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 static SkVector to_vector(const Sk2s& x) { 13 SkVector vector; 14 x.store(&vector); 15 return vector; 16 } 17 18 /** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 19 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 20 May also introduce overflow of fixed when we compute our setup. 21 */ 22 // #define DIRECT_EVAL_OF_POLYNOMIALS 23 24 //////////////////////////////////////////////////////////////////////// 25 26 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 27 SkScalar ab = a - b; 28 SkScalar bc = b - c; 29 if (ab < 0) { 30 bc = -bc; 31 } 32 return ab == 0 || bc < 0; 33 } 34 35 //////////////////////////////////////////////////////////////////////// 36 37 static bool is_unit_interval(SkScalar x) { 38 return x > 0 && x < SK_Scalar1; 39 } 40 41 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 42 SkASSERT(ratio); 43 44 if (numer < 0) { 45 numer = -numer; 46 denom = -denom; 47 } 48 49 if (denom == 0 || numer == 0 || numer >= denom) { 50 return 0; 51 } 52 53 SkScalar r = numer / denom; 54 if (SkScalarIsNaN(r)) { 55 return 0; 56 } 57 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 58 if (r == 0) { // catch underflow if numer <<<< denom 59 return 0; 60 } 61 *ratio = r; 62 return 1; 63 } 64 65 /** From Numerical Recipes in C. 66 67 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 68 x1 = Q / A 69 x2 = C / Q 70 */ 71 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 72 SkASSERT(roots); 73 74 if (A == 0) { 75 return valid_unit_divide(-C, B, roots); 76 } 77 78 SkScalar* r = roots; 79 80 SkScalar R = B*B - 4*A*C; 81 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 82 // if R is infinite, it's possible that it may still produce 83 // useful results if the operation was repeated in doubles 84 // the flipside is determining if the more precise answer 85 // isn't useful because surrounding machinery (e.g., subtracting 86 // the axis offset from C) already discards the extra precision 87 // more investigation and unit tests required... 88 return 0; 89 } 90 R = SkScalarSqrt(R); 91 92 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 93 r += valid_unit_divide(Q, A, r); 94 r += valid_unit_divide(C, Q, r); 95 if (r - roots == 2) { 96 if (roots[0] > roots[1]) 97 SkTSwap<SkScalar>(roots[0], roots[1]); 98 else if (roots[0] == roots[1]) // nearly-equal? 99 r -= 1; // skip the double root 100 } 101 return (int)(r - roots); 102 } 103 104 /////////////////////////////////////////////////////////////////////////////// 105 /////////////////////////////////////////////////////////////////////////////// 106 107 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 108 SkASSERT(src); 109 SkASSERT(t >= 0 && t <= SK_Scalar1); 110 111 if (pt) { 112 *pt = SkEvalQuadAt(src, t); 113 } 114 if (tangent) { 115 *tangent = SkEvalQuadTangentAt(src, t); 116 } 117 } 118 119 SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 120 return to_point(SkQuadCoeff(src).eval(t)); 121 } 122 123 SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 124 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 125 // zero tangent vector when t is 0 or 1, and the control point is equal 126 // to the end point. In this case, use the quad end points to compute the tangent. 127 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 128 return src[2] - src[0]; 129 } 130 SkASSERT(src); 131 SkASSERT(t >= 0 && t <= SK_Scalar1); 132 133 Sk2s P0 = from_point(src[0]); 134 Sk2s P1 = from_point(src[1]); 135 Sk2s P2 = from_point(src[2]); 136 137 Sk2s B = P1 - P0; 138 Sk2s A = P2 - P1 - B; 139 Sk2s T = A * Sk2s(t) + B; 140 141 return to_vector(T + T); 142 } 143 144 static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 145 return v0 + (v1 - v0) * t; 146 } 147 148 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 149 SkASSERT(t > 0 && t < SK_Scalar1); 150 151 Sk2s p0 = from_point(src[0]); 152 Sk2s p1 = from_point(src[1]); 153 Sk2s p2 = from_point(src[2]); 154 Sk2s tt(t); 155 156 Sk2s p01 = interp(p0, p1, tt); 157 Sk2s p12 = interp(p1, p2, tt); 158 159 dst[0] = to_point(p0); 160 dst[1] = to_point(p01); 161 dst[2] = to_point(interp(p01, p12, tt)); 162 dst[3] = to_point(p12); 163 dst[4] = to_point(p2); 164 } 165 166 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 167 SkChopQuadAt(src, dst, 0.5f); 168 } 169 170 /** Quad'(t) = At + B, where 171 A = 2(a - 2b + c) 172 B = 2(b - a) 173 Solve for t, only if it fits between 0 < t < 1 174 */ 175 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 176 /* At + B == 0 177 t = -B / A 178 */ 179 return valid_unit_divide(a - b, a - b - b + c, tValue); 180 } 181 182 static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 183 coords[2] = coords[6] = coords[4]; 184 } 185 186 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 187 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 188 */ 189 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 190 SkASSERT(src); 191 SkASSERT(dst); 192 193 SkScalar a = src[0].fY; 194 SkScalar b = src[1].fY; 195 SkScalar c = src[2].fY; 196 197 if (is_not_monotonic(a, b, c)) { 198 SkScalar tValue; 199 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 200 SkChopQuadAt(src, dst, tValue); 201 flatten_double_quad_extrema(&dst[0].fY); 202 return 1; 203 } 204 // if we get here, we need to force dst to be monotonic, even though 205 // we couldn't compute a unit_divide value (probably underflow). 206 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 207 } 208 dst[0].set(src[0].fX, a); 209 dst[1].set(src[1].fX, b); 210 dst[2].set(src[2].fX, c); 211 return 0; 212 } 213 214 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 215 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 216 */ 217 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 218 SkASSERT(src); 219 SkASSERT(dst); 220 221 SkScalar a = src[0].fX; 222 SkScalar b = src[1].fX; 223 SkScalar c = src[2].fX; 224 225 if (is_not_monotonic(a, b, c)) { 226 SkScalar tValue; 227 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 228 SkChopQuadAt(src, dst, tValue); 229 flatten_double_quad_extrema(&dst[0].fX); 230 return 1; 231 } 232 // if we get here, we need to force dst to be monotonic, even though 233 // we couldn't compute a unit_divide value (probably underflow). 234 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 235 } 236 dst[0].set(a, src[0].fY); 237 dst[1].set(b, src[1].fY); 238 dst[2].set(c, src[2].fY); 239 return 0; 240 } 241 242 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 243 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t 244 // F''(t) = 2 (a - 2b + c) 245 // 246 // A = 2 (b - a) 247 // B = 2 (a - 2b + c) 248 // 249 // Maximum curvature for a quadratic means solving 250 // Fx' Fx'' + Fy' Fy'' = 0 251 // 252 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 253 // 254 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 255 SkScalar Ax = src[1].fX - src[0].fX; 256 SkScalar Ay = src[1].fY - src[0].fY; 257 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 258 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 259 SkScalar t = 0; // 0 means don't chop 260 261 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 262 return t; 263 } 264 265 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 266 SkScalar t = SkFindQuadMaxCurvature(src); 267 if (t == 0) { 268 memcpy(dst, src, 3 * sizeof(SkPoint)); 269 return 1; 270 } else { 271 SkChopQuadAt(src, dst, t); 272 return 2; 273 } 274 } 275 276 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 277 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 278 Sk2s s0 = from_point(src[0]); 279 Sk2s s1 = from_point(src[1]); 280 Sk2s s2 = from_point(src[2]); 281 282 dst[0] = src[0]; 283 dst[1] = to_point(s0 + (s1 - s0) * scale); 284 dst[2] = to_point(s2 + (s1 - s2) * scale); 285 dst[3] = src[2]; 286 } 287 288 ////////////////////////////////////////////////////////////////////////////// 289 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 290 ////////////////////////////////////////////////////////////////////////////// 291 292 #ifdef SK_SUPPORT_LEGACY_EVAL_CUBIC 293 static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 294 SkASSERT(src); 295 SkASSERT(t >= 0 && t <= SK_Scalar1); 296 297 if (t == 0) { 298 return src[0]; 299 } 300 301 #ifdef DIRECT_EVAL_OF_POLYNOMIALS 302 SkScalar D = src[0]; 303 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 304 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 305 SkScalar C = 3*(src[2] - D); 306 307 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 308 #else 309 SkScalar ab = SkScalarInterp(src[0], src[2], t); 310 SkScalar bc = SkScalarInterp(src[2], src[4], t); 311 SkScalar cd = SkScalarInterp(src[4], src[6], t); 312 SkScalar abc = SkScalarInterp(ab, bc, t); 313 SkScalar bcd = SkScalarInterp(bc, cd, t); 314 return SkScalarInterp(abc, bcd, t); 315 #endif 316 } 317 #endif 318 319 static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) { 320 SkQuadCoeff coeff; 321 Sk2s P0 = from_point(src[0]); 322 Sk2s P1 = from_point(src[1]); 323 Sk2s P2 = from_point(src[2]); 324 Sk2s P3 = from_point(src[3]); 325 326 coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0; 327 coeff.fB = times_2(P2 - times_2(P1) + P0); 328 coeff.fC = P1 - P0; 329 return to_vector(coeff.eval(t)); 330 } 331 332 static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) { 333 Sk2s P0 = from_point(src[0]); 334 Sk2s P1 = from_point(src[1]); 335 Sk2s P2 = from_point(src[2]); 336 Sk2s P3 = from_point(src[3]); 337 Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0; 338 Sk2s B = P2 - times_2(P1) + P0; 339 340 return to_vector(A * Sk2s(t) + B); 341 } 342 343 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 344 SkVector* tangent, SkVector* curvature) { 345 SkASSERT(src); 346 SkASSERT(t >= 0 && t <= SK_Scalar1); 347 348 if (loc) { 349 #ifdef SK_SUPPORT_LEGACY_EVAL_CUBIC 350 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 351 #else 352 *loc = to_point(SkCubicCoeff(src).eval(t)); 353 #endif 354 } 355 if (tangent) { 356 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 357 // adjacent control point is equal to the end point. In this case, use the 358 // next control point or the end points to compute the tangent. 359 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 360 if (t == 0) { 361 *tangent = src[2] - src[0]; 362 } else { 363 *tangent = src[3] - src[1]; 364 } 365 if (!tangent->fX && !tangent->fY) { 366 *tangent = src[3] - src[0]; 367 } 368 } else { 369 *tangent = eval_cubic_derivative(src, t); 370 } 371 } 372 if (curvature) { 373 *curvature = eval_cubic_2ndDerivative(src, t); 374 } 375 } 376 377 /** Cubic'(t) = At^2 + Bt + C, where 378 A = 3(-a + 3(b - c) + d) 379 B = 6(a - 2b + c) 380 C = 3(b - a) 381 Solve for t, keeping only those that fit betwee 0 < t < 1 382 */ 383 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 384 SkScalar tValues[2]) { 385 // we divide A,B,C by 3 to simplify 386 SkScalar A = d - a + 3*(b - c); 387 SkScalar B = 2*(a - b - b + c); 388 SkScalar C = b - a; 389 390 return SkFindUnitQuadRoots(A, B, C, tValues); 391 } 392 393 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 394 SkASSERT(t > 0 && t < SK_Scalar1); 395 396 Sk2s p0 = from_point(src[0]); 397 Sk2s p1 = from_point(src[1]); 398 Sk2s p2 = from_point(src[2]); 399 Sk2s p3 = from_point(src[3]); 400 Sk2s tt(t); 401 402 Sk2s ab = interp(p0, p1, tt); 403 Sk2s bc = interp(p1, p2, tt); 404 Sk2s cd = interp(p2, p3, tt); 405 Sk2s abc = interp(ab, bc, tt); 406 Sk2s bcd = interp(bc, cd, tt); 407 Sk2s abcd = interp(abc, bcd, tt); 408 409 dst[0] = src[0]; 410 dst[1] = to_point(ab); 411 dst[2] = to_point(abc); 412 dst[3] = to_point(abcd); 413 dst[4] = to_point(bcd); 414 dst[5] = to_point(cd); 415 dst[6] = src[3]; 416 } 417 418 /* http://code.google.com/p/skia/issues/detail?id=32 419 420 This test code would fail when we didn't check the return result of 421 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 422 that after the first chop, the parameters to valid_unit_divide are equal 423 (thanks to finite float precision and rounding in the subtracts). Thus 424 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 425 up with 1.0, hence the need to check and just return the last cubic as 426 a degenerate clump of 4 points in the sampe place. 427 428 static void test_cubic() { 429 SkPoint src[4] = { 430 { 556.25000, 523.03003 }, 431 { 556.23999, 522.96002 }, 432 { 556.21997, 522.89001 }, 433 { 556.21997, 522.82001 } 434 }; 435 SkPoint dst[10]; 436 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 437 SkChopCubicAt(src, dst, tval, 2); 438 } 439 */ 440 441 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 442 const SkScalar tValues[], int roots) { 443 #ifdef SK_DEBUG 444 { 445 for (int i = 0; i < roots - 1; i++) 446 { 447 SkASSERT(is_unit_interval(tValues[i])); 448 SkASSERT(is_unit_interval(tValues[i+1])); 449 SkASSERT(tValues[i] < tValues[i+1]); 450 } 451 } 452 #endif 453 454 if (dst) { 455 if (roots == 0) { // nothing to chop 456 memcpy(dst, src, 4*sizeof(SkPoint)); 457 } else { 458 SkScalar t = tValues[0]; 459 SkPoint tmp[4]; 460 461 for (int i = 0; i < roots; i++) { 462 SkChopCubicAt(src, dst, t); 463 if (i == roots - 1) { 464 break; 465 } 466 467 dst += 3; 468 // have src point to the remaining cubic (after the chop) 469 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 470 src = tmp; 471 472 // watch out in case the renormalized t isn't in range 473 if (!valid_unit_divide(tValues[i+1] - tValues[i], 474 SK_Scalar1 - tValues[i], &t)) { 475 // if we can't, just create a degenerate cubic 476 dst[4] = dst[5] = dst[6] = src[3]; 477 break; 478 } 479 } 480 } 481 } 482 } 483 484 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 485 SkChopCubicAt(src, dst, 0.5f); 486 } 487 488 static void flatten_double_cubic_extrema(SkScalar coords[14]) { 489 coords[4] = coords[8] = coords[6]; 490 } 491 492 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 493 the resulting beziers are monotonic in Y. This is called by the scan 494 converter. Depending on what is returned, dst[] is treated as follows: 495 0 dst[0..3] is the original cubic 496 1 dst[0..3] and dst[3..6] are the two new cubics 497 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 498 If dst == null, it is ignored and only the count is returned. 499 */ 500 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 501 SkScalar tValues[2]; 502 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 503 src[3].fY, tValues); 504 505 SkChopCubicAt(src, dst, tValues, roots); 506 if (dst && roots > 0) { 507 // we do some cleanup to ensure our Y extrema are flat 508 flatten_double_cubic_extrema(&dst[0].fY); 509 if (roots == 2) { 510 flatten_double_cubic_extrema(&dst[3].fY); 511 } 512 } 513 return roots; 514 } 515 516 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 517 SkScalar tValues[2]; 518 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 519 src[3].fX, tValues); 520 521 SkChopCubicAt(src, dst, tValues, roots); 522 if (dst && roots > 0) { 523 // we do some cleanup to ensure our Y extrema are flat 524 flatten_double_cubic_extrema(&dst[0].fX); 525 if (roots == 2) { 526 flatten_double_cubic_extrema(&dst[3].fX); 527 } 528 } 529 return roots; 530 } 531 532 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 533 534 Inflection means that curvature is zero. 535 Curvature is [F' x F''] / [F'^3] 536 So we solve F'x X F''y - F'y X F''y == 0 537 After some canceling of the cubic term, we get 538 A = b - a 539 B = c - 2b + a 540 C = d - 3c + 3b - a 541 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 542 */ 543 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 544 SkScalar Ax = src[1].fX - src[0].fX; 545 SkScalar Ay = src[1].fY - src[0].fY; 546 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 547 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 548 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 549 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 550 551 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 552 Ax*Cy - Ay*Cx, 553 Ax*By - Ay*Bx, 554 tValues); 555 } 556 557 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 558 SkScalar tValues[2]; 559 int count = SkFindCubicInflections(src, tValues); 560 561 if (dst) { 562 if (count == 0) { 563 memcpy(dst, src, 4 * sizeof(SkPoint)); 564 } else { 565 SkChopCubicAt(src, dst, tValues, count); 566 } 567 } 568 return count + 1; 569 } 570 571 // See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3) 572 // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 573 // Classification: 574 // discr(I) > 0 Serpentine 575 // discr(I) = 0 Cusp 576 // discr(I) < 0 Loop 577 // d0 = d1 = 0 Quadratic 578 // d0 = d1 = d2 = 0 Line 579 // p0 = p1 = p2 = p3 Point 580 static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 581 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 582 return kPoint_SkCubicType; 583 } 584 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 585 if (discr > SK_ScalarNearlyZero) { 586 return kSerpentine_SkCubicType; 587 } else if (discr < -SK_ScalarNearlyZero) { 588 return kLoop_SkCubicType; 589 } else { 590 if (0.f == d[0] && 0.f == d[1]) { 591 return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType); 592 } else { 593 return kCusp_SkCubicType; 594 } 595 } 596 } 597 598 // Assumes the third component of points is 1. 599 // Calcs p0 . (p1 x p2) 600 static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 601 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 602 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 603 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 604 return (xComp + yComp + wComp); 605 } 606 607 // Calc coefficients of I(s,t) where roots of I are inflection points of curve 608 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 609 // d0 = a1 - 2*a2+3*a3 610 // d1 = -a2 + 3*a3 611 // d2 = 3*a3 612 // a1 = p0 . (p3 x p2) 613 // a2 = p1 . (p0 x p3) 614 // a3 = p2 . (p1 x p0) 615 // Places the values of d1, d2, d3 in array d passed in 616 static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 617 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 618 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 619 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 620 621 // need to scale a's or values in later calculations will grow to high 622 SkScalar max = SkScalarAbs(a1); 623 max = SkMaxScalar(max, SkScalarAbs(a2)); 624 max = SkMaxScalar(max, SkScalarAbs(a3)); 625 max = 1.f/max; 626 a1 = a1 * max; 627 a2 = a2 * max; 628 a3 = a3 * max; 629 630 d[2] = 3.f * a3; 631 d[1] = d[2] - a2; 632 d[0] = d[1] - a2 + a1; 633 } 634 635 SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) { 636 calc_cubic_inflection_func(src, d); 637 return classify_cubic(src, d); 638 } 639 640 template <typename T> void bubble_sort(T array[], int count) { 641 for (int i = count - 1; i > 0; --i) 642 for (int j = i; j > 0; --j) 643 if (array[j] < array[j-1]) 644 { 645 T tmp(array[j]); 646 array[j] = array[j-1]; 647 array[j-1] = tmp; 648 } 649 } 650 651 /** 652 * Given an array and count, remove all pair-wise duplicates from the array, 653 * keeping the existing sorting, and return the new count 654 */ 655 static int collaps_duplicates(SkScalar array[], int count) { 656 for (int n = count; n > 1; --n) { 657 if (array[0] == array[1]) { 658 for (int i = 1; i < n; ++i) { 659 array[i - 1] = array[i]; 660 } 661 count -= 1; 662 } else { 663 array += 1; 664 } 665 } 666 return count; 667 } 668 669 #ifdef SK_DEBUG 670 671 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 672 673 static void test_collaps_duplicates() { 674 static bool gOnce; 675 if (gOnce) { return; } 676 gOnce = true; 677 const SkScalar src0[] = { 0 }; 678 const SkScalar src1[] = { 0, 0 }; 679 const SkScalar src2[] = { 0, 1 }; 680 const SkScalar src3[] = { 0, 0, 0 }; 681 const SkScalar src4[] = { 0, 0, 1 }; 682 const SkScalar src5[] = { 0, 1, 1 }; 683 const SkScalar src6[] = { 0, 1, 2 }; 684 const struct { 685 const SkScalar* fData; 686 int fCount; 687 int fCollapsedCount; 688 } data[] = { 689 { TEST_COLLAPS_ENTRY(src0), 1 }, 690 { TEST_COLLAPS_ENTRY(src1), 1 }, 691 { TEST_COLLAPS_ENTRY(src2), 2 }, 692 { TEST_COLLAPS_ENTRY(src3), 1 }, 693 { TEST_COLLAPS_ENTRY(src4), 2 }, 694 { TEST_COLLAPS_ENTRY(src5), 2 }, 695 { TEST_COLLAPS_ENTRY(src6), 3 }, 696 }; 697 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 698 SkScalar dst[3]; 699 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 700 int count = collaps_duplicates(dst, data[i].fCount); 701 SkASSERT(data[i].fCollapsedCount == count); 702 for (int j = 1; j < count; ++j) { 703 SkASSERT(dst[j-1] < dst[j]); 704 } 705 } 706 } 707 #endif 708 709 static SkScalar SkScalarCubeRoot(SkScalar x) { 710 return SkScalarPow(x, 0.3333333f); 711 } 712 713 /* Solve coeff(t) == 0, returning the number of roots that 714 lie withing 0 < t < 1. 715 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 716 717 Eliminates repeated roots (so that all tValues are distinct, and are always 718 in increasing order. 719 */ 720 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 721 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 722 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 723 } 724 725 SkScalar a, b, c, Q, R; 726 727 { 728 SkASSERT(coeff[0] != 0); 729 730 SkScalar inva = SkScalarInvert(coeff[0]); 731 a = coeff[1] * inva; 732 b = coeff[2] * inva; 733 c = coeff[3] * inva; 734 } 735 Q = (a*a - b*3) / 9; 736 R = (2*a*a*a - 9*a*b + 27*c) / 54; 737 738 SkScalar Q3 = Q * Q * Q; 739 SkScalar R2MinusQ3 = R * R - Q3; 740 SkScalar adiv3 = a / 3; 741 742 SkScalar* roots = tValues; 743 SkScalar r; 744 745 if (R2MinusQ3 < 0) { // we have 3 real roots 746 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 747 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 748 749 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 750 if (is_unit_interval(r)) { 751 *roots++ = r; 752 } 753 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 754 if (is_unit_interval(r)) { 755 *roots++ = r; 756 } 757 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 758 if (is_unit_interval(r)) { 759 *roots++ = r; 760 } 761 SkDEBUGCODE(test_collaps_duplicates();) 762 763 // now sort the roots 764 int count = (int)(roots - tValues); 765 SkASSERT((unsigned)count <= 3); 766 bubble_sort(tValues, count); 767 count = collaps_duplicates(tValues, count); 768 roots = tValues + count; // so we compute the proper count below 769 } else { // we have 1 real root 770 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 771 A = SkScalarCubeRoot(A); 772 if (R > 0) { 773 A = -A; 774 } 775 if (A != 0) { 776 A += Q / A; 777 } 778 r = A - adiv3; 779 if (is_unit_interval(r)) { 780 *roots++ = r; 781 } 782 } 783 784 return (int)(roots - tValues); 785 } 786 787 /* Looking for F' dot F'' == 0 788 789 A = b - a 790 B = c - 2b + a 791 C = d - 3c + 3b - a 792 793 F' = 3Ct^2 + 6Bt + 3A 794 F'' = 6Ct + 6B 795 796 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 797 */ 798 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 799 SkScalar a = src[2] - src[0]; 800 SkScalar b = src[4] - 2 * src[2] + src[0]; 801 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 802 803 coeff[0] = c * c; 804 coeff[1] = 3 * b * c; 805 coeff[2] = 2 * b * b + c * a; 806 coeff[3] = a * b; 807 } 808 809 /* Looking for F' dot F'' == 0 810 811 A = b - a 812 B = c - 2b + a 813 C = d - 3c + 3b - a 814 815 F' = 3Ct^2 + 6Bt + 3A 816 F'' = 6Ct + 6B 817 818 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 819 */ 820 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 821 SkScalar coeffX[4], coeffY[4]; 822 int i; 823 824 formulate_F1DotF2(&src[0].fX, coeffX); 825 formulate_F1DotF2(&src[0].fY, coeffY); 826 827 for (i = 0; i < 4; i++) { 828 coeffX[i] += coeffY[i]; 829 } 830 831 SkScalar t[3]; 832 int count = solve_cubic_poly(coeffX, t); 833 int maxCount = 0; 834 835 // now remove extrema where the curvature is zero (mins) 836 // !!!! need a test for this !!!! 837 for (i = 0; i < count; i++) { 838 // if (not_min_curvature()) 839 if (t[i] > 0 && t[i] < SK_Scalar1) { 840 tValues[maxCount++] = t[i]; 841 } 842 } 843 return maxCount; 844 } 845 846 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 847 SkScalar tValues[3]) { 848 SkScalar t_storage[3]; 849 850 if (tValues == nullptr) { 851 tValues = t_storage; 852 } 853 854 int count = SkFindCubicMaxCurvature(src, tValues); 855 856 if (dst) { 857 if (count == 0) { 858 memcpy(dst, src, 4 * sizeof(SkPoint)); 859 } else { 860 SkChopCubicAt(src, dst, tValues, count); 861 } 862 } 863 return count + 1; 864 } 865 866 #include "../pathops/SkPathOpsCubic.h" 867 868 typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 869 870 static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 871 InterceptProc method) { 872 SkDCubic cubic; 873 double roots[3]; 874 int count = (cubic.set(src).*method)(intercept, roots); 875 if (count > 0) { 876 SkDCubicPair pair = cubic.chopAt(roots[0]); 877 for (int i = 0; i < 7; ++i) { 878 dst[i] = pair.pts[i].asSkPoint(); 879 } 880 return true; 881 } 882 return false; 883 } 884 885 bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 886 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 887 } 888 889 bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 890 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 891 } 892 893 /////////////////////////////////////////////////////////////////////////////// 894 // 895 // NURB representation for conics. Helpful explanations at: 896 // 897 // http://citeseerx.ist.psu.edu/viewdoc/ 898 // download?doi=10.1.1.44.5740&rep=rep1&type=ps 899 // and 900 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 901 // 902 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 903 // ------------------------------------------ 904 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 905 // 906 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 907 // ------------------------------------------------ 908 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 909 // 910 911 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 912 // 913 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 914 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 915 // t^0 : -2 P0 w + 2 P1 w 916 // 917 // We disregard magnitude, so we can freely ignore the denominator of F', and 918 // divide the numerator by 2 919 // 920 // coeff[0] for t^2 921 // coeff[1] for t^1 922 // coeff[2] for t^0 923 // 924 static void conic_deriv_coeff(const SkScalar src[], 925 SkScalar w, 926 SkScalar coeff[3]) { 927 const SkScalar P20 = src[4] - src[0]; 928 const SkScalar P10 = src[2] - src[0]; 929 const SkScalar wP10 = w * P10; 930 coeff[0] = w * P20 - P20; 931 coeff[1] = P20 - 2 * wP10; 932 coeff[2] = wP10; 933 } 934 935 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 936 SkScalar coeff[3]; 937 conic_deriv_coeff(src, w, coeff); 938 939 SkScalar tValues[2]; 940 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 941 SkASSERT(0 == roots || 1 == roots); 942 943 if (1 == roots) { 944 *t = tValues[0]; 945 return true; 946 } 947 return false; 948 } 949 950 struct SkP3D { 951 SkScalar fX, fY, fZ; 952 953 void set(SkScalar x, SkScalar y, SkScalar z) { 954 fX = x; fY = y; fZ = z; 955 } 956 957 void projectDown(SkPoint* dst) const { 958 dst->set(fX / fZ, fY / fZ); 959 } 960 }; 961 962 // We only interpolate one dimension at a time (the first, at +0, +3, +6). 963 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 964 SkScalar ab = SkScalarInterp(src[0], src[3], t); 965 SkScalar bc = SkScalarInterp(src[3], src[6], t); 966 dst[0] = ab; 967 dst[3] = SkScalarInterp(ab, bc, t); 968 dst[6] = bc; 969 } 970 971 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 972 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 973 dst[1].set(src[1].fX * w, src[1].fY * w, w); 974 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 975 } 976 977 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 978 SkP3D tmp[3], tmp2[3]; 979 980 ratquad_mapTo3D(fPts, fW, tmp); 981 982 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 983 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 984 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 985 986 dst[0].fPts[0] = fPts[0]; 987 tmp2[0].projectDown(&dst[0].fPts[1]); 988 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 989 tmp2[2].projectDown(&dst[1].fPts[1]); 990 dst[1].fPts[2] = fPts[2]; 991 992 // to put in "standard form", where w0 and w2 are both 1, we compute the 993 // new w1 as sqrt(w1*w1/w0*w2) 994 // or 995 // w1 /= sqrt(w0*w2) 996 // 997 // However, in our case, we know that for dst[0]: 998 // w0 == 1, and for dst[1], w2 == 1 999 // 1000 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1001 dst[0].fW = tmp2[0].fZ / root; 1002 dst[1].fW = tmp2[2].fZ / root; 1003 } 1004 1005 void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { 1006 if (0 == t1 || 1 == t2) { 1007 if (0 == t1 && 1 == t2) { 1008 *dst = *this; 1009 } else { 1010 SkConic pair[2]; 1011 this->chopAt(t1 ? t1 : t2, pair); 1012 *dst = pair[SkToBool(t1)]; 1013 } 1014 return; 1015 } 1016 SkConicCoeff coeff(*this); 1017 Sk2s tt1(t1); 1018 Sk2s aXY = coeff.fNumer.eval(tt1); 1019 Sk2s aZZ = coeff.fDenom.eval(tt1); 1020 Sk2s midTT((t1 + t2) / 2); 1021 Sk2s dXY = coeff.fNumer.eval(midTT); 1022 Sk2s dZZ = coeff.fDenom.eval(midTT); 1023 Sk2s tt2(t2); 1024 Sk2s cXY = coeff.fNumer.eval(tt2); 1025 Sk2s cZZ = coeff.fDenom.eval(tt2); 1026 Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); 1027 Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); 1028 dst->fPts[0] = to_point(aXY / aZZ); 1029 dst->fPts[1] = to_point(bXY / bZZ); 1030 dst->fPts[2] = to_point(cXY / cZZ); 1031 Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); 1032 dst->fW = ww[0]; 1033 } 1034 1035 SkPoint SkConic::evalAt(SkScalar t) const { 1036 return to_point(SkConicCoeff(*this).eval(t)); 1037 } 1038 1039 SkVector SkConic::evalTangentAt(SkScalar t) const { 1040 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1041 // and the control point is equal to the end point. 1042 // In this case, use the conic endpoints to compute the tangent. 1043 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1044 return fPts[2] - fPts[0]; 1045 } 1046 Sk2s p0 = from_point(fPts[0]); 1047 Sk2s p1 = from_point(fPts[1]); 1048 Sk2s p2 = from_point(fPts[2]); 1049 Sk2s ww(fW); 1050 1051 Sk2s p20 = p2 - p0; 1052 Sk2s p10 = p1 - p0; 1053 1054 Sk2s C = ww * p10; 1055 Sk2s A = ww * p20 - p20; 1056 Sk2s B = p20 - C - C; 1057 1058 return to_vector(SkQuadCoeff(A, B, C).eval(t)); 1059 } 1060 1061 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1062 SkASSERT(t >= 0 && t <= SK_Scalar1); 1063 1064 if (pt) { 1065 *pt = this->evalAt(t); 1066 } 1067 if (tangent) { 1068 *tangent = this->evalTangentAt(t); 1069 } 1070 } 1071 1072 static SkScalar subdivide_w_value(SkScalar w) { 1073 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1074 } 1075 1076 void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1077 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1078 SkScalar newW = subdivide_w_value(fW); 1079 1080 Sk2s p0 = from_point(fPts[0]); 1081 Sk2s p1 = from_point(fPts[1]); 1082 Sk2s p2 = from_point(fPts[2]); 1083 Sk2s ww(fW); 1084 1085 Sk2s wp1 = ww * p1; 1086 Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f); 1087 1088 dst[0].fPts[0] = fPts[0]; 1089 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1090 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1091 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1092 dst[1].fPts[2] = fPts[2]; 1093 1094 dst[0].fW = dst[1].fW = newW; 1095 } 1096 1097 /* 1098 * "High order approximation of conic sections by quadratic splines" 1099 * by Michael Floater, 1993 1100 */ 1101 #define AS_QUAD_ERROR_SETUP \ 1102 SkScalar a = fW - 1; \ 1103 SkScalar k = a / (4 * (2 + a)); \ 1104 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1105 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1106 1107 void SkConic::computeAsQuadError(SkVector* err) const { 1108 AS_QUAD_ERROR_SETUP 1109 err->set(x, y); 1110 } 1111 1112 bool SkConic::asQuadTol(SkScalar tol) const { 1113 AS_QUAD_ERROR_SETUP 1114 return (x * x + y * y) <= tol * tol; 1115 } 1116 1117 // Limit the number of suggested quads to approximate a conic 1118 #define kMaxConicToQuadPOW2 5 1119 1120 int SkConic::computeQuadPOW2(SkScalar tol) const { 1121 if (tol < 0 || !SkScalarIsFinite(tol)) { 1122 return 0; 1123 } 1124 1125 AS_QUAD_ERROR_SETUP 1126 1127 SkScalar error = SkScalarSqrt(x * x + y * y); 1128 int pow2; 1129 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1130 if (error <= tol) { 1131 break; 1132 } 1133 error *= 0.25f; 1134 } 1135 // float version -- using ceil gives the same results as the above. 1136 if (false) { 1137 SkScalar err = SkScalarSqrt(x * x + y * y); 1138 if (err <= tol) { 1139 return 0; 1140 } 1141 SkScalar tol2 = tol * tol; 1142 if (tol2 == 0) { 1143 return kMaxConicToQuadPOW2; 1144 } 1145 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1146 int altPow2 = SkScalarCeilToInt(fpow2); 1147 if (altPow2 != pow2) { 1148 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1149 } 1150 pow2 = altPow2; 1151 } 1152 return pow2; 1153 } 1154 1155 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1156 SkASSERT(level >= 0); 1157 1158 if (0 == level) { 1159 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1160 return pts + 2; 1161 } else { 1162 SkConic dst[2]; 1163 src.chop(dst); 1164 --level; 1165 pts = subdivide(dst[0], pts, level); 1166 return subdivide(dst[1], pts, level); 1167 } 1168 } 1169 1170 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1171 SkASSERT(pow2 >= 0); 1172 *pts = fPts[0]; 1173 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1174 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1175 return 1 << pow2; 1176 } 1177 1178 bool SkConic::findXExtrema(SkScalar* t) const { 1179 return conic_find_extrema(&fPts[0].fX, fW, t); 1180 } 1181 1182 bool SkConic::findYExtrema(SkScalar* t) const { 1183 return conic_find_extrema(&fPts[0].fY, fW, t); 1184 } 1185 1186 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1187 SkScalar t; 1188 if (this->findXExtrema(&t)) { 1189 this->chopAt(t, dst); 1190 // now clean-up the middle, since we know t was meant to be at 1191 // an X-extrema 1192 SkScalar value = dst[0].fPts[2].fX; 1193 dst[0].fPts[1].fX = value; 1194 dst[1].fPts[0].fX = value; 1195 dst[1].fPts[1].fX = value; 1196 return true; 1197 } 1198 return false; 1199 } 1200 1201 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1202 SkScalar t; 1203 if (this->findYExtrema(&t)) { 1204 this->chopAt(t, dst); 1205 // now clean-up the middle, since we know t was meant to be at 1206 // an Y-extrema 1207 SkScalar value = dst[0].fPts[2].fY; 1208 dst[0].fPts[1].fY = value; 1209 dst[1].fPts[0].fY = value; 1210 dst[1].fPts[1].fY = value; 1211 return true; 1212 } 1213 return false; 1214 } 1215 1216 void SkConic::computeTightBounds(SkRect* bounds) const { 1217 SkPoint pts[4]; 1218 pts[0] = fPts[0]; 1219 pts[1] = fPts[2]; 1220 int count = 2; 1221 1222 SkScalar t; 1223 if (this->findXExtrema(&t)) { 1224 this->evalAt(t, &pts[count++]); 1225 } 1226 if (this->findYExtrema(&t)) { 1227 this->evalAt(t, &pts[count++]); 1228 } 1229 bounds->set(pts, count); 1230 } 1231 1232 void SkConic::computeFastBounds(SkRect* bounds) const { 1233 bounds->set(fPts, 3); 1234 } 1235 1236 #if 0 // unimplemented 1237 bool SkConic::findMaxCurvature(SkScalar* t) const { 1238 // TODO: Implement me 1239 return false; 1240 } 1241 #endif 1242 1243 SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1244 const SkMatrix& matrix) { 1245 if (!matrix.hasPerspective()) { 1246 return w; 1247 } 1248 1249 SkP3D src[3], dst[3]; 1250 1251 ratquad_mapTo3D(pts, w, src); 1252 1253 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1254 1255 // w' = sqrt(w1*w1/w0*w2) 1256 SkScalar w0 = dst[0].fZ; 1257 SkScalar w1 = dst[1].fZ; 1258 SkScalar w2 = dst[2].fZ; 1259 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1260 return w; 1261 } 1262 1263 int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1264 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1265 // rotate by x,y so that uStart is (1.0) 1266 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1267 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1268 1269 SkScalar absY = SkScalarAbs(y); 1270 1271 // check for (effectively) coincident vectors 1272 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1273 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1274 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1275 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1276 return 0; 1277 } 1278 1279 if (dir == kCCW_SkRotationDirection) { 1280 y = -y; 1281 } 1282 1283 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1284 // 0 == [0 .. 90) 1285 // 1 == [90 ..180) 1286 // 2 == [180..270) 1287 // 3 == [270..360) 1288 // 1289 int quadrant = 0; 1290 if (0 == y) { 1291 quadrant = 2; // 180 1292 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1293 } else if (0 == x) { 1294 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1295 quadrant = y > 0 ? 1 : 3; // 90 : 270 1296 } else { 1297 if (y < 0) { 1298 quadrant += 2; 1299 } 1300 if ((x < 0) != (y < 0)) { 1301 quadrant += 1; 1302 } 1303 } 1304 1305 const SkPoint quadrantPts[] = { 1306 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1307 }; 1308 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1309 1310 int conicCount = quadrant; 1311 for (int i = 0; i < conicCount; ++i) { 1312 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1313 } 1314 1315 // Now compute any remaing (sub-90-degree) arc for the last conic 1316 const SkPoint finalP = { x, y }; 1317 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1318 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1319 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1320 1321 if (dot < 1) { 1322 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1323 // compute the bisector vector, and then rescale to be the off-curve point. 1324 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1325 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1326 // This is nice, since our computed weight is cos(theta/2) as well! 1327 // 1328 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1329 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1330 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1331 conicCount += 1; 1332 } 1333 1334 // now handle counter-clockwise and the initial unitStart rotation 1335 SkMatrix matrix; 1336 matrix.setSinCos(uStart.fY, uStart.fX); 1337 if (dir == kCCW_SkRotationDirection) { 1338 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1339 } 1340 if (userMatrix) { 1341 matrix.postConcat(*userMatrix); 1342 } 1343 for (int i = 0; i < conicCount; ++i) { 1344 matrix.mapPoints(dst[i].fPts, 3); 1345 } 1346 return conicCount; 1347 } 1348
