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 //////////////////////////////////////////////////////////////////////// 19 20 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 21 SkScalar ab = a - b; 22 SkScalar bc = b - c; 23 if (ab < 0) { 24 bc = -bc; 25 } 26 return ab == 0 || bc < 0; 27 } 28 29 //////////////////////////////////////////////////////////////////////// 30 31 static bool is_unit_interval(SkScalar x) { 32 return x > 0 && x < SK_Scalar1; 33 } 34 35 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 36 SkASSERT(ratio); 37 38 if (numer < 0) { 39 numer = -numer; 40 denom = -denom; 41 } 42 43 if (denom == 0 || numer == 0 || numer >= denom) { 44 return 0; 45 } 46 47 SkScalar r = numer / denom; 48 if (SkScalarIsNaN(r)) { 49 return 0; 50 } 51 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 52 if (r == 0) { // catch underflow if numer <<<< denom 53 return 0; 54 } 55 *ratio = r; 56 return 1; 57 } 58 59 /** From Numerical Recipes in C. 60 61 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 62 x1 = Q / A 63 x2 = C / Q 64 */ 65 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 66 SkASSERT(roots); 67 68 if (A == 0) { 69 return valid_unit_divide(-C, B, roots); 70 } 71 72 SkScalar* r = roots; 73 74 SkScalar R = B*B - 4*A*C; 75 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 76 // if R is infinite, it's possible that it may still produce 77 // useful results if the operation was repeated in doubles 78 // the flipside is determining if the more precise answer 79 // isn't useful because surrounding machinery (e.g., subtracting 80 // the axis offset from C) already discards the extra precision 81 // more investigation and unit tests required... 82 return 0; 83 } 84 R = SkScalarSqrt(R); 85 86 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 87 r += valid_unit_divide(Q, A, r); 88 r += valid_unit_divide(C, Q, r); 89 if (r - roots == 2) { 90 if (roots[0] > roots[1]) 91 SkTSwap<SkScalar>(roots[0], roots[1]); 92 else if (roots[0] == roots[1]) // nearly-equal? 93 r -= 1; // skip the double root 94 } 95 return (int)(r - roots); 96 } 97 98 /////////////////////////////////////////////////////////////////////////////// 99 /////////////////////////////////////////////////////////////////////////////// 100 101 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 102 SkASSERT(src); 103 SkASSERT(t >= 0 && t <= SK_Scalar1); 104 105 if (pt) { 106 *pt = SkEvalQuadAt(src, t); 107 } 108 if (tangent) { 109 *tangent = SkEvalQuadTangentAt(src, t); 110 } 111 } 112 113 SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 114 return to_point(SkQuadCoeff(src).eval(t)); 115 } 116 117 SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 118 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 119 // zero tangent vector when t is 0 or 1, and the control point is equal 120 // to the end point. In this case, use the quad end points to compute the tangent. 121 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 122 return src[2] - src[0]; 123 } 124 SkASSERT(src); 125 SkASSERT(t >= 0 && t <= SK_Scalar1); 126 127 Sk2s P0 = from_point(src[0]); 128 Sk2s P1 = from_point(src[1]); 129 Sk2s P2 = from_point(src[2]); 130 131 Sk2s B = P1 - P0; 132 Sk2s A = P2 - P1 - B; 133 Sk2s T = A * Sk2s(t) + B; 134 135 return to_vector(T + T); 136 } 137 138 static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 139 return v0 + (v1 - v0) * t; 140 } 141 142 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 143 SkASSERT(t > 0 && t < SK_Scalar1); 144 145 Sk2s p0 = from_point(src[0]); 146 Sk2s p1 = from_point(src[1]); 147 Sk2s p2 = from_point(src[2]); 148 Sk2s tt(t); 149 150 Sk2s p01 = interp(p0, p1, tt); 151 Sk2s p12 = interp(p1, p2, tt); 152 153 dst[0] = to_point(p0); 154 dst[1] = to_point(p01); 155 dst[2] = to_point(interp(p01, p12, tt)); 156 dst[3] = to_point(p12); 157 dst[4] = to_point(p2); 158 } 159 160 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 161 SkChopQuadAt(src, dst, 0.5f); 162 } 163 164 /** Quad'(t) = At + B, where 165 A = 2(a - 2b + c) 166 B = 2(b - a) 167 Solve for t, only if it fits between 0 < t < 1 168 */ 169 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 170 /* At + B == 0 171 t = -B / A 172 */ 173 return valid_unit_divide(a - b, a - b - b + c, tValue); 174 } 175 176 static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 177 coords[2] = coords[6] = coords[4]; 178 } 179 180 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 181 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 182 */ 183 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 184 SkASSERT(src); 185 SkASSERT(dst); 186 187 SkScalar a = src[0].fY; 188 SkScalar b = src[1].fY; 189 SkScalar c = src[2].fY; 190 191 if (is_not_monotonic(a, b, c)) { 192 SkScalar tValue; 193 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 194 SkChopQuadAt(src, dst, tValue); 195 flatten_double_quad_extrema(&dst[0].fY); 196 return 1; 197 } 198 // if we get here, we need to force dst to be monotonic, even though 199 // we couldn't compute a unit_divide value (probably underflow). 200 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 201 } 202 dst[0].set(src[0].fX, a); 203 dst[1].set(src[1].fX, b); 204 dst[2].set(src[2].fX, c); 205 return 0; 206 } 207 208 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 209 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 210 */ 211 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 212 SkASSERT(src); 213 SkASSERT(dst); 214 215 SkScalar a = src[0].fX; 216 SkScalar b = src[1].fX; 217 SkScalar c = src[2].fX; 218 219 if (is_not_monotonic(a, b, c)) { 220 SkScalar tValue; 221 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 222 SkChopQuadAt(src, dst, tValue); 223 flatten_double_quad_extrema(&dst[0].fX); 224 return 1; 225 } 226 // if we get here, we need to force dst to be monotonic, even though 227 // we couldn't compute a unit_divide value (probably underflow). 228 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 229 } 230 dst[0].set(a, src[0].fY); 231 dst[1].set(b, src[1].fY); 232 dst[2].set(c, src[2].fY); 233 return 0; 234 } 235 236 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 237 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t 238 // F''(t) = 2 (a - 2b + c) 239 // 240 // A = 2 (b - a) 241 // B = 2 (a - 2b + c) 242 // 243 // Maximum curvature for a quadratic means solving 244 // Fx' Fx'' + Fy' Fy'' = 0 245 // 246 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 247 // 248 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 249 SkScalar Ax = src[1].fX - src[0].fX; 250 SkScalar Ay = src[1].fY - src[0].fY; 251 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 252 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 253 SkScalar t = 0; // 0 means don't chop 254 255 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 256 return t; 257 } 258 259 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 260 SkScalar t = SkFindQuadMaxCurvature(src); 261 if (t == 0) { 262 memcpy(dst, src, 3 * sizeof(SkPoint)); 263 return 1; 264 } else { 265 SkChopQuadAt(src, dst, t); 266 return 2; 267 } 268 } 269 270 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 271 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 272 Sk2s s0 = from_point(src[0]); 273 Sk2s s1 = from_point(src[1]); 274 Sk2s s2 = from_point(src[2]); 275 276 dst[0] = src[0]; 277 dst[1] = to_point(s0 + (s1 - s0) * scale); 278 dst[2] = to_point(s2 + (s1 - s2) * scale); 279 dst[3] = src[2]; 280 } 281 282 ////////////////////////////////////////////////////////////////////////////// 283 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 284 ////////////////////////////////////////////////////////////////////////////// 285 286 static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) { 287 SkQuadCoeff coeff; 288 Sk2s P0 = from_point(src[0]); 289 Sk2s P1 = from_point(src[1]); 290 Sk2s P2 = from_point(src[2]); 291 Sk2s P3 = from_point(src[3]); 292 293 coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0; 294 coeff.fB = times_2(P2 - times_2(P1) + P0); 295 coeff.fC = P1 - P0; 296 return to_vector(coeff.eval(t)); 297 } 298 299 static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) { 300 Sk2s P0 = from_point(src[0]); 301 Sk2s P1 = from_point(src[1]); 302 Sk2s P2 = from_point(src[2]); 303 Sk2s P3 = from_point(src[3]); 304 Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0; 305 Sk2s B = P2 - times_2(P1) + P0; 306 307 return to_vector(A * Sk2s(t) + B); 308 } 309 310 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 311 SkVector* tangent, SkVector* curvature) { 312 SkASSERT(src); 313 SkASSERT(t >= 0 && t <= SK_Scalar1); 314 315 if (loc) { 316 *loc = to_point(SkCubicCoeff(src).eval(t)); 317 } 318 if (tangent) { 319 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 320 // adjacent control point is equal to the end point. In this case, use the 321 // next control point or the end points to compute the tangent. 322 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 323 if (t == 0) { 324 *tangent = src[2] - src[0]; 325 } else { 326 *tangent = src[3] - src[1]; 327 } 328 if (!tangent->fX && !tangent->fY) { 329 *tangent = src[3] - src[0]; 330 } 331 } else { 332 *tangent = eval_cubic_derivative(src, t); 333 } 334 } 335 if (curvature) { 336 *curvature = eval_cubic_2ndDerivative(src, t); 337 } 338 } 339 340 /** Cubic'(t) = At^2 + Bt + C, where 341 A = 3(-a + 3(b - c) + d) 342 B = 6(a - 2b + c) 343 C = 3(b - a) 344 Solve for t, keeping only those that fit betwee 0 < t < 1 345 */ 346 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 347 SkScalar tValues[2]) { 348 // we divide A,B,C by 3 to simplify 349 SkScalar A = d - a + 3*(b - c); 350 SkScalar B = 2*(a - b - b + c); 351 SkScalar C = b - a; 352 353 return SkFindUnitQuadRoots(A, B, C, tValues); 354 } 355 356 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 357 SkASSERT(t > 0 && t < SK_Scalar1); 358 359 Sk2s p0 = from_point(src[0]); 360 Sk2s p1 = from_point(src[1]); 361 Sk2s p2 = from_point(src[2]); 362 Sk2s p3 = from_point(src[3]); 363 Sk2s tt(t); 364 365 Sk2s ab = interp(p0, p1, tt); 366 Sk2s bc = interp(p1, p2, tt); 367 Sk2s cd = interp(p2, p3, tt); 368 Sk2s abc = interp(ab, bc, tt); 369 Sk2s bcd = interp(bc, cd, tt); 370 Sk2s abcd = interp(abc, bcd, tt); 371 372 dst[0] = src[0]; 373 dst[1] = to_point(ab); 374 dst[2] = to_point(abc); 375 dst[3] = to_point(abcd); 376 dst[4] = to_point(bcd); 377 dst[5] = to_point(cd); 378 dst[6] = src[3]; 379 } 380 381 /* http://code.google.com/p/skia/issues/detail?id=32 382 383 This test code would fail when we didn't check the return result of 384 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 385 that after the first chop, the parameters to valid_unit_divide are equal 386 (thanks to finite float precision and rounding in the subtracts). Thus 387 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 388 up with 1.0, hence the need to check and just return the last cubic as 389 a degenerate clump of 4 points in the sampe place. 390 391 static void test_cubic() { 392 SkPoint src[4] = { 393 { 556.25000, 523.03003 }, 394 { 556.23999, 522.96002 }, 395 { 556.21997, 522.89001 }, 396 { 556.21997, 522.82001 } 397 }; 398 SkPoint dst[10]; 399 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 400 SkChopCubicAt(src, dst, tval, 2); 401 } 402 */ 403 404 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 405 const SkScalar tValues[], int roots) { 406 #ifdef SK_DEBUG 407 { 408 for (int i = 0; i < roots - 1; i++) 409 { 410 SkASSERT(is_unit_interval(tValues[i])); 411 SkASSERT(is_unit_interval(tValues[i+1])); 412 SkASSERT(tValues[i] < tValues[i+1]); 413 } 414 } 415 #endif 416 417 if (dst) { 418 if (roots == 0) { // nothing to chop 419 memcpy(dst, src, 4*sizeof(SkPoint)); 420 } else { 421 SkScalar t = tValues[0]; 422 SkPoint tmp[4]; 423 424 for (int i = 0; i < roots; i++) { 425 SkChopCubicAt(src, dst, t); 426 if (i == roots - 1) { 427 break; 428 } 429 430 dst += 3; 431 // have src point to the remaining cubic (after the chop) 432 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 433 src = tmp; 434 435 // watch out in case the renormalized t isn't in range 436 if (!valid_unit_divide(tValues[i+1] - tValues[i], 437 SK_Scalar1 - tValues[i], &t)) { 438 // if we can't, just create a degenerate cubic 439 dst[4] = dst[5] = dst[6] = src[3]; 440 break; 441 } 442 } 443 } 444 } 445 } 446 447 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 448 SkChopCubicAt(src, dst, 0.5f); 449 } 450 451 static void flatten_double_cubic_extrema(SkScalar coords[14]) { 452 coords[4] = coords[8] = coords[6]; 453 } 454 455 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 456 the resulting beziers are monotonic in Y. This is called by the scan 457 converter. Depending on what is returned, dst[] is treated as follows: 458 0 dst[0..3] is the original cubic 459 1 dst[0..3] and dst[3..6] are the two new cubics 460 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 461 If dst == null, it is ignored and only the count is returned. 462 */ 463 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 464 SkScalar tValues[2]; 465 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 466 src[3].fY, tValues); 467 468 SkChopCubicAt(src, dst, tValues, roots); 469 if (dst && roots > 0) { 470 // we do some cleanup to ensure our Y extrema are flat 471 flatten_double_cubic_extrema(&dst[0].fY); 472 if (roots == 2) { 473 flatten_double_cubic_extrema(&dst[3].fY); 474 } 475 } 476 return roots; 477 } 478 479 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 480 SkScalar tValues[2]; 481 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 482 src[3].fX, tValues); 483 484 SkChopCubicAt(src, dst, tValues, roots); 485 if (dst && roots > 0) { 486 // we do some cleanup to ensure our Y extrema are flat 487 flatten_double_cubic_extrema(&dst[0].fX); 488 if (roots == 2) { 489 flatten_double_cubic_extrema(&dst[3].fX); 490 } 491 } 492 return roots; 493 } 494 495 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 496 497 Inflection means that curvature is zero. 498 Curvature is [F' x F''] / [F'^3] 499 So we solve F'x X F''y - F'y X F''y == 0 500 After some canceling of the cubic term, we get 501 A = b - a 502 B = c - 2b + a 503 C = d - 3c + 3b - a 504 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 505 */ 506 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 507 SkScalar Ax = src[1].fX - src[0].fX; 508 SkScalar Ay = src[1].fY - src[0].fY; 509 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 510 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 511 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 512 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 513 514 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 515 Ax*Cy - Ay*Cx, 516 Ax*By - Ay*Bx, 517 tValues); 518 } 519 520 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 521 SkScalar tValues[2]; 522 int count = SkFindCubicInflections(src, tValues); 523 524 if (dst) { 525 if (count == 0) { 526 memcpy(dst, src, 4 * sizeof(SkPoint)); 527 } else { 528 SkChopCubicAt(src, dst, tValues, count); 529 } 530 } 531 return count + 1; 532 } 533 534 // See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3) 535 // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 536 // Classification: 537 // discr(I) > 0 Serpentine 538 // discr(I) = 0 Cusp 539 // discr(I) < 0 Loop 540 // d0 = d1 = 0 Quadratic 541 // d0 = d1 = d2 = 0 Line 542 // p0 = p1 = p2 = p3 Point 543 static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 544 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 545 return kPoint_SkCubicType; 546 } 547 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 548 if (discr > SK_ScalarNearlyZero) { 549 return kSerpentine_SkCubicType; 550 } else if (discr < -SK_ScalarNearlyZero) { 551 return kLoop_SkCubicType; 552 } else { 553 if (SkScalarAbs(d[0]) < SK_ScalarNearlyZero && SkScalarAbs(d[1]) < SK_ScalarNearlyZero) { 554 return ((SkScalarAbs(d[2]) < SK_ScalarNearlyZero) ? kLine_SkCubicType 555 : kQuadratic_SkCubicType); 556 } else { 557 return kCusp_SkCubicType; 558 } 559 } 560 } 561 562 // Assumes the third component of points is 1. 563 // Calcs p0 . (p1 x p2) 564 static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 565 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 566 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 567 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 568 return (xComp + yComp + wComp); 569 } 570 571 // Calc coefficients of I(s,t) where roots of I are inflection points of curve 572 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 573 // d0 = a1 - 2*a2+3*a3 574 // d1 = -a2 + 3*a3 575 // d2 = 3*a3 576 // a1 = p0 . (p3 x p2) 577 // a2 = p1 . (p0 x p3) 578 // a3 = p2 . (p1 x p0) 579 // Places the values of d1, d2, d3 in array d passed in 580 static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 581 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 582 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 583 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 584 585 // need to scale a's or values in later calculations will grow to high 586 SkScalar max = SkScalarAbs(a1); 587 max = SkMaxScalar(max, SkScalarAbs(a2)); 588 max = SkMaxScalar(max, SkScalarAbs(a3)); 589 max = 1.f/max; 590 a1 = a1 * max; 591 a2 = a2 * max; 592 a3 = a3 * max; 593 594 d[2] = 3.f * a3; 595 d[1] = d[2] - a2; 596 d[0] = d[1] - a2 + a1; 597 } 598 599 SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) { 600 calc_cubic_inflection_func(src, d); 601 return classify_cubic(src, d); 602 } 603 604 template <typename T> void bubble_sort(T array[], int count) { 605 for (int i = count - 1; i > 0; --i) 606 for (int j = i; j > 0; --j) 607 if (array[j] < array[j-1]) 608 { 609 T tmp(array[j]); 610 array[j] = array[j-1]; 611 array[j-1] = tmp; 612 } 613 } 614 615 /** 616 * Given an array and count, remove all pair-wise duplicates from the array, 617 * keeping the existing sorting, and return the new count 618 */ 619 static int collaps_duplicates(SkScalar array[], int count) { 620 for (int n = count; n > 1; --n) { 621 if (array[0] == array[1]) { 622 for (int i = 1; i < n; ++i) { 623 array[i - 1] = array[i]; 624 } 625 count -= 1; 626 } else { 627 array += 1; 628 } 629 } 630 return count; 631 } 632 633 #ifdef SK_DEBUG 634 635 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 636 637 static void test_collaps_duplicates() { 638 static bool gOnce; 639 if (gOnce) { return; } 640 gOnce = true; 641 const SkScalar src0[] = { 0 }; 642 const SkScalar src1[] = { 0, 0 }; 643 const SkScalar src2[] = { 0, 1 }; 644 const SkScalar src3[] = { 0, 0, 0 }; 645 const SkScalar src4[] = { 0, 0, 1 }; 646 const SkScalar src5[] = { 0, 1, 1 }; 647 const SkScalar src6[] = { 0, 1, 2 }; 648 const struct { 649 const SkScalar* fData; 650 int fCount; 651 int fCollapsedCount; 652 } data[] = { 653 { TEST_COLLAPS_ENTRY(src0), 1 }, 654 { TEST_COLLAPS_ENTRY(src1), 1 }, 655 { TEST_COLLAPS_ENTRY(src2), 2 }, 656 { TEST_COLLAPS_ENTRY(src3), 1 }, 657 { TEST_COLLAPS_ENTRY(src4), 2 }, 658 { TEST_COLLAPS_ENTRY(src5), 2 }, 659 { TEST_COLLAPS_ENTRY(src6), 3 }, 660 }; 661 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 662 SkScalar dst[3]; 663 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 664 int count = collaps_duplicates(dst, data[i].fCount); 665 SkASSERT(data[i].fCollapsedCount == count); 666 for (int j = 1; j < count; ++j) { 667 SkASSERT(dst[j-1] < dst[j]); 668 } 669 } 670 } 671 #endif 672 673 static SkScalar SkScalarCubeRoot(SkScalar x) { 674 return SkScalarPow(x, 0.3333333f); 675 } 676 677 /* Solve coeff(t) == 0, returning the number of roots that 678 lie withing 0 < t < 1. 679 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 680 681 Eliminates repeated roots (so that all tValues are distinct, and are always 682 in increasing order. 683 */ 684 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 685 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 686 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 687 } 688 689 SkScalar a, b, c, Q, R; 690 691 { 692 SkASSERT(coeff[0] != 0); 693 694 SkScalar inva = SkScalarInvert(coeff[0]); 695 a = coeff[1] * inva; 696 b = coeff[2] * inva; 697 c = coeff[3] * inva; 698 } 699 Q = (a*a - b*3) / 9; 700 R = (2*a*a*a - 9*a*b + 27*c) / 54; 701 702 SkScalar Q3 = Q * Q * Q; 703 SkScalar R2MinusQ3 = R * R - Q3; 704 SkScalar adiv3 = a / 3; 705 706 SkScalar* roots = tValues; 707 SkScalar r; 708 709 if (R2MinusQ3 < 0) { // we have 3 real roots 710 // the divide/root can, due to finite precisions, be slightly outside of -1...1 711 SkScalar theta = SkScalarACos(SkScalarPin(R / SkScalarSqrt(Q3), -1, 1)); 712 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 713 714 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 715 if (is_unit_interval(r)) { 716 *roots++ = r; 717 } 718 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 719 if (is_unit_interval(r)) { 720 *roots++ = r; 721 } 722 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 723 if (is_unit_interval(r)) { 724 *roots++ = r; 725 } 726 SkDEBUGCODE(test_collaps_duplicates();) 727 728 // now sort the roots 729 int count = (int)(roots - tValues); 730 SkASSERT((unsigned)count <= 3); 731 bubble_sort(tValues, count); 732 count = collaps_duplicates(tValues, count); 733 roots = tValues + count; // so we compute the proper count below 734 } else { // we have 1 real root 735 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 736 A = SkScalarCubeRoot(A); 737 if (R > 0) { 738 A = -A; 739 } 740 if (A != 0) { 741 A += Q / A; 742 } 743 r = A - adiv3; 744 if (is_unit_interval(r)) { 745 *roots++ = r; 746 } 747 } 748 749 return (int)(roots - tValues); 750 } 751 752 /* Looking for F' dot F'' == 0 753 754 A = b - a 755 B = c - 2b + a 756 C = d - 3c + 3b - a 757 758 F' = 3Ct^2 + 6Bt + 3A 759 F'' = 6Ct + 6B 760 761 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 762 */ 763 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 764 SkScalar a = src[2] - src[0]; 765 SkScalar b = src[4] - 2 * src[2] + src[0]; 766 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 767 768 coeff[0] = c * c; 769 coeff[1] = 3 * b * c; 770 coeff[2] = 2 * b * b + c * a; 771 coeff[3] = a * b; 772 } 773 774 /* Looking for F' dot F'' == 0 775 776 A = b - a 777 B = c - 2b + a 778 C = d - 3c + 3b - a 779 780 F' = 3Ct^2 + 6Bt + 3A 781 F'' = 6Ct + 6B 782 783 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 784 */ 785 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 786 SkScalar coeffX[4], coeffY[4]; 787 int i; 788 789 formulate_F1DotF2(&src[0].fX, coeffX); 790 formulate_F1DotF2(&src[0].fY, coeffY); 791 792 for (i = 0; i < 4; i++) { 793 coeffX[i] += coeffY[i]; 794 } 795 796 SkScalar t[3]; 797 int count = solve_cubic_poly(coeffX, t); 798 int maxCount = 0; 799 800 // now remove extrema where the curvature is zero (mins) 801 // !!!! need a test for this !!!! 802 for (i = 0; i < count; i++) { 803 // if (not_min_curvature()) 804 if (t[i] > 0 && t[i] < SK_Scalar1) { 805 tValues[maxCount++] = t[i]; 806 } 807 } 808 return maxCount; 809 } 810 811 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 812 SkScalar tValues[3]) { 813 SkScalar t_storage[3]; 814 815 if (tValues == nullptr) { 816 tValues = t_storage; 817 } 818 819 int count = SkFindCubicMaxCurvature(src, tValues); 820 821 if (dst) { 822 if (count == 0) { 823 memcpy(dst, src, 4 * sizeof(SkPoint)); 824 } else { 825 SkChopCubicAt(src, dst, tValues, count); 826 } 827 } 828 return count + 1; 829 } 830 831 #include "../pathops/SkPathOpsCubic.h" 832 833 typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 834 835 static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 836 InterceptProc method) { 837 SkDCubic cubic; 838 double roots[3]; 839 int count = (cubic.set(src).*method)(intercept, roots); 840 if (count > 0) { 841 SkDCubicPair pair = cubic.chopAt(roots[0]); 842 for (int i = 0; i < 7; ++i) { 843 dst[i] = pair.pts[i].asSkPoint(); 844 } 845 return true; 846 } 847 return false; 848 } 849 850 bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 851 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 852 } 853 854 bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 855 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 856 } 857 858 /////////////////////////////////////////////////////////////////////////////// 859 // 860 // NURB representation for conics. Helpful explanations at: 861 // 862 // http://citeseerx.ist.psu.edu/viewdoc/ 863 // download?doi=10.1.1.44.5740&rep=rep1&type=ps 864 // and 865 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 866 // 867 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 868 // ------------------------------------------ 869 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 870 // 871 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 872 // ------------------------------------------------ 873 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 874 // 875 876 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 877 // 878 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 879 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 880 // t^0 : -2 P0 w + 2 P1 w 881 // 882 // We disregard magnitude, so we can freely ignore the denominator of F', and 883 // divide the numerator by 2 884 // 885 // coeff[0] for t^2 886 // coeff[1] for t^1 887 // coeff[2] for t^0 888 // 889 static void conic_deriv_coeff(const SkScalar src[], 890 SkScalar w, 891 SkScalar coeff[3]) { 892 const SkScalar P20 = src[4] - src[0]; 893 const SkScalar P10 = src[2] - src[0]; 894 const SkScalar wP10 = w * P10; 895 coeff[0] = w * P20 - P20; 896 coeff[1] = P20 - 2 * wP10; 897 coeff[2] = wP10; 898 } 899 900 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 901 SkScalar coeff[3]; 902 conic_deriv_coeff(src, w, coeff); 903 904 SkScalar tValues[2]; 905 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 906 SkASSERT(0 == roots || 1 == roots); 907 908 if (1 == roots) { 909 *t = tValues[0]; 910 return true; 911 } 912 return false; 913 } 914 915 struct SkP3D { 916 SkScalar fX, fY, fZ; 917 918 void set(SkScalar x, SkScalar y, SkScalar z) { 919 fX = x; fY = y; fZ = z; 920 } 921 922 void projectDown(SkPoint* dst) const { 923 dst->set(fX / fZ, fY / fZ); 924 } 925 }; 926 927 // We only interpolate one dimension at a time (the first, at +0, +3, +6). 928 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 929 SkScalar ab = SkScalarInterp(src[0], src[3], t); 930 SkScalar bc = SkScalarInterp(src[3], src[6], t); 931 dst[0] = ab; 932 dst[3] = SkScalarInterp(ab, bc, t); 933 dst[6] = bc; 934 } 935 936 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 937 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 938 dst[1].set(src[1].fX * w, src[1].fY * w, w); 939 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 940 } 941 942 // return false if infinity or NaN is generated; caller must check 943 bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 944 SkP3D tmp[3], tmp2[3]; 945 946 ratquad_mapTo3D(fPts, fW, tmp); 947 948 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 949 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 950 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 951 952 dst[0].fPts[0] = fPts[0]; 953 tmp2[0].projectDown(&dst[0].fPts[1]); 954 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 955 tmp2[2].projectDown(&dst[1].fPts[1]); 956 dst[1].fPts[2] = fPts[2]; 957 958 // to put in "standard form", where w0 and w2 are both 1, we compute the 959 // new w1 as sqrt(w1*w1/w0*w2) 960 // or 961 // w1 /= sqrt(w0*w2) 962 // 963 // However, in our case, we know that for dst[0]: 964 // w0 == 1, and for dst[1], w2 == 1 965 // 966 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 967 dst[0].fW = tmp2[0].fZ / root; 968 dst[1].fW = tmp2[2].fZ / root; 969 SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7); 970 SkASSERT(0 == offsetof(SkConic, fPts[0].fX)); 971 return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2); 972 } 973 974 void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { 975 if (0 == t1 || 1 == t2) { 976 if (0 == t1 && 1 == t2) { 977 *dst = *this; 978 return; 979 } else { 980 SkConic pair[2]; 981 if (this->chopAt(t1 ? t1 : t2, pair)) { 982 *dst = pair[SkToBool(t1)]; 983 return; 984 } 985 } 986 } 987 SkConicCoeff coeff(*this); 988 Sk2s tt1(t1); 989 Sk2s aXY = coeff.fNumer.eval(tt1); 990 Sk2s aZZ = coeff.fDenom.eval(tt1); 991 Sk2s midTT((t1 + t2) / 2); 992 Sk2s dXY = coeff.fNumer.eval(midTT); 993 Sk2s dZZ = coeff.fDenom.eval(midTT); 994 Sk2s tt2(t2); 995 Sk2s cXY = coeff.fNumer.eval(tt2); 996 Sk2s cZZ = coeff.fDenom.eval(tt2); 997 Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); 998 Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); 999 dst->fPts[0] = to_point(aXY / aZZ); 1000 dst->fPts[1] = to_point(bXY / bZZ); 1001 dst->fPts[2] = to_point(cXY / cZZ); 1002 Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); 1003 dst->fW = ww[0]; 1004 } 1005 1006 SkPoint SkConic::evalAt(SkScalar t) const { 1007 return to_point(SkConicCoeff(*this).eval(t)); 1008 } 1009 1010 SkVector SkConic::evalTangentAt(SkScalar t) const { 1011 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1012 // and the control point is equal to the end point. 1013 // In this case, use the conic endpoints to compute the tangent. 1014 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1015 return fPts[2] - fPts[0]; 1016 } 1017 Sk2s p0 = from_point(fPts[0]); 1018 Sk2s p1 = from_point(fPts[1]); 1019 Sk2s p2 = from_point(fPts[2]); 1020 Sk2s ww(fW); 1021 1022 Sk2s p20 = p2 - p0; 1023 Sk2s p10 = p1 - p0; 1024 1025 Sk2s C = ww * p10; 1026 Sk2s A = ww * p20 - p20; 1027 Sk2s B = p20 - C - C; 1028 1029 return to_vector(SkQuadCoeff(A, B, C).eval(t)); 1030 } 1031 1032 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1033 SkASSERT(t >= 0 && t <= SK_Scalar1); 1034 1035 if (pt) { 1036 *pt = this->evalAt(t); 1037 } 1038 if (tangent) { 1039 *tangent = this->evalTangentAt(t); 1040 } 1041 } 1042 1043 static SkScalar subdivide_w_value(SkScalar w) { 1044 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1045 } 1046 1047 void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1048 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1049 SkScalar newW = subdivide_w_value(fW); 1050 1051 Sk2s p0 = from_point(fPts[0]); 1052 Sk2s p1 = from_point(fPts[1]); 1053 Sk2s p2 = from_point(fPts[2]); 1054 Sk2s ww(fW); 1055 1056 Sk2s wp1 = ww * p1; 1057 Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f); 1058 1059 dst[0].fPts[0] = fPts[0]; 1060 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1061 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1062 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1063 dst[1].fPts[2] = fPts[2]; 1064 1065 dst[0].fW = dst[1].fW = newW; 1066 } 1067 1068 /* 1069 * "High order approximation of conic sections by quadratic splines" 1070 * by Michael Floater, 1993 1071 */ 1072 #define AS_QUAD_ERROR_SETUP \ 1073 SkScalar a = fW - 1; \ 1074 SkScalar k = a / (4 * (2 + a)); \ 1075 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1076 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1077 1078 void SkConic::computeAsQuadError(SkVector* err) const { 1079 AS_QUAD_ERROR_SETUP 1080 err->set(x, y); 1081 } 1082 1083 bool SkConic::asQuadTol(SkScalar tol) const { 1084 AS_QUAD_ERROR_SETUP 1085 return (x * x + y * y) <= tol * tol; 1086 } 1087 1088 // Limit the number of suggested quads to approximate a conic 1089 #define kMaxConicToQuadPOW2 5 1090 1091 int SkConic::computeQuadPOW2(SkScalar tol) const { 1092 if (tol < 0 || !SkScalarIsFinite(tol)) { 1093 return 0; 1094 } 1095 1096 AS_QUAD_ERROR_SETUP 1097 1098 SkScalar error = SkScalarSqrt(x * x + y * y); 1099 int pow2; 1100 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1101 if (error <= tol) { 1102 break; 1103 } 1104 error *= 0.25f; 1105 } 1106 // float version -- using ceil gives the same results as the above. 1107 if (false) { 1108 SkScalar err = SkScalarSqrt(x * x + y * y); 1109 if (err <= tol) { 1110 return 0; 1111 } 1112 SkScalar tol2 = tol * tol; 1113 if (tol2 == 0) { 1114 return kMaxConicToQuadPOW2; 1115 } 1116 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1117 int altPow2 = SkScalarCeilToInt(fpow2); 1118 if (altPow2 != pow2) { 1119 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1120 } 1121 pow2 = altPow2; 1122 } 1123 return pow2; 1124 } 1125 1126 // This was originally developed and tested for pathops: see SkOpTypes.h 1127 // returns true if (a <= b <= c) || (a >= b >= c) 1128 static bool between(SkScalar a, SkScalar b, SkScalar c) { 1129 return (a - b) * (c - b) <= 0; 1130 } 1131 1132 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1133 SkASSERT(level >= 0); 1134 1135 if (0 == level) { 1136 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1137 return pts + 2; 1138 } else { 1139 SkConic dst[2]; 1140 src.chop(dst); 1141 const SkScalar startY = src.fPts[0].fY; 1142 const SkScalar endY = src.fPts[2].fY; 1143 if (between(startY, src.fPts[1].fY, endY)) { 1144 // If the input is monotonic and the output is not, the scan converter hangs. 1145 // Ensure that the chopped conics maintain their y-order. 1146 SkScalar midY = dst[0].fPts[2].fY; 1147 if (!between(startY, midY, endY)) { 1148 // If the computed midpoint is outside the ends, move it to the closer one. 1149 SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY; 1150 dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY; 1151 } 1152 if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) { 1153 // If the 1st control is not between the start and end, put it at the start. 1154 // This also reduces the quad to a line. 1155 dst[0].fPts[1].fY = startY; 1156 } 1157 if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) { 1158 // If the 2nd control is not between the start and end, put it at the end. 1159 // This also reduces the quad to a line. 1160 dst[1].fPts[1].fY = endY; 1161 } 1162 // Verify that all five points are in order. 1163 SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)); 1164 SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY)); 1165 SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY)); 1166 } 1167 --level; 1168 pts = subdivide(dst[0], pts, level); 1169 return subdivide(dst[1], pts, level); 1170 } 1171 } 1172 1173 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1174 SkASSERT(pow2 >= 0); 1175 *pts = fPts[0]; 1176 SkDEBUGCODE(SkPoint* endPts); 1177 if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ... 1178 SkConic dst[2]; 1179 this->chop(dst); 1180 // check to see if the first chop generates a pair of lines 1181 if (dst[0].fPts[1].equalsWithinTolerance(dst[0].fPts[2]) 1182 && dst[1].fPts[0].equalsWithinTolerance(dst[1].fPts[1])) { 1183 pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines 1184 pts[4] = dst[1].fPts[2]; 1185 pow2 = 1; 1186 SkDEBUGCODE(endPts = &pts[5]); 1187 goto commonFinitePtCheck; 1188 } 1189 } 1190 SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2); 1191 commonFinitePtCheck: 1192 const int quadCount = 1 << pow2; 1193 const int ptCount = 2 * quadCount + 1; 1194 SkASSERT(endPts - pts == ptCount); 1195 if (!SkPointsAreFinite(pts, ptCount)) { 1196 // if we generated a non-finite, pin ourselves to the middle of the hull, 1197 // as our first and last are already on the first/last pts of the hull. 1198 for (int i = 1; i < ptCount - 1; ++i) { 1199 pts[i] = fPts[1]; 1200 } 1201 } 1202 return 1 << pow2; 1203 } 1204 1205 bool SkConic::findXExtrema(SkScalar* t) const { 1206 return conic_find_extrema(&fPts[0].fX, fW, t); 1207 } 1208 1209 bool SkConic::findYExtrema(SkScalar* t) const { 1210 return conic_find_extrema(&fPts[0].fY, fW, t); 1211 } 1212 1213 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1214 SkScalar t; 1215 if (this->findXExtrema(&t)) { 1216 if (!this->chopAt(t, dst)) { 1217 // if chop can't return finite values, don't chop 1218 return false; 1219 } 1220 // now clean-up the middle, since we know t was meant to be at 1221 // an X-extrema 1222 SkScalar value = dst[0].fPts[2].fX; 1223 dst[0].fPts[1].fX = value; 1224 dst[1].fPts[0].fX = value; 1225 dst[1].fPts[1].fX = value; 1226 return true; 1227 } 1228 return false; 1229 } 1230 1231 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1232 SkScalar t; 1233 if (this->findYExtrema(&t)) { 1234 if (!this->chopAt(t, dst)) { 1235 // if chop can't return finite values, don't chop 1236 return false; 1237 } 1238 // now clean-up the middle, since we know t was meant to be at 1239 // an Y-extrema 1240 SkScalar value = dst[0].fPts[2].fY; 1241 dst[0].fPts[1].fY = value; 1242 dst[1].fPts[0].fY = value; 1243 dst[1].fPts[1].fY = value; 1244 return true; 1245 } 1246 return false; 1247 } 1248 1249 void SkConic::computeTightBounds(SkRect* bounds) const { 1250 SkPoint pts[4]; 1251 pts[0] = fPts[0]; 1252 pts[1] = fPts[2]; 1253 int count = 2; 1254 1255 SkScalar t; 1256 if (this->findXExtrema(&t)) { 1257 this->evalAt(t, &pts[count++]); 1258 } 1259 if (this->findYExtrema(&t)) { 1260 this->evalAt(t, &pts[count++]); 1261 } 1262 bounds->set(pts, count); 1263 } 1264 1265 void SkConic::computeFastBounds(SkRect* bounds) const { 1266 bounds->set(fPts, 3); 1267 } 1268 1269 #if 0 // unimplemented 1270 bool SkConic::findMaxCurvature(SkScalar* t) const { 1271 // TODO: Implement me 1272 return false; 1273 } 1274 #endif 1275 1276 SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1277 const SkMatrix& matrix) { 1278 if (!matrix.hasPerspective()) { 1279 return w; 1280 } 1281 1282 SkP3D src[3], dst[3]; 1283 1284 ratquad_mapTo3D(pts, w, src); 1285 1286 matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3); 1287 1288 // w' = sqrt(w1*w1/w0*w2) 1289 SkScalar w0 = dst[0].fZ; 1290 SkScalar w1 = dst[1].fZ; 1291 SkScalar w2 = dst[2].fZ; 1292 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1293 return w; 1294 } 1295 1296 int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1297 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1298 // rotate by x,y so that uStart is (1.0) 1299 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1300 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1301 1302 SkScalar absY = SkScalarAbs(y); 1303 1304 // check for (effectively) coincident vectors 1305 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1306 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1307 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1308 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1309 return 0; 1310 } 1311 1312 if (dir == kCCW_SkRotationDirection) { 1313 y = -y; 1314 } 1315 1316 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1317 // 0 == [0 .. 90) 1318 // 1 == [90 ..180) 1319 // 2 == [180..270) 1320 // 3 == [270..360) 1321 // 1322 int quadrant = 0; 1323 if (0 == y) { 1324 quadrant = 2; // 180 1325 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1326 } else if (0 == x) { 1327 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1328 quadrant = y > 0 ? 1 : 3; // 90 : 270 1329 } else { 1330 if (y < 0) { 1331 quadrant += 2; 1332 } 1333 if ((x < 0) != (y < 0)) { 1334 quadrant += 1; 1335 } 1336 } 1337 1338 const SkPoint quadrantPts[] = { 1339 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1340 }; 1341 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1342 1343 int conicCount = quadrant; 1344 for (int i = 0; i < conicCount; ++i) { 1345 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1346 } 1347 1348 // Now compute any remaing (sub-90-degree) arc for the last conic 1349 const SkPoint finalP = { x, y }; 1350 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1351 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1352 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1353 1354 if (dot < 1) { 1355 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1356 // compute the bisector vector, and then rescale to be the off-curve point. 1357 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1358 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1359 // This is nice, since our computed weight is cos(theta/2) as well! 1360 // 1361 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1362 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1363 if (!lastQ.equalsWithinTolerance(offCurve)) { 1364 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1365 conicCount += 1; 1366 } 1367 } 1368 1369 // now handle counter-clockwise and the initial unitStart rotation 1370 SkMatrix matrix; 1371 matrix.setSinCos(uStart.fY, uStart.fX); 1372 if (dir == kCCW_SkRotationDirection) { 1373 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1374 } 1375 if (userMatrix) { 1376 matrix.postConcat(*userMatrix); 1377 } 1378 for (int i = 0; i < conicCount; ++i) { 1379 matrix.mapPoints(dst[i].fPts, 3); 1380 } 1381 return conicCount; 1382 } 1383