] && value < base[2])) // should root have been 1 1250 { 1251 dest[0] = quad[1]; 1252 dest[1].set(x, y); 1253 return true; 1254 } 1255 } 1256 return false; 1257} 1258 1259#ifdef SK_SCALAR_IS_FLOAT 1260 1261// Due to floating point issues (i.e., 1.0f - SK_ScalarRoot2Over2 != 1262// SK_ScalarRoot2Over2 - SK_ScalarTanPIOver8), the "correct" off curve 1263// control points cause the quadratic circle approximation to be concave. 1264// SK_OffEps is used to pull in the off-curve control points a bit 1265// to make the quadratic approximation convex. 1266// Pulling the off-curve controls points in is preferable to pushing some 1267// of the on-curve points off. 1268#define SK_OffEps 0.0001f 1269#else 1270#define SK_OffEps 0 1271#endif 1272 1273 1274static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1275 { SK_Scalar1, 0 }, 1276 { SK_Scalar1 - SK_OffEps, SK_ScalarTanPIOver8 - SK_OffEps }, 1277 { SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, 1278 { SK_ScalarTanPIOver8 - SK_OffEps, SK_Scalar1 - SK_OffEps }, 1279 1280 { 0, SK_Scalar1 }, 1281 { -SK_ScalarTanPIOver8 + SK_OffEps,SK_Scalar1 - SK_OffEps }, 1282 { -SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, 1283 { -SK_Scalar1 + SK_OffEps, SK_ScalarTanPIOver8 - SK_OffEps }, 1284 1285 { -SK_Scalar1, 0 }, 1286 { -SK_Scalar1 + SK_OffEps, -SK_ScalarTanPIOver8 + SK_OffEps }, 1287 { -SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, 1288 { -SK_ScalarTanPIOver8 + SK_OffEps,-SK_Scalar1 + SK_OffEps }, 1289 1290 { 0, -SK_Scalar1 }, 1291 { SK_ScalarTanPIOver8 - SK_OffEps, -SK_Scalar1 + SK_OffEps }, 1292 { SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, 1293 { SK_Scalar1 - SK_OffEps, -SK_ScalarTanPIOver8 + SK_OffEps }, 1294 1295 { SK_Scalar1, 0 } 1296}; 1297 1298int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1299 SkRotationDirection dir, const SkMatrix* userMatrix, 1300 SkPoint quadPoints[]) 1301{ 1302 // rotate by x,y so that uStart is (1.0) 1303 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1304 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1305 1306 SkScalar absX = SkScalarAbs(x); 1307 SkScalar absY = SkScalarAbs(y); 1308 1309 int pointCount; 1310 1311 // check for (effectively) coincident vectors 1312 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1313 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1314 if (absY <= SK_ScalarNearlyZero && x > 0 && 1315 ((y >= 0 && kCW_SkRotationDirection == dir) || 1316 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1317 1318 // just return the start-point 1319 quadPoints[0].set(SK_Scalar1, 0); 1320 pointCount = 1; 1321 } else { 1322 if (dir == kCCW_SkRotationDirection) 1323 y = -y; 1324 1325 // what octant (quadratic curve) is [xy] in? 1326 int oct = 0; 1327 bool sameSign = true; 1328 1329 if (0 == y) 1330 { 1331 oct = 4; // 180 1332 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1333 } 1334 else if (0 == x) 1335 { 1336 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1337 if (y > 0) 1338 oct = 2; // 90 1339 else 1340 oct = 6; // 270 1341 } 1342 else 1343 { 1344 if (y < 0) 1345 oct += 4; 1346 if ((x < 0) != (y < 0)) 1347 { 1348 oct += 2; 1349 sameSign = false; 1350 } 1351 if ((absX < absY) == sameSign) 1352 oct += 1; 1353 } 1354 1355 int wholeCount = oct << 1; 1356 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1357 1358 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1359 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) 1360 { 1361 wholeCount += 2; 1362 } 1363 pointCount = wholeCount + 1; 1364 } 1365 1366 // now handle counter-clockwise and the initial unitStart rotation 1367 SkMatrix matrix; 1368 matrix.setSinCos(uStart.fY, uStart.fX); 1369 if (dir == kCCW_SkRotationDirection) { 1370 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1371 } 1372 if (userMatrix) { 1373 matrix.postConcat(*userMatrix); 1374 } 1375 matrix.mapPoints(quadPoints, pointCount); 1376 return pointCount; 1377} 1378 1379/////////////////////////////////////////////////////////////////////////////// 1380 1381// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1382// ------------------------------------------ 1383// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1384// 1385// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1386// ------------------------------------------------ 1387// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1388// 1389 1390// Take the parametric specification for the conic (either X or Y) and return 1391// in coeff[] the coefficients for the simple quadratic polynomial 1392// coeff[0] for t^2 1393// coeff[1] for t 1394// coeff[2] for constant term 1395// 1396static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1397 SkASSERT(src); 1398 SkASSERT(t >= 0 && t <= SK_Scalar1); 1399 1400 SkScalar src2w = SkScalarMul(src[2], w); 1401 SkScalar C = src[0]; 1402 SkScalar A = src[4] - 2 * src2w + C; 1403 SkScalar B = 2 * (src2w - C); 1404 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1405 1406 B = 2 * (w - SK_Scalar1); 1407 C = SK_Scalar1; 1408 A = -B; 1409 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1410 1411 return SkScalarDiv(numer, denom); 1412} 1413 1414// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1415// 1416// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1417// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1418// t^0 : -2 P0 w + 2 P1 w 1419// 1420// We disregard magnitude, so we can freely ignore the denominator of F', and 1421// divide the numerator by 2 1422// 1423// coeff[0] for t^2 1424// coeff[1] for t^1 1425// coeff[2] for t^0 1426// 1427static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1428 const SkScalar P20 = src[4] - src[0]; 1429 const SkScalar P10 = src[2] - src[0]; 1430 const SkScalar wP10 = w * P10; 1431 coeff[0] = w * P20 - P20; 1432 coeff[1] = P20 - 2 * wP10; 1433 coeff[2] = wP10; 1434} 1435 1436static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1437 SkScalar coeff[3]; 1438 conic_deriv_coeff(coord, w, coeff); 1439 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1440} 1441 1442static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1443 SkScalar coeff[3]; 1444 conic_deriv_coeff(src, w, coeff); 1445 1446 SkScalar tValues[2]; 1447 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1448 SkASSERT(0 == roots || 1 == roots); 1449 1450 if (1 == roots) { 1451 *t = tValues[0]; 1452 return true; 1453 } 1454 return false; 1455} 1456 1457struct SkP3D { 1458 SkScalar fX, fY, fZ; 1459 1460 void set(SkScalar x, SkScalar y, SkScalar z) { 1461 fX = x; fY = y; fZ = z; 1462 } 1463 1464 void projectDown(SkPoint* dst) const { 1465 dst->set(fX / fZ, fY / fZ); 1466 } 1467}; 1468 1469// we just return the middle 3 points, since the first and last are dups of src 1470// 1471static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) { 1472 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1473 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1474 dst[0] = ab; 1475 dst[3] = SkScalarInterp(ab, bc, t); 1476 dst[6] = bc; 1477} 1478 1479static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1480 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1481 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1482 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1483} 1484 1485void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1486 SkASSERT(t >= 0 && t <= SK_Scalar1); 1487 1488 if (pt) { 1489 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1490 conic_eval_pos(&fPts[0].fY, fW, t)); 1491 } 1492 if (tangent) { 1493 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1494 conic_eval_tan(&fPts[0].fY, fW, t)); 1495 } 1496} 1497 1498void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1499 SkP3D tmp[3], tmp2[3]; 1500 1501 ratquad_mapTo3D(fPts, fW, tmp); 1502 1503 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1504 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1505 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1506 1507 dst[0].fPts[0] = fPts[0]; 1508 tmp2[0].projectDown(&dst[0].fPts[1]); 1509 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1510 tmp2[2].projectDown(&dst[1].fPts[1]); 1511 dst[1].fPts[2] = fPts[2]; 1512 1513 // to put in "standard form", where w0 and w2 are both 1, we compute the 1514 // new w1 as sqrt(w1*w1/w0*w2) 1515 // or 1516 // w1 /= sqrt(w0*w2) 1517 // 1518 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1519 // 1520 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1521 dst[0].fW = tmp2[0].fZ / root; 1522 dst[1].fW = tmp2[2].fZ / root; 1523} 1524 1525static SkScalar subdivide_w_value(SkScalar w) { 1526 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1527} 1528 1529void SkConic::chop(SkConic dst[2]) const { 1530 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1531 SkScalar p1x = fW * fPts[1].fX; 1532 SkScalar p1y = fW * fPts[1].fY; 1533 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1534 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1535 1536 dst[0].fPts[0] = fPts[0]; 1537 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1538 (fPts[0].fY + p1y) * scale); 1539 dst[0].fPts[2].set(mx, my); 1540 1541 dst[1].fPts[0].set(mx, my); 1542 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1543 (p1y + fPts[2].fY) * scale); 1544 dst[1].fPts[2] = fPts[2]; 1545 1546 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1547} 1548 1549/* 1550 * "High order approximation of conic sections by quadratic splines" 1551 * by Michael Floater, 1993 1552 */ 1553#define AS_QUAD_ERROR_SETUP \ 1554 SkScalar a = fW - 1; \ 1555 SkScalar k = a / (4 * (2 + a)); \ 1556 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1557 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1558 1559void SkConic::computeAsQuadError(SkVector* err) const { 1560 AS_QUAD_ERROR_SETUP 1561 err->set(x, y); 1562} 1563 1564bool SkConic::asQuadTol(SkScalar tol) const { 1565 AS_QUAD_ERROR_SETUP 1566 return (x * x + y * y) <= tol * tol; 1567} 1568 1569int SkConic::computeQuadPOW2(SkScalar tol) const { 1570 AS_QUAD_ERROR_SETUP 1571 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1572 1573 if (error <= 0) { 1574 return 0; 1575 } 1576 uint32_t ierr = (uint32_t)error; 1577 return (34 - SkCLZ(ierr)) >> 1; 1578} 1579 1580static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1581 SkASSERT(level >= 0); 1582 1583 if (0 == level) { 1584 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1585 return pts + 2; 1586 } else { 1587 SkConic dst[2]; 1588 src.chop(dst); 1589 --level; 1590 pts = subdivide(dst[0], pts, level); 1591 return subdivide(dst[1], pts, level); 1592 } 1593} 1594 1595int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1596 SkASSERT(pow2 >= 0); 1597 *pts = fPts[0]; 1598 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1599 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1600 return 1 << pow2; 1601} 1602 1603bool SkConic::findXExtrema(SkScalar* t) const { 1604 return conic_find_extrema(&fPts[0].fX, fW, t); 1605} 1606 1607bool SkConic::findYExtrema(SkScalar* t) const { 1608 return conic_find_extrema(&fPts[0].fY, fW, t); 1609} 1610 1611bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1612 SkScalar t; 1613 if (this->findXExtrema(&t)) { 1614 this->chopAt(t, dst); 1615 // now clean-up the middle, since we know t was meant to be at 1616 // an X-extrema 1617 SkScalar value = dst[0].fPts[2].fX; 1618 dst[0].fPts[1].fX = value; 1619 dst[1].fPts[0].fX = value; 1620 dst[1].fPts[1].fX = value; 1621 return true; 1622 } 1623 return false; 1624} 1625 1626bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1627 SkScalar t; 1628 if (this->findYExtrema(&t)) { 1629 this->chopAt(t, dst); 1630 // now clean-up the middle, since we know t was meant to be at 1631 // an Y-extrema 1632 SkScalar value = dst[0].fPts[2].fY; 1633 dst[0].fPts[1].fY = value; 1634 dst[1].fPts[0].fY = value; 1635 dst[1].fPts[1].fY = value; 1636 return true; 1637 } 1638 return false; 1639} 1640 1641void SkConic::computeTightBounds(SkRect* bounds) const { 1642 SkPoint pts[4]; 1643 pts[0] = fPts[0]; 1644 pts[1] = fPts[2]; 1645 int count = 2; 1646 1647 SkScalar t; 1648 if (this->findXExtrema(&t)) { 1649 this->evalAt(t, &pts[count++]); 1650 } 1651 if (this->findYExtrema(&t)) { 1652 this->evalAt(t, &pts[count++]); 1653 } 1654 bounds->set(pts, count); 1655} 1656 1657void SkConic::computeFastBounds(SkRect* bounds) const { 1658 bounds->set(fPts, 3); 1659} 1660