SkGeometry.cpp revision 9e1ec1a52985cce9db3a0d0e8d448b82a32e70cb
1 2/* 3 * Copyright 2006 The Android Open Source Project 4 * 5 * Use of this source code is governed by a BSD-style license that can be 6 * found in the LICENSE file. 7 */ 8 9 10#include "SkGeometry.h" 11#include "Sk64.h" 12#include "SkMatrix.h" 13 14bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) { 15 if (ambiguous) { 16 *ambiguous = false; 17 } 18 // Determine quick discards. 19 // Consider query line going exactly through point 0 to not 20 // intersect, for symmetry with SkXRayCrossesMonotonicCubic. 21 if (pt.fY == pts[0].fY) { 22 if (ambiguous) { 23 *ambiguous = true; 24 } 25 return false; 26 } 27 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) 28 return false; 29 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) 30 return false; 31 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) 32 return false; 33 // Determine degenerate cases 34 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) 35 return false; 36 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { 37 // We've already determined the query point lies within the 38 // vertical range of the line segment. 39 if (pt.fX <= pts[0].fX) { 40 if (ambiguous) { 41 *ambiguous = (pt.fY == pts[1].fY); 42 } 43 return true; 44 } 45 return false; 46 } 47 // Ambiguity check 48 if (pt.fY == pts[1].fY) { 49 if (pt.fX <= pts[1].fX) { 50 if (ambiguous) { 51 *ambiguous = true; 52 } 53 return true; 54 } 55 return false; 56 } 57 // Full line segment evaluation 58 SkScalar delta_y = pts[1].fY - pts[0].fY; 59 SkScalar delta_x = pts[1].fX - pts[0].fX; 60 SkScalar slope = SkScalarDiv(delta_y, delta_x); 61 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); 62 // Solve for x coordinate at y = pt.fY 63 SkScalar x = SkScalarDiv(pt.fY - b, slope); 64 return pt.fX <= x; 65} 66 67/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 68 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 69 May also introduce overflow of fixed when we compute our setup. 70*/ 71#ifdef SK_SCALAR_IS_FIXED 72 #define DIRECT_EVAL_OF_POLYNOMIALS 73#endif 74 75//////////////////////////////////////////////////////////////////////// 76 77#ifdef SK_SCALAR_IS_FIXED 78 static int is_not_monotonic(int a, int b, int c, int d) 79 { 80 return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31; 81 } 82 83 static int is_not_monotonic(int a, int b, int c) 84 { 85 return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31; 86 } 87#else 88 static int is_not_monotonic(float a, float b, float c) 89 { 90 float ab = a - b; 91 float bc = b - c; 92 if (ab < 0) 93 bc = -bc; 94 return ab == 0 || bc < 0; 95 } 96#endif 97 98//////////////////////////////////////////////////////////////////////// 99 100static bool is_unit_interval(SkScalar x) 101{ 102 return x > 0 && x < SK_Scalar1; 103} 104 105static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) 106{ 107 SkASSERT(ratio); 108 109 if (numer < 0) 110 { 111 numer = -numer; 112 denom = -denom; 113 } 114 115 if (denom == 0 || numer == 0 || numer >= denom) 116 return 0; 117 118 SkScalar r = SkScalarDiv(numer, denom); 119 if (SkScalarIsNaN(r)) { 120 return 0; 121 } 122 SkASSERT(r >= 0 && r < SK_Scalar1); 123 if (r == 0) // catch underflow if numer <<<< denom 124 return 0; 125 *ratio = r; 126 return 1; 127} 128 129/** From Numerical Recipes in C. 130 131 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 132 x1 = Q / A 133 x2 = C / Q 134*/ 135int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) 136{ 137 SkASSERT(roots); 138 139 if (A == 0) 140 return valid_unit_divide(-C, B, roots); 141 142 SkScalar* r = roots; 143 144#ifdef SK_SCALAR_IS_FLOAT 145 float R = B*B - 4*A*C; 146 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 147 return 0; 148 } 149 R = sk_float_sqrt(R); 150#else 151 Sk64 RR, tmp; 152 153 RR.setMul(B,B); 154 tmp.setMul(A,C); 155 tmp.shiftLeft(2); 156 RR.sub(tmp); 157 if (RR.isNeg()) 158 return 0; 159 SkFixed R = RR.getSqrt(); 160#endif 161 162 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 163 r += valid_unit_divide(Q, A, r); 164 r += valid_unit_divide(C, Q, r); 165 if (r - roots == 2) 166 { 167 if (roots[0] > roots[1]) 168 SkTSwap<SkScalar>(roots[0], roots[1]); 169 else if (roots[0] == roots[1]) // nearly-equal? 170 r -= 1; // skip the double root 171 } 172 return (int)(r - roots); 173} 174 175#ifdef SK_SCALAR_IS_FIXED 176/** Trim A/B/C down so that they are all <= 32bits 177 and then call SkFindUnitQuadRoots() 178*/ 179static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2]) 180{ 181 int na = A.shiftToMake32(); 182 int nb = B.shiftToMake32(); 183 int nc = C.shiftToMake32(); 184 185 int shift = SkMax32(na, SkMax32(nb, nc)); 186 SkASSERT(shift >= 0); 187 188 return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots); 189} 190#endif 191 192///////////////////////////////////////////////////////////////////////////////////// 193///////////////////////////////////////////////////////////////////////////////////// 194 195static SkScalar eval_quad(const SkScalar src[], SkScalar t) 196{ 197 SkASSERT(src); 198 SkASSERT(t >= 0 && t <= SK_Scalar1); 199 200#ifdef DIRECT_EVAL_OF_POLYNOMIALS 201 SkScalar C = src[0]; 202 SkScalar A = src[4] - 2 * src[2] + C; 203 SkScalar B = 2 * (src[2] - C); 204 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 205#else 206 SkScalar ab = SkScalarInterp(src[0], src[2], t); 207 SkScalar bc = SkScalarInterp(src[2], src[4], t); 208 return SkScalarInterp(ab, bc, t); 209#endif 210} 211 212static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) 213{ 214 SkScalar A = src[4] - 2 * src[2] + src[0]; 215 SkScalar B = src[2] - src[0]; 216 217 return 2 * SkScalarMulAdd(A, t, B); 218} 219 220static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) 221{ 222 SkScalar A = src[4] - 2 * src[2] + src[0]; 223 SkScalar B = src[2] - src[0]; 224 return A + 2 * B; 225} 226 227void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) 228{ 229 SkASSERT(src); 230 SkASSERT(t >= 0 && t <= SK_Scalar1); 231 232 if (pt) 233 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 234 if (tangent) 235 tangent->set(eval_quad_derivative(&src[0].fX, t), 236 eval_quad_derivative(&src[0].fY, t)); 237} 238 239void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) 240{ 241 SkASSERT(src); 242 243 if (pt) 244 { 245 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 246 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 247 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 248 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 249 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 250 } 251 if (tangent) 252 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 253 eval_quad_derivative_at_half(&src[0].fY)); 254} 255 256static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 257{ 258 SkScalar ab = SkScalarInterp(src[0], src[2], t); 259 SkScalar bc = SkScalarInterp(src[2], src[4], t); 260 261 dst[0] = src[0]; 262 dst[2] = ab; 263 dst[4] = SkScalarInterp(ab, bc, t); 264 dst[6] = bc; 265 dst[8] = src[4]; 266} 267 268void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) 269{ 270 SkASSERT(t > 0 && t < SK_Scalar1); 271 272 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 273 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 274} 275 276void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) 277{ 278 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 279 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 280 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 281 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 282 283 dst[0] = src[0]; 284 dst[1].set(x01, y01); 285 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 286 dst[3].set(x12, y12); 287 dst[4] = src[2]; 288} 289 290/** Quad'(t) = At + B, where 291 A = 2(a - 2b + c) 292 B = 2(b - a) 293 Solve for t, only if it fits between 0 < t < 1 294*/ 295int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) 296{ 297 /* At + B == 0 298 t = -B / A 299 */ 300#ifdef SK_SCALAR_IS_FIXED 301 return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue); 302#else 303 return valid_unit_divide(a - b, a - b - b + c, tValue); 304#endif 305} 306 307static inline void flatten_double_quad_extrema(SkScalar coords[14]) 308{ 309 coords[2] = coords[6] = coords[4]; 310} 311 312/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 313 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 314 */ 315int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) 316{ 317 SkASSERT(src); 318 SkASSERT(dst); 319 320#if 0 321 static bool once = true; 322 if (once) 323 { 324 once = false; 325 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; 326 SkPoint d[6]; 327 328 int n = SkChopQuadAtYExtrema(s, d); 329 SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY); 330 } 331#endif 332 333 SkScalar a = src[0].fY; 334 SkScalar b = src[1].fY; 335 SkScalar c = src[2].fY; 336 337 if (is_not_monotonic(a, b, c)) 338 { 339 SkScalar tValue; 340 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) 341 { 342 SkChopQuadAt(src, dst, tValue); 343 flatten_double_quad_extrema(&dst[0].fY); 344 return 1; 345 } 346 // if we get here, we need to force dst to be monotonic, even though 347 // we couldn't compute a unit_divide value (probably underflow). 348 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 349 } 350 dst[0].set(src[0].fX, a); 351 dst[1].set(src[1].fX, b); 352 dst[2].set(src[2].fX, c); 353 return 0; 354} 355 356/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 357 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 358 */ 359int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) 360{ 361 SkASSERT(src); 362 SkASSERT(dst); 363 364 SkScalar a = src[0].fX; 365 SkScalar b = src[1].fX; 366 SkScalar c = src[2].fX; 367 368 if (is_not_monotonic(a, b, c)) { 369 SkScalar tValue; 370 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 371 SkChopQuadAt(src, dst, tValue); 372 flatten_double_quad_extrema(&dst[0].fX); 373 return 1; 374 } 375 // if we get here, we need to force dst to be monotonic, even though 376 // we couldn't compute a unit_divide value (probably underflow). 377 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 378 } 379 dst[0].set(a, src[0].fY); 380 dst[1].set(b, src[1].fY); 381 dst[2].set(c, src[2].fY); 382 return 0; 383} 384 385// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 386// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 387// F''(t) = 2 (a - 2b + c) 388// 389// A = 2 (b - a) 390// B = 2 (a - 2b + c) 391// 392// Maximum curvature for a quadratic means solving 393// Fx' Fx'' + Fy' Fy'' = 0 394// 395// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 396// 397int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) 398{ 399 SkScalar Ax = src[1].fX - src[0].fX; 400 SkScalar Ay = src[1].fY - src[0].fY; 401 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 402 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 403 SkScalar t = 0; // 0 means don't chop 404 405#ifdef SK_SCALAR_IS_FLOAT 406 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 407#else 408 // !!! should I use SkFloat here? seems like it 409 Sk64 numer, denom, tmp; 410 411 numer.setMul(Ax, -Bx); 412 tmp.setMul(Ay, -By); 413 numer.add(tmp); 414 415 if (numer.isPos()) // do nothing if numer <= 0 416 { 417 denom.setMul(Bx, Bx); 418 tmp.setMul(By, By); 419 denom.add(tmp); 420 SkASSERT(!denom.isNeg()); 421 if (numer < denom) 422 { 423 t = numer.getFixedDiv(denom); 424 SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!) 425 if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability 426 t = 0; // ignore the chop 427 } 428 } 429#endif 430 431 if (t == 0) 432 { 433 memcpy(dst, src, 3 * sizeof(SkPoint)); 434 return 1; 435 } 436 else 437 { 438 SkChopQuadAt(src, dst, t); 439 return 2; 440 } 441} 442 443#ifdef SK_SCALAR_IS_FLOAT 444 #define SK_ScalarTwoThirds (0.666666666f) 445#else 446 #define SK_ScalarTwoThirds ((SkFixed)(43691)) 447#endif 448 449void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 450 const SkScalar scale = SK_ScalarTwoThirds; 451 dst[0] = src[0]; 452 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 453 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 454 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 455 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 456 dst[3] = src[2]; 457} 458 459//////////////////////////////////////////////////////////////////////////////////////// 460///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 461//////////////////////////////////////////////////////////////////////////////////////// 462 463static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) 464{ 465 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 466 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 467 coeff[2] = 3*(pt[2] - pt[0]); 468 coeff[3] = pt[0]; 469} 470 471void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) 472{ 473 SkASSERT(pts); 474 475 if (cx) 476 get_cubic_coeff(&pts[0].fX, cx); 477 if (cy) 478 get_cubic_coeff(&pts[0].fY, cy); 479} 480 481static SkScalar eval_cubic(const SkScalar src[], SkScalar t) 482{ 483 SkASSERT(src); 484 SkASSERT(t >= 0 && t <= SK_Scalar1); 485 486 if (t == 0) 487 return src[0]; 488 489#ifdef DIRECT_EVAL_OF_POLYNOMIALS 490 SkScalar D = src[0]; 491 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 492 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 493 SkScalar C = 3*(src[2] - D); 494 495 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 496#else 497 SkScalar ab = SkScalarInterp(src[0], src[2], t); 498 SkScalar bc = SkScalarInterp(src[2], src[4], t); 499 SkScalar cd = SkScalarInterp(src[4], src[6], t); 500 SkScalar abc = SkScalarInterp(ab, bc, t); 501 SkScalar bcd = SkScalarInterp(bc, cd, t); 502 return SkScalarInterp(abc, bcd, t); 503#endif 504} 505 506/** return At^2 + Bt + C 507*/ 508static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) 509{ 510 SkASSERT(t >= 0 && t <= SK_Scalar1); 511 512 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 513} 514 515static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) 516{ 517 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 518 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 519 SkScalar C = src[2] - src[0]; 520 521 return eval_quadratic(A, B, C, t); 522} 523 524static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) 525{ 526 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 527 SkScalar B = src[4] - 2 * src[2] + src[0]; 528 529 return SkScalarMulAdd(A, t, B); 530} 531 532void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature) 533{ 534 SkASSERT(src); 535 SkASSERT(t >= 0 && t <= SK_Scalar1); 536 537 if (loc) 538 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 539 if (tangent) 540 tangent->set(eval_cubic_derivative(&src[0].fX, t), 541 eval_cubic_derivative(&src[0].fY, t)); 542 if (curvature) 543 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 544 eval_cubic_2ndDerivative(&src[0].fY, t)); 545} 546 547/** Cubic'(t) = At^2 + Bt + C, where 548 A = 3(-a + 3(b - c) + d) 549 B = 6(a - 2b + c) 550 C = 3(b - a) 551 Solve for t, keeping only those that fit betwee 0 < t < 1 552*/ 553int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2]) 554{ 555#ifdef SK_SCALAR_IS_FIXED 556 if (!is_not_monotonic(a, b, c, d)) 557 return 0; 558#endif 559 560 // we divide A,B,C by 3 to simplify 561 SkScalar A = d - a + 3*(b - c); 562 SkScalar B = 2*(a - b - b + c); 563 SkScalar C = b - a; 564 565 return SkFindUnitQuadRoots(A, B, C, tValues); 566} 567 568static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t) 569{ 570 SkScalar ab = SkScalarInterp(src[0], src[2], t); 571 SkScalar bc = SkScalarInterp(src[2], src[4], t); 572 SkScalar cd = SkScalarInterp(src[4], src[6], t); 573 SkScalar abc = SkScalarInterp(ab, bc, t); 574 SkScalar bcd = SkScalarInterp(bc, cd, t); 575 SkScalar abcd = SkScalarInterp(abc, bcd, t); 576 577 dst[0] = src[0]; 578 dst[2] = ab; 579 dst[4] = abc; 580 dst[6] = abcd; 581 dst[8] = bcd; 582 dst[10] = cd; 583 dst[12] = src[6]; 584} 585 586void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) 587{ 588 SkASSERT(t > 0 && t < SK_Scalar1); 589 590 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 591 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 592} 593 594/* http://code.google.com/p/skia/issues/detail?id=32 595 596 This test code would fail when we didn't check the return result of 597 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 598 that after the first chop, the parameters to valid_unit_divide are equal 599 (thanks to finite float precision and rounding in the subtracts). Thus 600 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 601 up with 1.0, hence the need to check and just return the last cubic as 602 a degenerate clump of 4 points in the sampe place. 603 604 static void test_cubic() { 605 SkPoint src[4] = { 606 { 556.25000, 523.03003 }, 607 { 556.23999, 522.96002 }, 608 { 556.21997, 522.89001 }, 609 { 556.21997, 522.82001 } 610 }; 611 SkPoint dst[10]; 612 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 613 SkChopCubicAt(src, dst, tval, 2); 614 } 615 */ 616 617void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots) 618{ 619#ifdef SK_DEBUG 620 { 621 for (int i = 0; i < roots - 1; i++) 622 { 623 SkASSERT(is_unit_interval(tValues[i])); 624 SkASSERT(is_unit_interval(tValues[i+1])); 625 SkASSERT(tValues[i] < tValues[i+1]); 626 } 627 } 628#endif 629 630 if (dst) 631 { 632 if (roots == 0) // nothing to chop 633 memcpy(dst, src, 4*sizeof(SkPoint)); 634 else 635 { 636 SkScalar t = tValues[0]; 637 SkPoint tmp[4]; 638 639 for (int i = 0; i < roots; i++) 640 { 641 SkChopCubicAt(src, dst, t); 642 if (i == roots - 1) 643 break; 644 645 dst += 3; 646 // have src point to the remaining cubic (after the chop) 647 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 648 src = tmp; 649 650 // watch out in case the renormalized t isn't in range 651 if (!valid_unit_divide(tValues[i+1] - tValues[i], 652 SK_Scalar1 - tValues[i], &t)) { 653 // if we can't, just create a degenerate cubic 654 dst[4] = dst[5] = dst[6] = src[3]; 655 break; 656 } 657 } 658 } 659 } 660} 661 662void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) 663{ 664 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 665 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 666 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 667 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 668 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 669 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 670 671 SkScalar x012 = SkScalarAve(x01, x12); 672 SkScalar y012 = SkScalarAve(y01, y12); 673 SkScalar x123 = SkScalarAve(x12, x23); 674 SkScalar y123 = SkScalarAve(y12, y23); 675 676 dst[0] = src[0]; 677 dst[1].set(x01, y01); 678 dst[2].set(x012, y012); 679 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 680 dst[4].set(x123, y123); 681 dst[5].set(x23, y23); 682 dst[6] = src[3]; 683} 684 685static void flatten_double_cubic_extrema(SkScalar coords[14]) 686{ 687 coords[4] = coords[8] = coords[6]; 688} 689 690/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 691 the resulting beziers are monotonic in Y. This is called by the scan converter. 692 Depending on what is returned, dst[] is treated as follows 693 0 dst[0..3] is the original cubic 694 1 dst[0..3] and dst[3..6] are the two new cubics 695 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 696 If dst == null, it is ignored and only the count is returned. 697*/ 698int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 699 SkScalar tValues[2]; 700 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 701 src[3].fY, tValues); 702 703 SkChopCubicAt(src, dst, tValues, roots); 704 if (dst && roots > 0) { 705 // we do some cleanup to ensure our Y extrema are flat 706 flatten_double_cubic_extrema(&dst[0].fY); 707 if (roots == 2) { 708 flatten_double_cubic_extrema(&dst[3].fY); 709 } 710 } 711 return roots; 712} 713 714int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 715 SkScalar tValues[2]; 716 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 717 src[3].fX, tValues); 718 719 SkChopCubicAt(src, dst, tValues, roots); 720 if (dst && roots > 0) { 721 // we do some cleanup to ensure our Y extrema are flat 722 flatten_double_cubic_extrema(&dst[0].fX); 723 if (roots == 2) { 724 flatten_double_cubic_extrema(&dst[3].fX); 725 } 726 } 727 return roots; 728} 729 730/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 731 732 Inflection means that curvature is zero. 733 Curvature is [F' x F''] / [F'^3] 734 So we solve F'x X F''y - F'y X F''y == 0 735 After some canceling of the cubic term, we get 736 A = b - a 737 B = c - 2b + a 738 C = d - 3c + 3b - a 739 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 740*/ 741int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) 742{ 743 SkScalar Ax = src[1].fX - src[0].fX; 744 SkScalar Ay = src[1].fY - src[0].fY; 745 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 746 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 747 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 748 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 749 int count; 750 751#ifdef SK_SCALAR_IS_FLOAT 752 count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); 753#else 754 Sk64 A, B, C, tmp; 755 756 A.setMul(Bx, Cy); 757 tmp.setMul(By, Cx); 758 A.sub(tmp); 759 760 B.setMul(Ax, Cy); 761 tmp.setMul(Ay, Cx); 762 B.sub(tmp); 763 764 C.setMul(Ax, By); 765 tmp.setMul(Ay, Bx); 766 C.sub(tmp); 767 768 count = Sk64FindFixedQuadRoots(A, B, C, tValues); 769#endif 770 771 return count; 772} 773 774int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) 775{ 776 SkScalar tValues[2]; 777 int count = SkFindCubicInflections(src, tValues); 778 779 if (dst) 780 { 781 if (count == 0) 782 memcpy(dst, src, 4 * sizeof(SkPoint)); 783 else 784 SkChopCubicAt(src, dst, tValues, count); 785 } 786 return count + 1; 787} 788 789template <typename T> void bubble_sort(T array[], int count) 790{ 791 for (int i = count - 1; i > 0; --i) 792 for (int j = i; j > 0; --j) 793 if (array[j] < array[j-1]) 794 { 795 T tmp(array[j]); 796 array[j] = array[j-1]; 797 array[j-1] = tmp; 798 } 799} 800 801#include "SkFP.h" 802 803// newton refinement 804#if 0 805static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) 806{ 807 // x1 = x0 - f(t) / f'(t) 808 809 SkFP T = SkScalarToFloat(root); 810 SkFP N, D; 811 812 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] 813 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); 814 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); 815 D = SkFPAdd(D, coeff[2]); 816 817 if (D == 0) 818 return root; 819 820 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] 821 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); 822 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); 823 N = SkFPAdd(N, SkFPMul(T, coeff[2])); 824 N = SkFPAdd(N, coeff[3]); 825 826 if (N) 827 { 828 SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); 829 830 if (delta) 831 root -= delta; 832 } 833 return root; 834} 835#endif 836 837/** 838 * Given an array and count, remove all pair-wise duplicates from the array, 839 * keeping the existing sorting, and return the new count 840 */ 841static int collaps_duplicates(float array[], int count) { 842 for (int n = count; n > 1; --n) { 843 if (array[0] == array[1]) { 844 for (int i = 1; i < n; ++i) { 845 array[i - 1] = array[i]; 846 } 847 count -= 1; 848 } else { 849 array += 1; 850 } 851 } 852 return count; 853} 854 855#ifdef SK_DEBUG 856 857#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 858 859static void test_collaps_duplicates() { 860 static bool gOnce; 861 if (gOnce) { return; } 862 gOnce = true; 863 const float src0[] = { 0 }; 864 const float src1[] = { 0, 0 }; 865 const float src2[] = { 0, 1 }; 866 const float src3[] = { 0, 0, 0 }; 867 const float src4[] = { 0, 0, 1 }; 868 const float src5[] = { 0, 1, 1 }; 869 const float src6[] = { 0, 1, 2 }; 870 const struct { 871 const float* fData; 872 int fCount; 873 int fCollapsedCount; 874 } data[] = { 875 { TEST_COLLAPS_ENTRY(src0), 1 }, 876 { TEST_COLLAPS_ENTRY(src1), 1 }, 877 { TEST_COLLAPS_ENTRY(src2), 2 }, 878 { TEST_COLLAPS_ENTRY(src3), 1 }, 879 { TEST_COLLAPS_ENTRY(src4), 2 }, 880 { TEST_COLLAPS_ENTRY(src5), 2 }, 881 { TEST_COLLAPS_ENTRY(src6), 3 }, 882 }; 883 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 884 float dst[3]; 885 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 886 int count = collaps_duplicates(dst, data[i].fCount); 887 SkASSERT(data[i].fCollapsedCount == count); 888 for (int j = 1; j < count; ++j) { 889 SkASSERT(dst[j-1] < dst[j]); 890 } 891 } 892} 893#endif 894 895#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop 896#pragma warning ( disable : 4702 ) 897#endif 898 899/* Solve coeff(t) == 0, returning the number of roots that 900 lie withing 0 < t < 1. 901 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 902 903 Eliminates repeated roots (so that all tValues are distinct, and are always 904 in increasing order. 905*/ 906static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3]) 907{ 908#ifndef SK_SCALAR_IS_FLOAT 909 return 0; // this is not yet implemented for software float 910#endif 911 912 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic 913 { 914 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 915 } 916 917 SkFP a, b, c, Q, R; 918 919 { 920 SkASSERT(coeff[0] != 0); 921 922 SkFP inva = SkFPInvert(coeff[0]); 923 a = SkFPMul(coeff[1], inva); 924 b = SkFPMul(coeff[2], inva); 925 c = SkFPMul(coeff[3], inva); 926 } 927 Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); 928// R = (2*a*a*a - 9*a*b + 27*c) / 54; 929 R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); 930 R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); 931 R = SkFPAdd(R, SkFPMulInt(c, 27)); 932 R = SkFPDivInt(R, 54); 933 934 SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); 935 SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); 936 SkFP adiv3 = SkFPDivInt(a, 3); 937 938 SkScalar* roots = tValues; 939 SkScalar r; 940 941 if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots 942 { 943#ifdef SK_SCALAR_IS_FLOAT 944 float theta = sk_float_acos(R / sk_float_sqrt(Q3)); 945 float neg2RootQ = -2 * sk_float_sqrt(Q); 946 947 r = neg2RootQ * sk_float_cos(theta/3) - adiv3; 948 if (is_unit_interval(r)) 949 *roots++ = r; 950 951 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; 952 if (is_unit_interval(r)) 953 *roots++ = r; 954 955 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; 956 if (is_unit_interval(r)) 957 *roots++ = r; 958 959 SkDEBUGCODE(test_collaps_duplicates();) 960 961 // now sort the roots 962 int count = (int)(roots - tValues); 963 SkASSERT((unsigned)count <= 3); 964 bubble_sort(tValues, count); 965 count = collaps_duplicates(tValues, count); 966 roots = tValues + count; // so we compute the proper count below 967#endif 968 } 969 else // we have 1 real root 970 { 971 SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3)); 972 A = SkFPCubeRoot(A); 973 if (SkFPGT(R, 0)) 974 A = SkFPNeg(A); 975 976 if (A != 0) 977 A = SkFPAdd(A, SkFPDiv(Q, A)); 978 r = SkFPToScalar(SkFPSub(A, adiv3)); 979 if (is_unit_interval(r)) 980 *roots++ = r; 981 } 982 983 return (int)(roots - tValues); 984} 985 986/* Looking for F' dot F'' == 0 987 988 A = b - a 989 B = c - 2b + a 990 C = d - 3c + 3b - a 991 992 F' = 3Ct^2 + 6Bt + 3A 993 F'' = 6Ct + 6B 994 995 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 996*/ 997static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4]) 998{ 999 SkScalar a = src[2] - src[0]; 1000 SkScalar b = src[4] - 2 * src[2] + src[0]; 1001 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 1002 1003 SkFP A = SkScalarToFP(a); 1004 SkFP B = SkScalarToFP(b); 1005 SkFP C = SkScalarToFP(c); 1006 1007 coeff[0] = SkFPMul(C, C); 1008 coeff[1] = SkFPMulInt(SkFPMul(B, C), 3); 1009 coeff[2] = SkFPMulInt(SkFPMul(B, B), 2); 1010 coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A)); 1011 coeff[3] = SkFPMul(A, B); 1012} 1013 1014// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 1015//#define kMinTValueForChopping (SK_Scalar1 / 256) 1016#define kMinTValueForChopping 0 1017 1018/* Looking for F' dot F'' == 0 1019 1020 A = b - a 1021 B = c - 2b + a 1022 C = d - 3c + 3b - a 1023 1024 F' = 3Ct^2 + 6Bt + 3A 1025 F'' = 6Ct + 6B 1026 1027 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 1028*/ 1029int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) 1030{ 1031 SkFP coeffX[4], coeffY[4]; 1032 int i; 1033 1034 formulate_F1DotF2(&src[0].fX, coeffX); 1035 formulate_F1DotF2(&src[0].fY, coeffY); 1036 1037 for (i = 0; i < 4; i++) 1038 coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]); 1039 1040 SkScalar t[3]; 1041 int count = solve_cubic_polynomial(coeffX, t); 1042 int maxCount = 0; 1043 1044 // now remove extrema where the curvature is zero (mins) 1045 // !!!! need a test for this !!!! 1046 for (i = 0; i < count; i++) 1047 { 1048 // if (not_min_curvature()) 1049 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) 1050 tValues[maxCount++] = t[i]; 1051 } 1052 return maxCount; 1053} 1054 1055int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3]) 1056{ 1057 SkScalar t_storage[3]; 1058 1059 if (tValues == NULL) 1060 tValues = t_storage; 1061 1062 int count = SkFindCubicMaxCurvature(src, tValues); 1063 1064 if (dst) 1065 { 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 1191/* Find t value for quadratic [a, b, c] = d. 1192 Return 0 if there is no solution within [0, 1) 1193*/ 1194static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) 1195{ 1196 // At^2 + Bt + C = d 1197 SkScalar A = a - 2 * b + c; 1198 SkScalar B = 2 * (b - a); 1199 SkScalar C = a - d; 1200 1201 SkScalar roots[2]; 1202 int count = SkFindUnitQuadRoots(A, B, C, roots); 1203 1204 SkASSERT(count <= 1); 1205 return count == 1 ? roots[0] : 0; 1206} 1207 1208/* given a quad-curve and a point (x,y), chop the quad at that point and place 1209 the new off-curve point and endpoint into 'dest'. The new end point is used 1210 (rather than (x,y)) to compensate for numerical inaccuracies. 1211 Should only return false if the computed pos is the start of the curve 1212 (i.e. root == 0) 1213*/ 1214static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest) 1215{ 1216 const SkScalar* base; 1217 SkScalar value; 1218 1219 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1220 base = &quad[0].fX; 1221 value = x; 1222 } else { 1223 base = &quad[0].fY; 1224 value = y; 1225 } 1226 1227 // note: this returns 0 if it thinks value is out of range, meaning the 1228 // root might return something outside of [0, 1) 1229 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1230 1231 if (t > 0) 1232 { 1233 SkPoint tmp[5]; 1234 SkChopQuadAt(quad, tmp, t); 1235 dest[0] = tmp[1]; 1236 dest[1] = tmp[2]; 1237 return true; 1238 } else { 1239 /* t == 0 means either the value triggered a root outside of [0, 1) 1240 For our purposes, we can ignore the <= 0 roots, but we want to 1241 catch the >= 1 roots (which given our caller, will basically mean 1242 a root of 1, give-or-take numerical instability). If we are in the 1243 >= 1 case, return the existing offCurve point. 1244 1245 The test below checks to see if we are close to the "end" of the 1246 curve (near base[4]). Rather than specifying a tolerance, I just 1247 check to see if value is on to the right/left of the middle point 1248 (depending on the direction/sign of the end points). 1249 */ 1250 if ((base[0] < base[4] && value > base[2]) || 1251 (base[0] > base[4] && value < base[2])) // should root have been 1 1252 { 1253 dest[0] = quad[1]; 1254 dest[1].set(x, y); 1255 return true; 1256 } 1257 } 1258 return false; 1259} 1260 1261#ifdef SK_SCALAR_IS_FLOAT 1262 1263// Due to floating point issues (i.e., 1.0f - SK_ScalarRoot2Over2 != 1264// SK_ScalarRoot2Over2 - SK_ScalarTanPIOver8) a cruder root2over2 1265// approximation is required to make the quad circle points convex. The 1266// root of the problem is that with the root2over2 value in SkScalar.h 1267// the arcs really are ever so slightly concave. Some alternative fixes 1268// to this problem (besides just arbitrarily pushing out the mid-point as 1269// is done here) are: 1270// Adjust all the points (not just the middle) to both better approximate 1271// the curve and remain convex 1272// Switch over to using cubics rather then quads 1273// Use a different method to create the mid-point (e.g., compute 1274// the two side points, average them, then move it out as needed 1275#define SK_ScalarRoot2Over2_QuadCircle 0.7072f 1276 1277#else 1278 #define SK_ScalarRoot2Over2_QuadCircle SK_FixedRoot2Over2 1279#endif 1280 1281 1282static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1283 { SK_Scalar1, 0 }, 1284 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1285 { SK_ScalarRoot2Over2_QuadCircle, SK_ScalarRoot2Over2_QuadCircle }, 1286 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1287 1288 { 0, SK_Scalar1 }, 1289 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1290 { -SK_ScalarRoot2Over2_QuadCircle, SK_ScalarRoot2Over2_QuadCircle }, 1291 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1292 1293 { -SK_Scalar1, 0 }, 1294 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1295 { -SK_ScalarRoot2Over2_QuadCircle, -SK_ScalarRoot2Over2_QuadCircle }, 1296 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1297 1298 { 0, -SK_Scalar1 }, 1299 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1300 { SK_ScalarRoot2Over2_QuadCircle, -SK_ScalarRoot2Over2_QuadCircle }, 1301 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1302 1303 { SK_Scalar1, 0 } 1304}; 1305 1306int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1307 SkRotationDirection dir, const SkMatrix* userMatrix, 1308 SkPoint quadPoints[]) 1309{ 1310 // rotate by x,y so that uStart is (1.0) 1311 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1312 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1313 1314 SkScalar absX = SkScalarAbs(x); 1315 SkScalar absY = SkScalarAbs(y); 1316 1317 int pointCount; 1318 1319 // check for (effectively) coincident vectors 1320 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1321 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1322 if (absY <= SK_ScalarNearlyZero && x > 0 && 1323 ((y >= 0 && kCW_SkRotationDirection == dir) || 1324 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1325 1326 // just return the start-point 1327 quadPoints[0].set(SK_Scalar1, 0); 1328 pointCount = 1; 1329 } else { 1330 if (dir == kCCW_SkRotationDirection) 1331 y = -y; 1332 1333 // what octant (quadratic curve) is [xy] in? 1334 int oct = 0; 1335 bool sameSign = true; 1336 1337 if (0 == y) 1338 { 1339 oct = 4; // 180 1340 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1341 } 1342 else if (0 == x) 1343 { 1344 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1345 if (y > 0) 1346 oct = 2; // 90 1347 else 1348 oct = 6; // 270 1349 } 1350 else 1351 { 1352 if (y < 0) 1353 oct += 4; 1354 if ((x < 0) != (y < 0)) 1355 { 1356 oct += 2; 1357 sameSign = false; 1358 } 1359 if ((absX < absY) == sameSign) 1360 oct += 1; 1361 } 1362 1363 int wholeCount = oct << 1; 1364 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1365 1366 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1367 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) 1368 { 1369 wholeCount += 2; 1370 } 1371 pointCount = wholeCount + 1; 1372 } 1373 1374 // now handle counter-clockwise and the initial unitStart rotation 1375 SkMatrix matrix; 1376 matrix.setSinCos(uStart.fY, uStart.fX); 1377 if (dir == kCCW_SkRotationDirection) { 1378 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1379 } 1380 if (userMatrix) { 1381 matrix.postConcat(*userMatrix); 1382 } 1383 matrix.mapPoints(quadPoints, pointCount); 1384 return pointCount; 1385} 1386 1387/////////////////////////////////////////////////////////////////////////////// 1388 1389// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1390// ------------------------------------------ 1391// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1392// 1393// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1394// ------------------------------------------------ 1395// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1396// 1397 1398// Take the parametric specification for the conic (either X or Y) and return 1399// in coeff[] the coefficients for the simple quadratic polynomial 1400// coeff[0] for t^2 1401// coeff[1] for t 1402// coeff[2] for constant term 1403// 1404static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1405 SkASSERT(src); 1406 SkASSERT(t >= 0 && t <= SK_Scalar1); 1407 1408 SkScalar src2w = SkScalarMul(src[2], w); 1409 SkScalar C = src[0]; 1410 SkScalar A = src[4] - 2 * src2w + C; 1411 SkScalar B = 2 * (src2w - C); 1412 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1413 1414 B = 2 * (w - SK_Scalar1); 1415 C = SK_Scalar1; 1416 A = -B; 1417 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1418 1419 return SkScalarDiv(numer, denom); 1420} 1421 1422// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1423// 1424// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1425// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1426// t^0 : -2 P0 w + 2 P1 w 1427// 1428// We disregard magnitude, so we can freely ignore the denominator of F', and 1429// divide the numerator by 2 1430// 1431// coeff[0] for t^2 1432// coeff[1] for t^1 1433// coeff[2] for t^0 1434// 1435static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1436 const SkScalar P20 = src[4] - src[0]; 1437 const SkScalar P10 = src[2] - src[0]; 1438 const SkScalar wP10 = w * P10; 1439 coeff[0] = w * P20 - P20; 1440 coeff[1] = P20 - 2 * wP10; 1441 coeff[2] = wP10; 1442} 1443 1444static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1445 SkScalar coeff[3]; 1446 conic_deriv_coeff(coord, w, coeff); 1447 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1448} 1449 1450static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1451 SkScalar coeff[3]; 1452 conic_deriv_coeff(src, w, coeff); 1453 1454 SkScalar tValues[2]; 1455 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1456 SkASSERT(0 == roots || 1 == roots); 1457 1458 if (1 == roots) { 1459 *t = tValues[0]; 1460 return true; 1461 } 1462 return false; 1463} 1464 1465struct SkP3D { 1466 SkScalar fX, fY, fZ; 1467 1468 void set(SkScalar x, SkScalar y, SkScalar z) { 1469 fX = x; fY = y; fZ = z; 1470 } 1471 1472 void projectDown(SkPoint* dst) const { 1473 dst->set(fX / fZ, fY / fZ); 1474 } 1475}; 1476 1477// we just return the middle 3 points, since the first and last are dups of src 1478// 1479static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) { 1480 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1481 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1482 dst[0] = ab; 1483 dst[3] = SkScalarInterp(ab, bc, t); 1484 dst[6] = bc; 1485} 1486 1487static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1488 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1489 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1490 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1491} 1492 1493void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1494 SkASSERT(t >= 0 && t <= SK_Scalar1); 1495 1496 if (pt) { 1497 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1498 conic_eval_pos(&fPts[0].fY, fW, t)); 1499 } 1500 if (tangent) { 1501 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1502 conic_eval_tan(&fPts[0].fY, fW, t)); 1503 } 1504} 1505 1506void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1507 SkP3D tmp[3], tmp2[3]; 1508 1509 ratquad_mapTo3D(fPts, fW, tmp); 1510 1511 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1512 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1513 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1514 1515 dst[0].fPts[0] = fPts[0]; 1516 tmp2[0].projectDown(&dst[0].fPts[1]); 1517 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1518 tmp2[2].projectDown(&dst[1].fPts[1]); 1519 dst[1].fPts[2] = fPts[2]; 1520 1521 // to put in "standard form", where w0 and w2 are both 1, we compute the 1522 // new w1 as sqrt(w1*w1/w0*w2) 1523 // or 1524 // w1 /= sqrt(w0*w2) 1525 // 1526 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1527 // 1528 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1529 dst[0].fW = tmp2[0].fZ / root; 1530 dst[1].fW = tmp2[2].fZ / root; 1531} 1532 1533static SkScalar subdivide_w_value(SkScalar w) { 1534 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1535} 1536 1537void SkConic::chop(SkConic dst[2]) const { 1538 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1539 SkScalar p1x = fW * fPts[1].fX; 1540 SkScalar p1y = fW * fPts[1].fY; 1541 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1542 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1543 1544 dst[0].fPts[0] = fPts[0]; 1545 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1546 (fPts[0].fY + p1y) * scale); 1547 dst[0].fPts[2].set(mx, my); 1548 1549 dst[1].fPts[0].set(mx, my); 1550 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1551 (p1y + fPts[2].fY) * scale); 1552 dst[1].fPts[2] = fPts[2]; 1553 1554 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1555} 1556 1557/* 1558 * "High order approximation of conic sections by quadratic splines" 1559 * by Michael Floater, 1993 1560 */ 1561#define AS_QUAD_ERROR_SETUP \ 1562 SkScalar a = fW - 1; \ 1563 SkScalar k = a / (4 * (2 + a)); \ 1564 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1565 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1566 1567void SkConic::computeAsQuadError(SkVector* err) const { 1568 AS_QUAD_ERROR_SETUP 1569 err->set(x, y); 1570} 1571 1572bool SkConic::asQuadTol(SkScalar tol) const { 1573 AS_QUAD_ERROR_SETUP 1574 return (x * x + y * y) <= tol * tol; 1575} 1576 1577int SkConic::computeQuadPOW2(SkScalar tol) const { 1578 AS_QUAD_ERROR_SETUP 1579 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1580 1581 if (error <= 0) { 1582 return 0; 1583 } 1584 uint32_t ierr = (uint32_t)error; 1585 return (34 - SkCLZ(ierr)) >> 1; 1586} 1587 1588static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1589 SkASSERT(level >= 0); 1590 1591 if (0 == level) { 1592 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1593 return pts + 2; 1594 } else { 1595 SkConic dst[2]; 1596 src.chop(dst); 1597 --level; 1598 pts = subdivide(dst[0], pts, level); 1599 return subdivide(dst[1], pts, level); 1600 } 1601} 1602 1603int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1604 SkASSERT(pow2 >= 0); 1605 *pts = fPts[0]; 1606 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1607 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1608 return 1 << pow2; 1609} 1610 1611bool SkConic::findXExtrema(SkScalar* t) const { 1612 return conic_find_extrema(&fPts[0].fX, fW, t); 1613} 1614 1615bool SkConic::findYExtrema(SkScalar* t) const { 1616 return conic_find_extrema(&fPts[0].fY, fW, t); 1617} 1618 1619bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1620 SkScalar t; 1621 if (this->findXExtrema(&t)) { 1622 this->chopAt(t, dst); 1623 // now clean-up the middle, since we know t was meant to be at 1624 // an X-extrema 1625 SkScalar value = dst[0].fPts[2].fX; 1626 dst[0].fPts[1].fX = value; 1627 dst[1].fPts[0].fX = value; 1628 dst[1].fPts[1].fX = value; 1629 return true; 1630 } 1631 return false; 1632} 1633 1634bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1635 SkScalar t; 1636 if (this->findYExtrema(&t)) { 1637 this->chopAt(t, dst); 1638 // now clean-up the middle, since we know t was meant to be at 1639 // an Y-extrema 1640 SkScalar value = dst[0].fPts[2].fY; 1641 dst[0].fPts[1].fY = value; 1642 dst[1].fPts[0].fY = value; 1643 dst[1].fPts[1].fY = value; 1644 return true; 1645 } 1646 return false; 1647} 1648 1649void SkConic::computeTightBounds(SkRect* bounds) const { 1650 SkPoint pts[4]; 1651 pts[0] = fPts[0]; 1652 pts[1] = fPts[2]; 1653 int count = 2; 1654 1655 SkScalar t; 1656 if (this->findXExtrema(&t)) { 1657 this->evalAt(t, &pts[count++]); 1658 } 1659 if (this->findYExtrema(&t)) { 1660 this->evalAt(t, &pts[count++]); 1661 } 1662 bounds->set(pts, count); 1663} 1664 1665void SkConic::computeFastBounds(SkRect* bounds) const { 1666 bounds->set(fPts, 3); 1667} 1668