1 /* 2 * Copyright 2006 The Android Open Source Project 3 * 4 * Use of this source code is governed by a BSD-style license that can be 5 * found in the LICENSE file. 6 */ 7 8 #include "SkGeometry.h" 9 #include "SkMatrix.h" 10 #include "SkNx.h" 11 12 #if 0 13 static Sk2s from_point(const SkPoint& point) { 14 return Sk2s::Load(&point.fX); 15 } 16 17 static SkPoint to_point(const Sk2s& x) { 18 SkPoint point; 19 x.store(&point.fX); 20 return point; 21 } 22 #endif 23 24 static SkVector to_vector(const Sk2s& x) { 25 SkVector vector; 26 x.store(&vector.fX); 27 return vector; 28 } 29 30 /** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 31 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 32 May also introduce overflow of fixed when we compute our setup. 33 */ 34 // #define DIRECT_EVAL_OF_POLYNOMIALS 35 36 //////////////////////////////////////////////////////////////////////// 37 38 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 39 SkScalar ab = a - b; 40 SkScalar bc = b - c; 41 if (ab < 0) { 42 bc = -bc; 43 } 44 return ab == 0 || bc < 0; 45 } 46 47 //////////////////////////////////////////////////////////////////////// 48 49 static bool is_unit_interval(SkScalar x) { 50 return x > 0 && x < SK_Scalar1; 51 } 52 53 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 54 SkASSERT(ratio); 55 56 if (numer < 0) { 57 numer = -numer; 58 denom = -denom; 59 } 60 61 if (denom == 0 || numer == 0 || numer >= denom) { 62 return 0; 63 } 64 65 SkScalar r = numer / denom; 66 if (SkScalarIsNaN(r)) { 67 return 0; 68 } 69 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 70 if (r == 0) { // catch underflow if numer <<<< denom 71 return 0; 72 } 73 *ratio = r; 74 return 1; 75 } 76 77 /** From Numerical Recipes in C. 78 79 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 80 x1 = Q / A 81 x2 = C / Q 82 */ 83 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 84 SkASSERT(roots); 85 86 if (A == 0) { 87 return valid_unit_divide(-C, B, roots); 88 } 89 90 SkScalar* r = roots; 91 92 SkScalar R = B*B - 4*A*C; 93 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 94 return 0; 95 } 96 R = SkScalarSqrt(R); 97 98 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 99 r += valid_unit_divide(Q, A, r); 100 r += valid_unit_divide(C, Q, r); 101 if (r - roots == 2) { 102 if (roots[0] > roots[1]) 103 SkTSwap<SkScalar>(roots[0], roots[1]); 104 else if (roots[0] == roots[1]) // nearly-equal? 105 r -= 1; // skip the double root 106 } 107 return (int)(r - roots); 108 } 109 110 /////////////////////////////////////////////////////////////////////////////// 111 /////////////////////////////////////////////////////////////////////////////// 112 113 static Sk2s quad_poly_eval(const Sk2s& A, const Sk2s& B, const Sk2s& C, const Sk2s& t) { 114 return (A * t + B) * t + C; 115 } 116 117 static SkScalar eval_quad(const SkScalar src[], SkScalar t) { 118 SkASSERT(src); 119 SkASSERT(t >= 0 && t <= SK_Scalar1); 120 121 #ifdef DIRECT_EVAL_OF_POLYNOMIALS 122 SkScalar C = src[0]; 123 SkScalar A = src[4] - 2 * src[2] + C; 124 SkScalar B = 2 * (src[2] - C); 125 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 126 #else 127 SkScalar ab = SkScalarInterp(src[0], src[2], t); 128 SkScalar bc = SkScalarInterp(src[2], src[4], t); 129 return SkScalarInterp(ab, bc, t); 130 #endif 131 } 132 133 static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) { 134 SkScalar A = src[4] - 2 * src[2] + src[0]; 135 SkScalar B = src[2] - src[0]; 136 137 return 2 * SkScalarMulAdd(A, t, B); 138 } 139 140 void SkQuadToCoeff(const SkPoint pts[3], SkPoint coeff[3]) { 141 Sk2s p0 = from_point(pts[0]); 142 Sk2s p1 = from_point(pts[1]); 143 Sk2s p2 = from_point(pts[2]); 144 145 Sk2s p1minus2 = p1 - p0; 146 147 coeff[0] = to_point(p2 - p1 - p1 + p0); // A * t^2 148 coeff[1] = to_point(p1minus2 + p1minus2); // B * t 149 coeff[2] = pts[0]; // C 150 } 151 152 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 153 SkASSERT(src); 154 SkASSERT(t >= 0 && t <= SK_Scalar1); 155 156 if (pt) { 157 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 158 } 159 if (tangent) { 160 tangent->set(eval_quad_derivative(&src[0].fX, t), 161 eval_quad_derivative(&src[0].fY, t)); 162 } 163 } 164 165 SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 166 SkASSERT(src); 167 SkASSERT(t >= 0 && t <= SK_Scalar1); 168 169 const Sk2s t2(t); 170 171 Sk2s P0 = from_point(src[0]); 172 Sk2s P1 = from_point(src[1]); 173 Sk2s P2 = from_point(src[2]); 174 175 Sk2s B = P1 - P0; 176 Sk2s A = P2 - P1 - B; 177 178 return to_point((A * t2 + B+B) * t2 + P0); 179 } 180 181 SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 182 SkASSERT(src); 183 SkASSERT(t >= 0 && t <= SK_Scalar1); 184 185 Sk2s P0 = from_point(src[0]); 186 Sk2s P1 = from_point(src[1]); 187 Sk2s P2 = from_point(src[2]); 188 189 Sk2s B = P1 - P0; 190 Sk2s A = P2 - P1 - B; 191 Sk2s T = A * Sk2s(t) + B; 192 193 return to_vector(T + T); 194 } 195 196 static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 197 return v0 + (v1 - v0) * t; 198 } 199 200 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 201 SkASSERT(t > 0 && t < SK_Scalar1); 202 203 Sk2s p0 = from_point(src[0]); 204 Sk2s p1 = from_point(src[1]); 205 Sk2s p2 = from_point(src[2]); 206 Sk2s tt(t); 207 208 Sk2s p01 = interp(p0, p1, tt); 209 Sk2s p12 = interp(p1, p2, tt); 210 211 dst[0] = to_point(p0); 212 dst[1] = to_point(p01); 213 dst[2] = to_point(interp(p01, p12, tt)); 214 dst[3] = to_point(p12); 215 dst[4] = to_point(p2); 216 } 217 218 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 219 SkChopQuadAt(src, dst, 0.5f); return; 220 } 221 222 /** Quad'(t) = At + B, where 223 A = 2(a - 2b + c) 224 B = 2(b - a) 225 Solve for t, only if it fits between 0 < t < 1 226 */ 227 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 228 /* At + B == 0 229 t = -B / A 230 */ 231 return valid_unit_divide(a - b, a - b - b + c, tValue); 232 } 233 234 static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 235 coords[2] = coords[6] = coords[4]; 236 } 237 238 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 239 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 240 */ 241 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 242 SkASSERT(src); 243 SkASSERT(dst); 244 245 SkScalar a = src[0].fY; 246 SkScalar b = src[1].fY; 247 SkScalar c = src[2].fY; 248 249 if (is_not_monotonic(a, b, c)) { 250 SkScalar tValue; 251 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 252 SkChopQuadAt(src, dst, tValue); 253 flatten_double_quad_extrema(&dst[0].fY); 254 return 1; 255 } 256 // if we get here, we need to force dst to be monotonic, even though 257 // we couldn't compute a unit_divide value (probably underflow). 258 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 259 } 260 dst[0].set(src[0].fX, a); 261 dst[1].set(src[1].fX, b); 262 dst[2].set(src[2].fX, c); 263 return 0; 264 } 265 266 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 267 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 268 */ 269 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 270 SkASSERT(src); 271 SkASSERT(dst); 272 273 SkScalar a = src[0].fX; 274 SkScalar b = src[1].fX; 275 SkScalar c = src[2].fX; 276 277 if (is_not_monotonic(a, b, c)) { 278 SkScalar tValue; 279 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 280 SkChopQuadAt(src, dst, tValue); 281 flatten_double_quad_extrema(&dst[0].fX); 282 return 1; 283 } 284 // if we get here, we need to force dst to be monotonic, even though 285 // we couldn't compute a unit_divide value (probably underflow). 286 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 287 } 288 dst[0].set(a, src[0].fY); 289 dst[1].set(b, src[1].fY); 290 dst[2].set(c, src[2].fY); 291 return 0; 292 } 293 294 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 295 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t 296 // F''(t) = 2 (a - 2b + c) 297 // 298 // A = 2 (b - a) 299 // B = 2 (a - 2b + c) 300 // 301 // Maximum curvature for a quadratic means solving 302 // Fx' Fx'' + Fy' Fy'' = 0 303 // 304 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 305 // 306 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 307 SkScalar Ax = src[1].fX - src[0].fX; 308 SkScalar Ay = src[1].fY - src[0].fY; 309 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 310 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 311 SkScalar t = 0; // 0 means don't chop 312 313 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 314 return t; 315 } 316 317 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 318 SkScalar t = SkFindQuadMaxCurvature(src); 319 if (t == 0) { 320 memcpy(dst, src, 3 * sizeof(SkPoint)); 321 return 1; 322 } else { 323 SkChopQuadAt(src, dst, t); 324 return 2; 325 } 326 } 327 328 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 329 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 330 Sk2s s0 = from_point(src[0]); 331 Sk2s s1 = from_point(src[1]); 332 Sk2s s2 = from_point(src[2]); 333 334 dst[0] = src[0]; 335 dst[1] = to_point(s0 + (s1 - s0) * scale); 336 dst[2] = to_point(s2 + (s1 - s2) * scale); 337 dst[3] = src[2]; 338 } 339 340 ////////////////////////////////////////////////////////////////////////////// 341 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 342 ////////////////////////////////////////////////////////////////////////////// 343 344 static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 345 SkASSERT(src); 346 SkASSERT(t >= 0 && t <= SK_Scalar1); 347 348 if (t == 0) { 349 return src[0]; 350 } 351 352 #ifdef DIRECT_EVAL_OF_POLYNOMIALS 353 SkScalar D = src[0]; 354 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 355 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 356 SkScalar C = 3*(src[2] - D); 357 358 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 359 #else 360 SkScalar ab = SkScalarInterp(src[0], src[2], t); 361 SkScalar bc = SkScalarInterp(src[2], src[4], t); 362 SkScalar cd = SkScalarInterp(src[4], src[6], t); 363 SkScalar abc = SkScalarInterp(ab, bc, t); 364 SkScalar bcd = SkScalarInterp(bc, cd, t); 365 return SkScalarInterp(abc, bcd, t); 366 #endif 367 } 368 369 /** return At^2 + Bt + C 370 */ 371 static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { 372 SkASSERT(t >= 0 && t <= SK_Scalar1); 373 374 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 375 } 376 377 static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { 378 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 379 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 380 SkScalar C = src[2] - src[0]; 381 382 return eval_quadratic(A, B, C, t); 383 } 384 385 static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { 386 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 387 SkScalar B = src[4] - 2 * src[2] + src[0]; 388 389 return SkScalarMulAdd(A, t, B); 390 } 391 392 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 393 SkVector* tangent, SkVector* curvature) { 394 SkASSERT(src); 395 SkASSERT(t >= 0 && t <= SK_Scalar1); 396 397 if (loc) { 398 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 399 } 400 if (tangent) { 401 tangent->set(eval_cubic_derivative(&src[0].fX, t), 402 eval_cubic_derivative(&src[0].fY, t)); 403 } 404 if (curvature) { 405 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 406 eval_cubic_2ndDerivative(&src[0].fY, t)); 407 } 408 } 409 410 /** Cubic'(t) = At^2 + Bt + C, where 411 A = 3(-a + 3(b - c) + d) 412 B = 6(a - 2b + c) 413 C = 3(b - a) 414 Solve for t, keeping only those that fit betwee 0 < t < 1 415 */ 416 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 417 SkScalar tValues[2]) { 418 // we divide A,B,C by 3 to simplify 419 SkScalar A = d - a + 3*(b - c); 420 SkScalar B = 2*(a - b - b + c); 421 SkScalar C = b - a; 422 423 return SkFindUnitQuadRoots(A, B, C, tValues); 424 } 425 426 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 427 SkASSERT(t > 0 && t < SK_Scalar1); 428 429 Sk2s p0 = from_point(src[0]); 430 Sk2s p1 = from_point(src[1]); 431 Sk2s p2 = from_point(src[2]); 432 Sk2s p3 = from_point(src[3]); 433 Sk2s tt(t); 434 435 Sk2s ab = interp(p0, p1, tt); 436 Sk2s bc = interp(p1, p2, tt); 437 Sk2s cd = interp(p2, p3, tt); 438 Sk2s abc = interp(ab, bc, tt); 439 Sk2s bcd = interp(bc, cd, tt); 440 Sk2s abcd = interp(abc, bcd, tt); 441 442 dst[0] = src[0]; 443 dst[1] = to_point(ab); 444 dst[2] = to_point(abc); 445 dst[3] = to_point(abcd); 446 dst[4] = to_point(bcd); 447 dst[5] = to_point(cd); 448 dst[6] = src[3]; 449 } 450 451 void SkCubicToCoeff(const SkPoint pts[4], SkPoint coeff[4]) { 452 Sk2s p0 = from_point(pts[0]); 453 Sk2s p1 = from_point(pts[1]); 454 Sk2s p2 = from_point(pts[2]); 455 Sk2s p3 = from_point(pts[3]); 456 457 const Sk2s three(3); 458 Sk2s p1minusp2 = p1 - p2; 459 460 Sk2s D = p0; 461 Sk2s A = p3 + three * p1minusp2 - D; 462 Sk2s B = three * (D - p1minusp2 - p1); 463 Sk2s C = three * (p1 - D); 464 465 coeff[0] = to_point(A); 466 coeff[1] = to_point(B); 467 coeff[2] = to_point(C); 468 coeff[3] = to_point(D); 469 } 470 471 /* http://code.google.com/p/skia/issues/detail?id=32 472 473 This test code would fail when we didn't check the return result of 474 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 475 that after the first chop, the parameters to valid_unit_divide are equal 476 (thanks to finite float precision and rounding in the subtracts). Thus 477 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 478 up with 1.0, hence the need to check and just return the last cubic as 479 a degenerate clump of 4 points in the sampe place. 480 481 static void test_cubic() { 482 SkPoint src[4] = { 483 { 556.25000, 523.03003 }, 484 { 556.23999, 522.96002 }, 485 { 556.21997, 522.89001 }, 486 { 556.21997, 522.82001 } 487 }; 488 SkPoint dst[10]; 489 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 490 SkChopCubicAt(src, dst, tval, 2); 491 } 492 */ 493 494 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 495 const SkScalar tValues[], int roots) { 496 #ifdef SK_DEBUG 497 { 498 for (int i = 0; i < roots - 1; i++) 499 { 500 SkASSERT(is_unit_interval(tValues[i])); 501 SkASSERT(is_unit_interval(tValues[i+1])); 502 SkASSERT(tValues[i] < tValues[i+1]); 503 } 504 } 505 #endif 506 507 if (dst) { 508 if (roots == 0) { // nothing to chop 509 memcpy(dst, src, 4*sizeof(SkPoint)); 510 } else { 511 SkScalar t = tValues[0]; 512 SkPoint tmp[4]; 513 514 for (int i = 0; i < roots; i++) { 515 SkChopCubicAt(src, dst, t); 516 if (i == roots - 1) { 517 break; 518 } 519 520 dst += 3; 521 // have src point to the remaining cubic (after the chop) 522 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 523 src = tmp; 524 525 // watch out in case the renormalized t isn't in range 526 if (!valid_unit_divide(tValues[i+1] - tValues[i], 527 SK_Scalar1 - tValues[i], &t)) { 528 // if we can't, just create a degenerate cubic 529 dst[4] = dst[5] = dst[6] = src[3]; 530 break; 531 } 532 } 533 } 534 } 535 } 536 537 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 538 SkChopCubicAt(src, dst, 0.5f); 539 } 540 541 static void flatten_double_cubic_extrema(SkScalar coords[14]) { 542 coords[4] = coords[8] = coords[6]; 543 } 544 545 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 546 the resulting beziers are monotonic in Y. This is called by the scan 547 converter. Depending on what is returned, dst[] is treated as follows: 548 0 dst[0..3] is the original cubic 549 1 dst[0..3] and dst[3..6] are the two new cubics 550 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 551 If dst == null, it is ignored and only the count is returned. 552 */ 553 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 554 SkScalar tValues[2]; 555 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 556 src[3].fY, tValues); 557 558 SkChopCubicAt(src, dst, tValues, roots); 559 if (dst && roots > 0) { 560 // we do some cleanup to ensure our Y extrema are flat 561 flatten_double_cubic_extrema(&dst[0].fY); 562 if (roots == 2) { 563 flatten_double_cubic_extrema(&dst[3].fY); 564 } 565 } 566 return roots; 567 } 568 569 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 570 SkScalar tValues[2]; 571 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 572 src[3].fX, tValues); 573 574 SkChopCubicAt(src, dst, tValues, roots); 575 if (dst && roots > 0) { 576 // we do some cleanup to ensure our Y extrema are flat 577 flatten_double_cubic_extrema(&dst[0].fX); 578 if (roots == 2) { 579 flatten_double_cubic_extrema(&dst[3].fX); 580 } 581 } 582 return roots; 583 } 584 585 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 586 587 Inflection means that curvature is zero. 588 Curvature is [F' x F''] / [F'^3] 589 So we solve F'x X F''y - F'y X F''y == 0 590 After some canceling of the cubic term, we get 591 A = b - a 592 B = c - 2b + a 593 C = d - 3c + 3b - a 594 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 595 */ 596 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 597 SkScalar Ax = src[1].fX - src[0].fX; 598 SkScalar Ay = src[1].fY - src[0].fY; 599 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 600 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 601 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 602 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 603 604 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 605 Ax*Cy - Ay*Cx, 606 Ax*By - Ay*Bx, 607 tValues); 608 } 609 610 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 611 SkScalar tValues[2]; 612 int count = SkFindCubicInflections(src, tValues); 613 614 if (dst) { 615 if (count == 0) { 616 memcpy(dst, src, 4 * sizeof(SkPoint)); 617 } else { 618 SkChopCubicAt(src, dst, tValues, count); 619 } 620 } 621 return count + 1; 622 } 623 624 // See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3) 625 // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 626 // Classification: 627 // discr(I) > 0 Serpentine 628 // discr(I) = 0 Cusp 629 // discr(I) < 0 Loop 630 // d0 = d1 = 0 Quadratic 631 // d0 = d1 = d2 = 0 Line 632 // p0 = p1 = p2 = p3 Point 633 static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 634 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 635 return kPoint_SkCubicType; 636 } 637 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 638 if (discr > SK_ScalarNearlyZero) { 639 return kSerpentine_SkCubicType; 640 } else if (discr < -SK_ScalarNearlyZero) { 641 return kLoop_SkCubicType; 642 } else { 643 if (0.f == d[0] && 0.f == d[1]) { 644 return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType); 645 } else { 646 return kCusp_SkCubicType; 647 } 648 } 649 } 650 651 // Assumes the third component of points is 1. 652 // Calcs p0 . (p1 x p2) 653 static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 654 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 655 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 656 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 657 return (xComp + yComp + wComp); 658 } 659 660 // Calc coefficients of I(s,t) where roots of I are inflection points of curve 661 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 662 // d0 = a1 - 2*a2+3*a3 663 // d1 = -a2 + 3*a3 664 // d2 = 3*a3 665 // a1 = p0 . (p3 x p2) 666 // a2 = p1 . (p0 x p3) 667 // a3 = p2 . (p1 x p0) 668 // Places the values of d1, d2, d3 in array d passed in 669 static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 670 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 671 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 672 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 673 674 // need to scale a's or values in later calculations will grow to high 675 SkScalar max = SkScalarAbs(a1); 676 max = SkMaxScalar(max, SkScalarAbs(a2)); 677 max = SkMaxScalar(max, SkScalarAbs(a3)); 678 max = 1.f/max; 679 a1 = a1 * max; 680 a2 = a2 * max; 681 a3 = a3 * max; 682 683 d[2] = 3.f * a3; 684 d[1] = d[2] - a2; 685 d[0] = d[1] - a2 + a1; 686 } 687 688 SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) { 689 calc_cubic_inflection_func(src, d); 690 return classify_cubic(src, d); 691 } 692 693 template <typename T> void bubble_sort(T array[], int count) { 694 for (int i = count - 1; i > 0; --i) 695 for (int j = i; j > 0; --j) 696 if (array[j] < array[j-1]) 697 { 698 T tmp(array[j]); 699 array[j] = array[j-1]; 700 array[j-1] = tmp; 701 } 702 } 703 704 /** 705 * Given an array and count, remove all pair-wise duplicates from the array, 706 * keeping the existing sorting, and return the new count 707 */ 708 static int collaps_duplicates(SkScalar array[], int count) { 709 for (int n = count; n > 1; --n) { 710 if (array[0] == array[1]) { 711 for (int i = 1; i < n; ++i) { 712 array[i - 1] = array[i]; 713 } 714 count -= 1; 715 } else { 716 array += 1; 717 } 718 } 719 return count; 720 } 721 722 #ifdef SK_DEBUG 723 724 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 725 726 static void test_collaps_duplicates() { 727 static bool gOnce; 728 if (gOnce) { return; } 729 gOnce = true; 730 const SkScalar src0[] = { 0 }; 731 const SkScalar src1[] = { 0, 0 }; 732 const SkScalar src2[] = { 0, 1 }; 733 const SkScalar src3[] = { 0, 0, 0 }; 734 const SkScalar src4[] = { 0, 0, 1 }; 735 const SkScalar src5[] = { 0, 1, 1 }; 736 const SkScalar src6[] = { 0, 1, 2 }; 737 const struct { 738 const SkScalar* fData; 739 int fCount; 740 int fCollapsedCount; 741 } data[] = { 742 { TEST_COLLAPS_ENTRY(src0), 1 }, 743 { TEST_COLLAPS_ENTRY(src1), 1 }, 744 { TEST_COLLAPS_ENTRY(src2), 2 }, 745 { TEST_COLLAPS_ENTRY(src3), 1 }, 746 { TEST_COLLAPS_ENTRY(src4), 2 }, 747 { TEST_COLLAPS_ENTRY(src5), 2 }, 748 { TEST_COLLAPS_ENTRY(src6), 3 }, 749 }; 750 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 751 SkScalar dst[3]; 752 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 753 int count = collaps_duplicates(dst, data[i].fCount); 754 SkASSERT(data[i].fCollapsedCount == count); 755 for (int j = 1; j < count; ++j) { 756 SkASSERT(dst[j-1] < dst[j]); 757 } 758 } 759 } 760 #endif 761 762 static SkScalar SkScalarCubeRoot(SkScalar x) { 763 return SkScalarPow(x, 0.3333333f); 764 } 765 766 /* Solve coeff(t) == 0, returning the number of roots that 767 lie withing 0 < t < 1. 768 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 769 770 Eliminates repeated roots (so that all tValues are distinct, and are always 771 in increasing order. 772 */ 773 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 774 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 775 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 776 } 777 778 SkScalar a, b, c, Q, R; 779 780 { 781 SkASSERT(coeff[0] != 0); 782 783 SkScalar inva = SkScalarInvert(coeff[0]); 784 a = coeff[1] * inva; 785 b = coeff[2] * inva; 786 c = coeff[3] * inva; 787 } 788 Q = (a*a - b*3) / 9; 789 R = (2*a*a*a - 9*a*b + 27*c) / 54; 790 791 SkScalar Q3 = Q * Q * Q; 792 SkScalar R2MinusQ3 = R * R - Q3; 793 SkScalar adiv3 = a / 3; 794 795 SkScalar* roots = tValues; 796 SkScalar r; 797 798 if (R2MinusQ3 < 0) { // we have 3 real roots 799 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 800 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 801 802 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 803 if (is_unit_interval(r)) { 804 *roots++ = r; 805 } 806 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 807 if (is_unit_interval(r)) { 808 *roots++ = r; 809 } 810 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 811 if (is_unit_interval(r)) { 812 *roots++ = r; 813 } 814 SkDEBUGCODE(test_collaps_duplicates();) 815 816 // now sort the roots 817 int count = (int)(roots - tValues); 818 SkASSERT((unsigned)count <= 3); 819 bubble_sort(tValues, count); 820 count = collaps_duplicates(tValues, count); 821 roots = tValues + count; // so we compute the proper count below 822 } else { // we have 1 real root 823 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 824 A = SkScalarCubeRoot(A); 825 if (R > 0) { 826 A = -A; 827 } 828 if (A != 0) { 829 A += Q / A; 830 } 831 r = A - adiv3; 832 if (is_unit_interval(r)) { 833 *roots++ = r; 834 } 835 } 836 837 return (int)(roots - tValues); 838 } 839 840 /* Looking for F' dot F'' == 0 841 842 A = b - a 843 B = c - 2b + a 844 C = d - 3c + 3b - a 845 846 F' = 3Ct^2 + 6Bt + 3A 847 F'' = 6Ct + 6B 848 849 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 850 */ 851 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 852 SkScalar a = src[2] - src[0]; 853 SkScalar b = src[4] - 2 * src[2] + src[0]; 854 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 855 856 coeff[0] = c * c; 857 coeff[1] = 3 * b * c; 858 coeff[2] = 2 * b * b + c * a; 859 coeff[3] = a * b; 860 } 861 862 /* Looking for F' dot F'' == 0 863 864 A = b - a 865 B = c - 2b + a 866 C = d - 3c + 3b - a 867 868 F' = 3Ct^2 + 6Bt + 3A 869 F'' = 6Ct + 6B 870 871 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 872 */ 873 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 874 SkScalar coeffX[4], coeffY[4]; 875 int i; 876 877 formulate_F1DotF2(&src[0].fX, coeffX); 878 formulate_F1DotF2(&src[0].fY, coeffY); 879 880 for (i = 0; i < 4; i++) { 881 coeffX[i] += coeffY[i]; 882 } 883 884 SkScalar t[3]; 885 int count = solve_cubic_poly(coeffX, t); 886 int maxCount = 0; 887 888 // now remove extrema where the curvature is zero (mins) 889 // !!!! need a test for this !!!! 890 for (i = 0; i < count; i++) { 891 // if (not_min_curvature()) 892 if (t[i] > 0 && t[i] < SK_Scalar1) { 893 tValues[maxCount++] = t[i]; 894 } 895 } 896 return maxCount; 897 } 898 899 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 900 SkScalar tValues[3]) { 901 SkScalar t_storage[3]; 902 903 if (tValues == NULL) { 904 tValues = t_storage; 905 } 906 907 int count = SkFindCubicMaxCurvature(src, tValues); 908 909 if (dst) { 910 if (count == 0) { 911 memcpy(dst, src, 4 * sizeof(SkPoint)); 912 } else { 913 SkChopCubicAt(src, dst, tValues, count); 914 } 915 } 916 return count + 1; 917 } 918 919 #include "../pathops/SkPathOpsCubic.h" 920 921 typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 922 923 static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 924 InterceptProc method) { 925 SkDCubic cubic; 926 double roots[3]; 927 int count = (cubic.set(src).*method)(intercept, roots); 928 if (count > 0) { 929 SkDCubicPair pair = cubic.chopAt(roots[0]); 930 for (int i = 0; i < 7; ++i) { 931 dst[i] = pair.pts[i].asSkPoint(); 932 } 933 return true; 934 } 935 return false; 936 } 937 938 bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 939 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 940 } 941 942 bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 943 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 944 } 945 946 /////////////////////////////////////////////////////////////////////////////// 947 948 /* Find t value for quadratic [a, b, c] = d. 949 Return 0 if there is no solution within [0, 1) 950 */ 951 static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { 952 // At^2 + Bt + C = d 953 SkScalar A = a - 2 * b + c; 954 SkScalar B = 2 * (b - a); 955 SkScalar C = a - d; 956 957 SkScalar roots[2]; 958 int count = SkFindUnitQuadRoots(A, B, C, roots); 959 960 SkASSERT(count <= 1); 961 return count == 1 ? roots[0] : 0; 962 } 963 964 /* given a quad-curve and a point (x,y), chop the quad at that point and place 965 the new off-curve point and endpoint into 'dest'. 966 Should only return false if the computed pos is the start of the curve 967 (i.e. root == 0) 968 */ 969 static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, 970 SkPoint* dest) { 971 const SkScalar* base; 972 SkScalar value; 973 974 if (SkScalarAbs(x) < SkScalarAbs(y)) { 975 base = &quad[0].fX; 976 value = x; 977 } else { 978 base = &quad[0].fY; 979 value = y; 980 } 981 982 // note: this returns 0 if it thinks value is out of range, meaning the 983 // root might return something outside of [0, 1) 984 SkScalar t = quad_solve(base[0], base[2], base[4], value); 985 986 if (t > 0) { 987 SkPoint tmp[5]; 988 SkChopQuadAt(quad, tmp, t); 989 dest[0] = tmp[1]; 990 dest[1].set(x, y); 991 return true; 992 } else { 993 /* t == 0 means either the value triggered a root outside of [0, 1) 994 For our purposes, we can ignore the <= 0 roots, but we want to 995 catch the >= 1 roots (which given our caller, will basically mean 996 a root of 1, give-or-take numerical instability). If we are in the 997 >= 1 case, return the existing offCurve point. 998 999 The test below checks to see if we are close to the "end" of the 1000 curve (near base[4]). Rather than specifying a tolerance, I just 1001 check to see if value is on to the right/left of the middle point 1002 (depending on the direction/sign of the end points). 1003 */ 1004 if ((base[0] < base[4] && value > base[2]) || 1005 (base[0] > base[4] && value < base[2])) // should root have been 1 1006 { 1007 dest[0] = quad[1]; 1008 dest[1].set(x, y); 1009 return true; 1010 } 1011 } 1012 return false; 1013 } 1014 1015 static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1016 // The mid point of the quadratic arc approximation is half way between the two 1017 // control points. The float epsilon adjustment moves the on curve point out by 1018 // two bits, distributing the convex test error between the round rect 1019 // approximation and the convex cross product sign equality test. 1020 #define SK_MID_RRECT_OFFSET \ 1021 (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1022 { SK_Scalar1, 0 }, 1023 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1024 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1025 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1026 1027 { 0, SK_Scalar1 }, 1028 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1029 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1030 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1031 1032 { -SK_Scalar1, 0 }, 1033 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1034 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1035 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1036 1037 { 0, -SK_Scalar1 }, 1038 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1039 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1040 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1041 1042 { SK_Scalar1, 0 } 1043 #undef SK_MID_RRECT_OFFSET 1044 }; 1045 1046 int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1047 SkRotationDirection dir, const SkMatrix* userMatrix, 1048 SkPoint quadPoints[]) { 1049 // rotate by x,y so that uStart is (1.0) 1050 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1051 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1052 1053 SkScalar absX = SkScalarAbs(x); 1054 SkScalar absY = SkScalarAbs(y); 1055 1056 int pointCount; 1057 1058 // check for (effectively) coincident vectors 1059 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1060 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1061 if (absY <= SK_ScalarNearlyZero && x > 0 && 1062 ((y >= 0 && kCW_SkRotationDirection == dir) || 1063 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1064 1065 // just return the start-point 1066 quadPoints[0].set(SK_Scalar1, 0); 1067 pointCount = 1; 1068 } else { 1069 if (dir == kCCW_SkRotationDirection) { 1070 y = -y; 1071 } 1072 // what octant (quadratic curve) is [xy] in? 1073 int oct = 0; 1074 bool sameSign = true; 1075 1076 if (0 == y) { 1077 oct = 4; // 180 1078 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1079 } else if (0 == x) { 1080 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1081 oct = y > 0 ? 2 : 6; // 90 : 270 1082 } else { 1083 if (y < 0) { 1084 oct += 4; 1085 } 1086 if ((x < 0) != (y < 0)) { 1087 oct += 2; 1088 sameSign = false; 1089 } 1090 if ((absX < absY) == sameSign) { 1091 oct += 1; 1092 } 1093 } 1094 1095 int wholeCount = oct << 1; 1096 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1097 1098 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1099 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { 1100 wholeCount += 2; 1101 } 1102 pointCount = wholeCount + 1; 1103 } 1104 1105 // now handle counter-clockwise and the initial unitStart rotation 1106 SkMatrix matrix; 1107 matrix.setSinCos(uStart.fY, uStart.fX); 1108 if (dir == kCCW_SkRotationDirection) { 1109 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1110 } 1111 if (userMatrix) { 1112 matrix.postConcat(*userMatrix); 1113 } 1114 matrix.mapPoints(quadPoints, pointCount); 1115 return pointCount; 1116 } 1117 1118 1119 /////////////////////////////////////////////////////////////////////////////// 1120 // 1121 // NURB representation for conics. Helpful explanations at: 1122 // 1123 // http://citeseerx.ist.psu.edu/viewdoc/ 1124 // download?doi=10.1.1.44.5740&rep=rep1&type=ps 1125 // and 1126 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 1127 // 1128 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1129 // ------------------------------------------ 1130 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1131 // 1132 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1133 // ------------------------------------------------ 1134 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1135 // 1136 1137 static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1138 SkASSERT(src); 1139 SkASSERT(t >= 0 && t <= SK_Scalar1); 1140 1141 SkScalar src2w = SkScalarMul(src[2], w); 1142 SkScalar C = src[0]; 1143 SkScalar A = src[4] - 2 * src2w + C; 1144 SkScalar B = 2 * (src2w - C); 1145 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1146 1147 B = 2 * (w - SK_Scalar1); 1148 C = SK_Scalar1; 1149 A = -B; 1150 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1151 1152 return numer / denom; 1153 } 1154 1155 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1156 // 1157 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1158 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1159 // t^0 : -2 P0 w + 2 P1 w 1160 // 1161 // We disregard magnitude, so we can freely ignore the denominator of F', and 1162 // divide the numerator by 2 1163 // 1164 // coeff[0] for t^2 1165 // coeff[1] for t^1 1166 // coeff[2] for t^0 1167 // 1168 static void conic_deriv_coeff(const SkScalar src[], 1169 SkScalar w, 1170 SkScalar coeff[3]) { 1171 const SkScalar P20 = src[4] - src[0]; 1172 const SkScalar P10 = src[2] - src[0]; 1173 const SkScalar wP10 = w * P10; 1174 coeff[0] = w * P20 - P20; 1175 coeff[1] = P20 - 2 * wP10; 1176 coeff[2] = wP10; 1177 } 1178 1179 static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1180 SkScalar coeff[3]; 1181 conic_deriv_coeff(coord, w, coeff); 1182 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1183 } 1184 1185 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1186 SkScalar coeff[3]; 1187 conic_deriv_coeff(src, w, coeff); 1188 1189 SkScalar tValues[2]; 1190 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1191 SkASSERT(0 == roots || 1 == roots); 1192 1193 if (1 == roots) { 1194 *t = tValues[0]; 1195 return true; 1196 } 1197 return false; 1198 } 1199 1200 struct SkP3D { 1201 SkScalar fX, fY, fZ; 1202 1203 void set(SkScalar x, SkScalar y, SkScalar z) { 1204 fX = x; fY = y; fZ = z; 1205 } 1206 1207 void projectDown(SkPoint* dst) const { 1208 dst->set(fX / fZ, fY / fZ); 1209 } 1210 }; 1211 1212 // We only interpolate one dimension at a time (the first, at +0, +3, +6). 1213 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 1214 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1215 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1216 dst[0] = ab; 1217 dst[3] = SkScalarInterp(ab, bc, t); 1218 dst[6] = bc; 1219 } 1220 1221 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1222 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1223 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1224 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1225 } 1226 1227 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1228 SkASSERT(t >= 0 && t <= SK_Scalar1); 1229 1230 if (pt) { 1231 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1232 conic_eval_pos(&fPts[0].fY, fW, t)); 1233 } 1234 if (tangent) { 1235 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1236 conic_eval_tan(&fPts[0].fY, fW, t)); 1237 } 1238 } 1239 1240 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1241 SkP3D tmp[3], tmp2[3]; 1242 1243 ratquad_mapTo3D(fPts, fW, tmp); 1244 1245 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1246 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1247 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1248 1249 dst[0].fPts[0] = fPts[0]; 1250 tmp2[0].projectDown(&dst[0].fPts[1]); 1251 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1252 tmp2[2].projectDown(&dst[1].fPts[1]); 1253 dst[1].fPts[2] = fPts[2]; 1254 1255 // to put in "standard form", where w0 and w2 are both 1, we compute the 1256 // new w1 as sqrt(w1*w1/w0*w2) 1257 // or 1258 // w1 /= sqrt(w0*w2) 1259 // 1260 // However, in our case, we know that for dst[0]: 1261 // w0 == 1, and for dst[1], w2 == 1 1262 // 1263 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1264 dst[0].fW = tmp2[0].fZ / root; 1265 dst[1].fW = tmp2[2].fZ / root; 1266 } 1267 1268 static Sk2s times_2(const Sk2s& value) { 1269 return value + value; 1270 } 1271 1272 SkPoint SkConic::evalAt(SkScalar t) const { 1273 Sk2s p0 = from_point(fPts[0]); 1274 Sk2s p1 = from_point(fPts[1]); 1275 Sk2s p2 = from_point(fPts[2]); 1276 Sk2s tt(t); 1277 Sk2s ww(fW); 1278 Sk2s one(1); 1279 1280 Sk2s p1w = p1 * ww; 1281 Sk2s C = p0; 1282 Sk2s A = p2 - times_2(p1w) + p0; 1283 Sk2s B = times_2(p1w - C); 1284 Sk2s numer = quad_poly_eval(A, B, C, tt); 1285 1286 B = times_2(ww - one); 1287 A = -B; 1288 Sk2s denom = quad_poly_eval(A, B, one, tt); 1289 1290 return to_point(numer / denom); 1291 } 1292 1293 SkVector SkConic::evalTangentAt(SkScalar t) const { 1294 Sk2s p0 = from_point(fPts[0]); 1295 Sk2s p1 = from_point(fPts[1]); 1296 Sk2s p2 = from_point(fPts[2]); 1297 Sk2s ww(fW); 1298 1299 Sk2s p20 = p2 - p0; 1300 Sk2s p10 = p1 - p0; 1301 1302 Sk2s C = ww * p10; 1303 Sk2s A = ww * p20 - p20; 1304 Sk2s B = p20 - C - C; 1305 1306 return to_vector(quad_poly_eval(A, B, C, Sk2s(t))); 1307 } 1308 1309 static SkScalar subdivide_w_value(SkScalar w) { 1310 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1311 } 1312 1313 static Sk2s twice(const Sk2s& value) { 1314 return value + value; 1315 } 1316 1317 void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1318 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1319 SkScalar newW = subdivide_w_value(fW); 1320 1321 Sk2s p0 = from_point(fPts[0]); 1322 Sk2s p1 = from_point(fPts[1]); 1323 Sk2s p2 = from_point(fPts[2]); 1324 Sk2s ww(fW); 1325 1326 Sk2s wp1 = ww * p1; 1327 Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f); 1328 1329 dst[0].fPts[0] = fPts[0]; 1330 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1331 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1332 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1333 dst[1].fPts[2] = fPts[2]; 1334 1335 dst[0].fW = dst[1].fW = newW; 1336 } 1337 1338 /* 1339 * "High order approximation of conic sections by quadratic splines" 1340 * by Michael Floater, 1993 1341 */ 1342 #define AS_QUAD_ERROR_SETUP \ 1343 SkScalar a = fW - 1; \ 1344 SkScalar k = a / (4 * (2 + a)); \ 1345 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1346 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1347 1348 void SkConic::computeAsQuadError(SkVector* err) const { 1349 AS_QUAD_ERROR_SETUP 1350 err->set(x, y); 1351 } 1352 1353 bool SkConic::asQuadTol(SkScalar tol) const { 1354 AS_QUAD_ERROR_SETUP 1355 return (x * x + y * y) <= tol * tol; 1356 } 1357 1358 // Limit the number of suggested quads to approximate a conic 1359 #define kMaxConicToQuadPOW2 5 1360 1361 int SkConic::computeQuadPOW2(SkScalar tol) const { 1362 if (tol < 0 || !SkScalarIsFinite(tol)) { 1363 return 0; 1364 } 1365 1366 AS_QUAD_ERROR_SETUP 1367 1368 SkScalar error = SkScalarSqrt(x * x + y * y); 1369 int pow2; 1370 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1371 if (error <= tol) { 1372 break; 1373 } 1374 error *= 0.25f; 1375 } 1376 // float version -- using ceil gives the same results as the above. 1377 if (false) { 1378 SkScalar err = SkScalarSqrt(x * x + y * y); 1379 if (err <= tol) { 1380 return 0; 1381 } 1382 SkScalar tol2 = tol * tol; 1383 if (tol2 == 0) { 1384 return kMaxConicToQuadPOW2; 1385 } 1386 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1387 int altPow2 = SkScalarCeilToInt(fpow2); 1388 if (altPow2 != pow2) { 1389 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1390 } 1391 pow2 = altPow2; 1392 } 1393 return pow2; 1394 } 1395 1396 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1397 SkASSERT(level >= 0); 1398 1399 if (0 == level) { 1400 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1401 return pts + 2; 1402 } else { 1403 SkConic dst[2]; 1404 src.chop(dst); 1405 --level; 1406 pts = subdivide(dst[0], pts, level); 1407 return subdivide(dst[1], pts, level); 1408 } 1409 } 1410 1411 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1412 SkASSERT(pow2 >= 0); 1413 *pts = fPts[0]; 1414 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1415 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1416 return 1 << pow2; 1417 } 1418 1419 bool SkConic::findXExtrema(SkScalar* t) const { 1420 return conic_find_extrema(&fPts[0].fX, fW, t); 1421 } 1422 1423 bool SkConic::findYExtrema(SkScalar* t) const { 1424 return conic_find_extrema(&fPts[0].fY, fW, t); 1425 } 1426 1427 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1428 SkScalar t; 1429 if (this->findXExtrema(&t)) { 1430 this->chopAt(t, dst); 1431 // now clean-up the middle, since we know t was meant to be at 1432 // an X-extrema 1433 SkScalar value = dst[0].fPts[2].fX; 1434 dst[0].fPts[1].fX = value; 1435 dst[1].fPts[0].fX = value; 1436 dst[1].fPts[1].fX = value; 1437 return true; 1438 } 1439 return false; 1440 } 1441 1442 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1443 SkScalar t; 1444 if (this->findYExtrema(&t)) { 1445 this->chopAt(t, dst); 1446 // now clean-up the middle, since we know t was meant to be at 1447 // an Y-extrema 1448 SkScalar value = dst[0].fPts[2].fY; 1449 dst[0].fPts[1].fY = value; 1450 dst[1].fPts[0].fY = value; 1451 dst[1].fPts[1].fY = value; 1452 return true; 1453 } 1454 return false; 1455 } 1456 1457 void SkConic::computeTightBounds(SkRect* bounds) const { 1458 SkPoint pts[4]; 1459 pts[0] = fPts[0]; 1460 pts[1] = fPts[2]; 1461 int count = 2; 1462 1463 SkScalar t; 1464 if (this->findXExtrema(&t)) { 1465 this->evalAt(t, &pts[count++]); 1466 } 1467 if (this->findYExtrema(&t)) { 1468 this->evalAt(t, &pts[count++]); 1469 } 1470 bounds->set(pts, count); 1471 } 1472 1473 void SkConic::computeFastBounds(SkRect* bounds) const { 1474 bounds->set(fPts, 3); 1475 } 1476 1477 bool SkConic::findMaxCurvature(SkScalar* t) const { 1478 // TODO: Implement me 1479 return false; 1480 } 1481 1482 SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1483 const SkMatrix& matrix) { 1484 if (!matrix.hasPerspective()) { 1485 return w; 1486 } 1487 1488 SkP3D src[3], dst[3]; 1489 1490 ratquad_mapTo3D(pts, w, src); 1491 1492 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1493 1494 // w' = sqrt(w1*w1/w0*w2) 1495 SkScalar w0 = dst[0].fZ; 1496 SkScalar w1 = dst[1].fZ; 1497 SkScalar w2 = dst[2].fZ; 1498 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1499 return w; 1500 } 1501 1502 int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1503 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1504 // rotate by x,y so that uStart is (1.0) 1505 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1506 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1507 1508 SkScalar absY = SkScalarAbs(y); 1509 1510 // check for (effectively) coincident vectors 1511 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1512 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1513 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1514 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1515 return 0; 1516 } 1517 1518 if (dir == kCCW_SkRotationDirection) { 1519 y = -y; 1520 } 1521 1522 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1523 // 0 == [0 .. 90) 1524 // 1 == [90 ..180) 1525 // 2 == [180..270) 1526 // 3 == [270..360) 1527 // 1528 int quadrant = 0; 1529 if (0 == y) { 1530 quadrant = 2; // 180 1531 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1532 } else if (0 == x) { 1533 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1534 quadrant = y > 0 ? 1 : 3; // 90 : 270 1535 } else { 1536 if (y < 0) { 1537 quadrant += 2; 1538 } 1539 if ((x < 0) != (y < 0)) { 1540 quadrant += 1; 1541 } 1542 } 1543 1544 const SkPoint quadrantPts[] = { 1545 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1546 }; 1547 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1548 1549 int conicCount = quadrant; 1550 for (int i = 0; i < conicCount; ++i) { 1551 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1552 } 1553 1554 // Now compute any remaing (sub-90-degree) arc for the last conic 1555 const SkPoint finalP = { x, y }; 1556 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1557 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1558 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1559 1560 if (dot < 1) { 1561 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1562 // compute the bisector vector, and then rescale to be the off-curve point. 1563 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1564 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1565 // This is nice, since our computed weight is cos(theta/2) as well! 1566 // 1567 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1568 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1569 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1570 conicCount += 1; 1571 } 1572 1573 // now handle counter-clockwise and the initial unitStart rotation 1574 SkMatrix matrix; 1575 matrix.setSinCos(uStart.fY, uStart.fX); 1576 if (dir == kCCW_SkRotationDirection) { 1577 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1578 } 1579 if (userMatrix) { 1580 matrix.postConcat(*userMatrix); 1581 } 1582 for (int i = 0; i < conicCount; ++i) { 1583 matrix.mapPoints(dst[i].fPts, 3); 1584 } 1585 return conicCount; 1586 } 1587
