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 14 bool 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 //////////////////////////////////////////////////////////////////////// 99 100 static bool is_unit_interval(SkScalar x) 101 { 102 return x > 0 && x < SK_Scalar1; 103 } 104 105 static 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 */ 135 int 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 */ 179 static 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 ///////////////////////////////////////////////////////////////////////////////////// 194 195 static 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 } 211 212 static 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 } 219 220 static 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 } 226 227 void 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 } 238 239 void 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 } 255 256 static 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 } 267 268 void 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 } 275 276 void 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 */ 295 int 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 } 306 307 static 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 */ 315 int 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 */ 359 int 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 // 397 float 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 } 431 432 int 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 449 450 void 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 //////////////////////////////////////////////////////////////////////////////////////// 463 464 static 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 } 471 472 void 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 } 481 482 static 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 */ 509 static 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 } 515 516 static 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 } 524 525 static 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 } 532 533 void 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 */ 554 int 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 } 568 569 static 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 } 586 587 void 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 */ 617 618 void 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 } 662 663 void 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 } 685 686 static 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 */ 699 int 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 } 714 715 int 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 */ 742 int 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 } 774 775 int 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 } 789 790 template <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 0 806 static 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 */ 842 static 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) 859 860 static 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 */ 907 static 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 */ 998 static 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 */ 1030 int 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 1056 int 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 1074 bool 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 = 0; 1138 } else { 1139 upper_t = 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 1161 int 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 */ 1193 static 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 */ 1212 static 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 static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1260 // The mid point of the quadratic arc approximation is half way between the two 1261 // control points. The float epsilon adjustment moves the on curve point out by 1262 // two bits, distributing the convex test error between the round rect approximation 1263 // and the convex cross product sign equality test. 1264 #define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1265 { SK_Scalar1, 0 }, 1266 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1267 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1268 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1269 1270 { 0, SK_Scalar1 }, 1271 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1272 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1273 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1274 1275 { -SK_Scalar1, 0 }, 1276 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1277 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1278 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1279 1280 { 0, -SK_Scalar1 }, 1281 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1282 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1283 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1284 1285 { SK_Scalar1, 0 } 1286 #undef SK_MID_RRECT_OFFSET 1287 }; 1288 1289 int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1290 SkRotationDirection dir, const SkMatrix* userMatrix, 1291 SkPoint quadPoints[]) 1292 { 1293 // rotate by x,y so that uStart is (1.0) 1294 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1295 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1296 1297 SkScalar absX = SkScalarAbs(x); 1298 SkScalar absY = SkScalarAbs(y); 1299 1300 int pointCount; 1301 1302 // check for (effectively) coincident vectors 1303 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1304 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1305 if (absY <= SK_ScalarNearlyZero && x > 0 && 1306 ((y >= 0 && kCW_SkRotationDirection == dir) || 1307 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1308 1309 // just return the start-point 1310 quadPoints[0].set(SK_Scalar1, 0); 1311 pointCount = 1; 1312 } else { 1313 if (dir == kCCW_SkRotationDirection) 1314 y = -y; 1315 1316 // what octant (quadratic curve) is [xy] in? 1317 int oct = 0; 1318 bool sameSign = true; 1319 1320 if (0 == y) 1321 { 1322 oct = 4; // 180 1323 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1324 } 1325 else if (0 == x) 1326 { 1327 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1328 if (y > 0) 1329 oct = 2; // 90 1330 else 1331 oct = 6; // 270 1332 } 1333 else 1334 { 1335 if (y < 0) 1336 oct += 4; 1337 if ((x < 0) != (y < 0)) 1338 { 1339 oct += 2; 1340 sameSign = false; 1341 } 1342 if ((absX < absY) == sameSign) 1343 oct += 1; 1344 } 1345 1346 int wholeCount = oct << 1; 1347 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1348 1349 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1350 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) 1351 { 1352 wholeCount += 2; 1353 } 1354 pointCount = wholeCount + 1; 1355 } 1356 1357 // now handle counter-clockwise and the initial unitStart rotation 1358 SkMatrix matrix; 1359 matrix.setSinCos(uStart.fY, uStart.fX); 1360 if (dir == kCCW_SkRotationDirection) { 1361 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1362 } 1363 if (userMatrix) { 1364 matrix.postConcat(*userMatrix); 1365 } 1366 matrix.mapPoints(quadPoints, pointCount); 1367 return pointCount; 1368 } 1369 1370 /////////////////////////////////////////////////////////////////////////////// 1371 1372 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1373 // ------------------------------------------ 1374 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1375 // 1376 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1377 // ------------------------------------------------ 1378 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1379 // 1380 1381 // Take the parametric specification for the conic (either X or Y) and return 1382 // in coeff[] the coefficients for the simple quadratic polynomial 1383 // coeff[0] for t^2 1384 // coeff[1] for t 1385 // coeff[2] for constant term 1386 // 1387 static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1388 SkASSERT(src); 1389 SkASSERT(t >= 0 && t <= SK_Scalar1); 1390 1391 SkScalar src2w = SkScalarMul(src[2], w); 1392 SkScalar C = src[0]; 1393 SkScalar A = src[4] - 2 * src2w + C; 1394 SkScalar B = 2 * (src2w - C); 1395 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1396 1397 B = 2 * (w - SK_Scalar1); 1398 C = SK_Scalar1; 1399 A = -B; 1400 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1401 1402 return SkScalarDiv(numer, denom); 1403 } 1404 1405 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1406 // 1407 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1408 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1409 // t^0 : -2 P0 w + 2 P1 w 1410 // 1411 // We disregard magnitude, so we can freely ignore the denominator of F', and 1412 // divide the numerator by 2 1413 // 1414 // coeff[0] for t^2 1415 // coeff[1] for t^1 1416 // coeff[2] for t^0 1417 // 1418 static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) { 1419 const SkScalar P20 = src[4] - src[0]; 1420 const SkScalar P10 = src[2] - src[0]; 1421 const SkScalar wP10 = w * P10; 1422 coeff[0] = w * P20 - P20; 1423 coeff[1] = P20 - 2 * wP10; 1424 coeff[2] = wP10; 1425 } 1426 1427 static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1428 SkScalar coeff[3]; 1429 conic_deriv_coeff(coord, w, coeff); 1430 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1431 } 1432 1433 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1434 SkScalar coeff[3]; 1435 conic_deriv_coeff(src, w, coeff); 1436 1437 SkScalar tValues[2]; 1438 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1439 SkASSERT(0 == roots || 1 == roots); 1440 1441 if (1 == roots) { 1442 *t = tValues[0]; 1443 return true; 1444 } 1445 return false; 1446 } 1447 1448 struct SkP3D { 1449 SkScalar fX, fY, fZ; 1450 1451 void set(SkScalar x, SkScalar y, SkScalar z) { 1452 fX = x; fY = y; fZ = z; 1453 } 1454 1455 void projectDown(SkPoint* dst) const { 1456 dst->set(fX / fZ, fY / fZ); 1457 } 1458 }; 1459 1460 // we just return the middle 3 points, since the first and last are dups of src 1461 // 1462 static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) { 1463 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1464 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1465 dst[0] = ab; 1466 dst[3] = SkScalarInterp(ab, bc, t); 1467 dst[6] = bc; 1468 } 1469 1470 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1471 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1472 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1473 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1474 } 1475 1476 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1477 SkASSERT(t >= 0 && t <= SK_Scalar1); 1478 1479 if (pt) { 1480 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1481 conic_eval_pos(&fPts[0].fY, fW, t)); 1482 } 1483 if (tangent) { 1484 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1485 conic_eval_tan(&fPts[0].fY, fW, t)); 1486 } 1487 } 1488 1489 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1490 SkP3D tmp[3], tmp2[3]; 1491 1492 ratquad_mapTo3D(fPts, fW, tmp); 1493 1494 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1495 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1496 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1497 1498 dst[0].fPts[0] = fPts[0]; 1499 tmp2[0].projectDown(&dst[0].fPts[1]); 1500 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1501 tmp2[2].projectDown(&dst[1].fPts[1]); 1502 dst[1].fPts[2] = fPts[2]; 1503 1504 // to put in "standard form", where w0 and w2 are both 1, we compute the 1505 // new w1 as sqrt(w1*w1/w0*w2) 1506 // or 1507 // w1 /= sqrt(w0*w2) 1508 // 1509 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1 1510 // 1511 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1512 dst[0].fW = tmp2[0].fZ / root; 1513 dst[1].fW = tmp2[2].fZ / root; 1514 } 1515 1516 static SkScalar subdivide_w_value(SkScalar w) { 1517 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1518 } 1519 1520 void SkConic::chop(SkConic dst[2]) const { 1521 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1522 SkScalar p1x = fW * fPts[1].fX; 1523 SkScalar p1y = fW * fPts[1].fY; 1524 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1525 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1526 1527 dst[0].fPts[0] = fPts[0]; 1528 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1529 (fPts[0].fY + p1y) * scale); 1530 dst[0].fPts[2].set(mx, my); 1531 1532 dst[1].fPts[0].set(mx, my); 1533 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1534 (p1y + fPts[2].fY) * scale); 1535 dst[1].fPts[2] = fPts[2]; 1536 1537 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1538 } 1539 1540 /* 1541 * "High order approximation of conic sections by quadratic splines" 1542 * by Michael Floater, 1993 1543 */ 1544 #define AS_QUAD_ERROR_SETUP \ 1545 SkScalar a = fW - 1; \ 1546 SkScalar k = a / (4 * (2 + a)); \ 1547 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1548 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1549 1550 void SkConic::computeAsQuadError(SkVector* err) const { 1551 AS_QUAD_ERROR_SETUP 1552 err->set(x, y); 1553 } 1554 1555 bool SkConic::asQuadTol(SkScalar tol) const { 1556 AS_QUAD_ERROR_SETUP 1557 return (x * x + y * y) <= tol * tol; 1558 } 1559 1560 int SkConic::computeQuadPOW2(SkScalar tol) const { 1561 AS_QUAD_ERROR_SETUP 1562 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1563 1564 if (error <= 0) { 1565 return 0; 1566 } 1567 uint32_t ierr = (uint32_t)error; 1568 return (34 - SkCLZ(ierr)) >> 1; 1569 } 1570 1571 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1572 SkASSERT(level >= 0); 1573 1574 if (0 == level) { 1575 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1576 return pts + 2; 1577 } else { 1578 SkConic dst[2]; 1579 src.chop(dst); 1580 --level; 1581 pts = subdivide(dst[0], pts, level); 1582 return subdivide(dst[1], pts, level); 1583 } 1584 } 1585 1586 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1587 SkASSERT(pow2 >= 0); 1588 *pts = fPts[0]; 1589 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1590 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1591 return 1 << pow2; 1592 } 1593 1594 bool SkConic::findXExtrema(SkScalar* t) const { 1595 return conic_find_extrema(&fPts[0].fX, fW, t); 1596 } 1597 1598 bool SkConic::findYExtrema(SkScalar* t) const { 1599 return conic_find_extrema(&fPts[0].fY, fW, t); 1600 } 1601 1602 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1603 SkScalar t; 1604 if (this->findXExtrema(&t)) { 1605 this->chopAt(t, dst); 1606 // now clean-up the middle, since we know t was meant to be at 1607 // an X-extrema 1608 SkScalar value = dst[0].fPts[2].fX; 1609 dst[0].fPts[1].fX = value; 1610 dst[1].fPts[0].fX = value; 1611 dst[1].fPts[1].fX = value; 1612 return true; 1613 } 1614 return false; 1615 } 1616 1617 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1618 SkScalar t; 1619 if (this->findYExtrema(&t)) { 1620 this->chopAt(t, dst); 1621 // now clean-up the middle, since we know t was meant to be at 1622 // an Y-extrema 1623 SkScalar value = dst[0].fPts[2].fY; 1624 dst[0].fPts[1].fY = value; 1625 dst[1].fPts[0].fY = value; 1626 dst[1].fPts[1].fY = value; 1627 return true; 1628 } 1629 return false; 1630 } 1631 1632 void SkConic::computeTightBounds(SkRect* bounds) const { 1633 SkPoint pts[4]; 1634 pts[0] = fPts[0]; 1635 pts[1] = fPts[2]; 1636 int count = 2; 1637 1638 SkScalar t; 1639 if (this->findXExtrema(&t)) { 1640 this->evalAt(t, &pts[count++]); 1641 } 1642 if (this->findYExtrema(&t)) { 1643 this->evalAt(t, &pts[count++]); 1644 } 1645 bounds->set(pts, count); 1646 } 1647 1648 void SkConic::computeFastBounds(SkRect* bounds) const { 1649 bounds->set(fPts, 3); 1650 } 1651
