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#include "SkPoint3.h" 12#include "SkPointPriv.h" 13 14static SkVector to_vector(const Sk2s& x) { 15 SkVector vector; 16 x.store(&vector); 17 return vector; 18} 19 20//////////////////////////////////////////////////////////////////////// 21 22static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 23 SkScalar ab = a - b; 24 SkScalar bc = b - c; 25 if (ab < 0) { 26 bc = -bc; 27 } 28 return ab == 0 || bc < 0; 29} 30 31//////////////////////////////////////////////////////////////////////// 32 33static bool is_unit_interval(SkScalar x) { 34 return x > 0 && x < SK_Scalar1; 35} 36 37static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 38 SkASSERT(ratio); 39 40 if (numer < 0) { 41 numer = -numer; 42 denom = -denom; 43 } 44 45 if (denom == 0 || numer == 0 || numer >= denom) { 46 return 0; 47 } 48 49 SkScalar r = numer / denom; 50 if (SkScalarIsNaN(r)) { 51 return 0; 52 } 53 SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); 54 if (r == 0) { // catch underflow if numer <<<< denom 55 return 0; 56 } 57 *ratio = r; 58 return 1; 59} 60 61/** From Numerical Recipes in C. 62 63 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 64 x1 = Q / A 65 x2 = C / Q 66*/ 67int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 68 SkASSERT(roots); 69 70 if (A == 0) { 71 return valid_unit_divide(-C, B, roots); 72 } 73 74 SkScalar* r = roots; 75 76 SkScalar R = B*B - 4*A*C; 77 if (R < 0 || !SkScalarIsFinite(R)) { // complex roots 78 // if R is infinite, it's possible that it may still produce 79 // useful results if the operation was repeated in doubles 80 // the flipside is determining if the more precise answer 81 // isn't useful because surrounding machinery (e.g., subtracting 82 // the axis offset from C) already discards the extra precision 83 // more investigation and unit tests required... 84 return 0; 85 } 86 R = SkScalarSqrt(R); 87 88 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 89 r += valid_unit_divide(Q, A, r); 90 r += valid_unit_divide(C, Q, r); 91 if (r - roots == 2) { 92 if (roots[0] > roots[1]) 93 SkTSwap<SkScalar>(roots[0], roots[1]); 94 else if (roots[0] == roots[1]) // nearly-equal? 95 r -= 1; // skip the double root 96 } 97 return (int)(r - roots); 98} 99 100/////////////////////////////////////////////////////////////////////////////// 101/////////////////////////////////////////////////////////////////////////////// 102 103void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { 104 SkASSERT(src); 105 SkASSERT(t >= 0 && t <= SK_Scalar1); 106 107 if (pt) { 108 *pt = SkEvalQuadAt(src, t); 109 } 110 if (tangent) { 111 *tangent = SkEvalQuadTangentAt(src, t); 112 } 113} 114 115SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { 116 return to_point(SkQuadCoeff(src).eval(t)); 117} 118 119SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { 120 // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a 121 // zero tangent vector when t is 0 or 1, and the control point is equal 122 // to the end point. In this case, use the quad end points to compute the tangent. 123 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { 124 return src[2] - src[0]; 125 } 126 SkASSERT(src); 127 SkASSERT(t >= 0 && t <= SK_Scalar1); 128 129 Sk2s P0 = from_point(src[0]); 130 Sk2s P1 = from_point(src[1]); 131 Sk2s P2 = from_point(src[2]); 132 133 Sk2s B = P1 - P0; 134 Sk2s A = P2 - P1 - B; 135 Sk2s T = A * Sk2s(t) + B; 136 137 return to_vector(T + T); 138} 139 140static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { 141 return v0 + (v1 - v0) * t; 142} 143 144void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 145 SkASSERT(t > 0 && t < SK_Scalar1); 146 147 Sk2s p0 = from_point(src[0]); 148 Sk2s p1 = from_point(src[1]); 149 Sk2s p2 = from_point(src[2]); 150 Sk2s tt(t); 151 152 Sk2s p01 = interp(p0, p1, tt); 153 Sk2s p12 = interp(p1, p2, tt); 154 155 dst[0] = to_point(p0); 156 dst[1] = to_point(p01); 157 dst[2] = to_point(interp(p01, p12, tt)); 158 dst[3] = to_point(p12); 159 dst[4] = to_point(p2); 160} 161 162void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 163 SkChopQuadAt(src, dst, 0.5f); 164} 165 166/** Quad'(t) = At + B, where 167 A = 2(a - 2b + c) 168 B = 2(b - a) 169 Solve for t, only if it fits between 0 < t < 1 170*/ 171int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 172 /* At + B == 0 173 t = -B / A 174 */ 175 return valid_unit_divide(a - b, a - b - b + c, tValue); 176} 177 178static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 179 coords[2] = coords[6] = coords[4]; 180} 181 182/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 183 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 184 */ 185int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 186 SkASSERT(src); 187 SkASSERT(dst); 188 189 SkScalar a = src[0].fY; 190 SkScalar b = src[1].fY; 191 SkScalar c = src[2].fY; 192 193 if (is_not_monotonic(a, b, c)) { 194 SkScalar tValue; 195 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 196 SkChopQuadAt(src, dst, tValue); 197 flatten_double_quad_extrema(&dst[0].fY); 198 return 1; 199 } 200 // if we get here, we need to force dst to be monotonic, even though 201 // we couldn't compute a unit_divide value (probably underflow). 202 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 203 } 204 dst[0].set(src[0].fX, a); 205 dst[1].set(src[1].fX, b); 206 dst[2].set(src[2].fX, c); 207 return 0; 208} 209 210/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 211 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 212 */ 213int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 214 SkASSERT(src); 215 SkASSERT(dst); 216 217 SkScalar a = src[0].fX; 218 SkScalar b = src[1].fX; 219 SkScalar c = src[2].fX; 220 221 if (is_not_monotonic(a, b, c)) { 222 SkScalar tValue; 223 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 224 SkChopQuadAt(src, dst, tValue); 225 flatten_double_quad_extrema(&dst[0].fX); 226 return 1; 227 } 228 // if we get here, we need to force dst to be monotonic, even though 229 // we couldn't compute a unit_divide value (probably underflow). 230 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 231 } 232 dst[0].set(a, src[0].fY); 233 dst[1].set(b, src[1].fY); 234 dst[2].set(c, src[2].fY); 235 return 0; 236} 237 238// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 239// F'(t) = 2 (b - a) + 2 (a - 2b + c) t 240// F''(t) = 2 (a - 2b + c) 241// 242// A = 2 (b - a) 243// B = 2 (a - 2b + c) 244// 245// Maximum curvature for a quadratic means solving 246// Fx' Fx'' + Fy' Fy'' = 0 247// 248// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 249// 250SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 251 SkScalar Ax = src[1].fX - src[0].fX; 252 SkScalar Ay = src[1].fY - src[0].fY; 253 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 254 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 255 SkScalar t = 0; // 0 means don't chop 256 257 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 258 return t; 259} 260 261int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 262 SkScalar t = SkFindQuadMaxCurvature(src); 263 if (t == 0) { 264 memcpy(dst, src, 3 * sizeof(SkPoint)); 265 return 1; 266 } else { 267 SkChopQuadAt(src, dst, t); 268 return 2; 269 } 270} 271 272void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 273 Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); 274 Sk2s s0 = from_point(src[0]); 275 Sk2s s1 = from_point(src[1]); 276 Sk2s s2 = from_point(src[2]); 277 278 dst[0] = src[0]; 279 dst[1] = to_point(s0 + (s1 - s0) * scale); 280 dst[2] = to_point(s2 + (s1 - s2) * scale); 281 dst[3] = src[2]; 282} 283 284////////////////////////////////////////////////////////////////////////////// 285///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 286////////////////////////////////////////////////////////////////////////////// 287 288static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) { 289 SkQuadCoeff coeff; 290 Sk2s P0 = from_point(src[0]); 291 Sk2s P1 = from_point(src[1]); 292 Sk2s P2 = from_point(src[2]); 293 Sk2s P3 = from_point(src[3]); 294 295 coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0; 296 coeff.fB = times_2(P2 - times_2(P1) + P0); 297 coeff.fC = P1 - P0; 298 return to_vector(coeff.eval(t)); 299} 300 301static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) { 302 Sk2s P0 = from_point(src[0]); 303 Sk2s P1 = from_point(src[1]); 304 Sk2s P2 = from_point(src[2]); 305 Sk2s P3 = from_point(src[3]); 306 Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0; 307 Sk2s B = P2 - times_2(P1) + P0; 308 309 return to_vector(A * Sk2s(t) + B); 310} 311 312void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 313 SkVector* tangent, SkVector* curvature) { 314 SkASSERT(src); 315 SkASSERT(t >= 0 && t <= SK_Scalar1); 316 317 if (loc) { 318 *loc = to_point(SkCubicCoeff(src).eval(t)); 319 } 320 if (tangent) { 321 // The derivative equation returns a zero tangent vector when t is 0 or 1, and the 322 // adjacent control point is equal to the end point. In this case, use the 323 // next control point or the end points to compute the tangent. 324 if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { 325 if (t == 0) { 326 *tangent = src[2] - src[0]; 327 } else { 328 *tangent = src[3] - src[1]; 329 } 330 if (!tangent->fX && !tangent->fY) { 331 *tangent = src[3] - src[0]; 332 } 333 } else { 334 *tangent = eval_cubic_derivative(src, t); 335 } 336 } 337 if (curvature) { 338 *curvature = eval_cubic_2ndDerivative(src, t); 339 } 340} 341 342/** Cubic'(t) = At^2 + Bt + C, where 343 A = 3(-a + 3(b - c) + d) 344 B = 6(a - 2b + c) 345 C = 3(b - a) 346 Solve for t, keeping only those that fit betwee 0 < t < 1 347*/ 348int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 349 SkScalar tValues[2]) { 350 // we divide A,B,C by 3 to simplify 351 SkScalar A = d - a + 3*(b - c); 352 SkScalar B = 2*(a - b - b + c); 353 SkScalar C = b - a; 354 355 return SkFindUnitQuadRoots(A, B, C, tValues); 356} 357 358void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 359 SkASSERT(t > 0 && t < SK_Scalar1); 360 361 Sk2s p0 = from_point(src[0]); 362 Sk2s p1 = from_point(src[1]); 363 Sk2s p2 = from_point(src[2]); 364 Sk2s p3 = from_point(src[3]); 365 Sk2s tt(t); 366 367 Sk2s ab = interp(p0, p1, tt); 368 Sk2s bc = interp(p1, p2, tt); 369 Sk2s cd = interp(p2, p3, tt); 370 Sk2s abc = interp(ab, bc, tt); 371 Sk2s bcd = interp(bc, cd, tt); 372 Sk2s abcd = interp(abc, bcd, tt); 373 374 dst[0] = src[0]; 375 dst[1] = to_point(ab); 376 dst[2] = to_point(abc); 377 dst[3] = to_point(abcd); 378 dst[4] = to_point(bcd); 379 dst[5] = to_point(cd); 380 dst[6] = src[3]; 381} 382 383/* http://code.google.com/p/skia/issues/detail?id=32 384 385 This test code would fail when we didn't check the return result of 386 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 387 that after the first chop, the parameters to valid_unit_divide are equal 388 (thanks to finite float precision and rounding in the subtracts). Thus 389 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 390 up with 1.0, hence the need to check and just return the last cubic as 391 a degenerate clump of 4 points in the sampe place. 392 393 static void test_cubic() { 394 SkPoint src[4] = { 395 { 556.25000, 523.03003 }, 396 { 556.23999, 522.96002 }, 397 { 556.21997, 522.89001 }, 398 { 556.21997, 522.82001 } 399 }; 400 SkPoint dst[10]; 401 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 402 SkChopCubicAt(src, dst, tval, 2); 403 } 404 */ 405 406void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 407 const SkScalar tValues[], int roots) { 408#ifdef SK_DEBUG 409 { 410 for (int i = 0; i < roots - 1; i++) 411 { 412 SkASSERT(is_unit_interval(tValues[i])); 413 SkASSERT(is_unit_interval(tValues[i+1])); 414 SkASSERT(tValues[i] < tValues[i+1]); 415 } 416 } 417#endif 418 419 if (dst) { 420 if (roots == 0) { // nothing to chop 421 memcpy(dst, src, 4*sizeof(SkPoint)); 422 } else { 423 SkScalar t = tValues[0]; 424 SkPoint tmp[4]; 425 426 for (int i = 0; i < roots; i++) { 427 SkChopCubicAt(src, dst, t); 428 if (i == roots - 1) { 429 break; 430 } 431 432 dst += 3; 433 // have src point to the remaining cubic (after the chop) 434 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 435 src = tmp; 436 437 // watch out in case the renormalized t isn't in range 438 if (!valid_unit_divide(tValues[i+1] - tValues[i], 439 SK_Scalar1 - tValues[i], &t)) { 440 // if we can't, just create a degenerate cubic 441 dst[4] = dst[5] = dst[6] = src[3]; 442 break; 443 } 444 } 445 } 446 } 447} 448 449void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 450 SkChopCubicAt(src, dst, 0.5f); 451} 452 453static void flatten_double_cubic_extrema(SkScalar coords[14]) { 454 coords[4] = coords[8] = coords[6]; 455} 456 457/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 458 the resulting beziers are monotonic in Y. This is called by the scan 459 converter. Depending on what is returned, dst[] is treated as follows: 460 0 dst[0..3] is the original cubic 461 1 dst[0..3] and dst[3..6] are the two new cubics 462 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 463 If dst == null, it is ignored and only the count is returned. 464*/ 465int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 466 SkScalar tValues[2]; 467 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 468 src[3].fY, tValues); 469 470 SkChopCubicAt(src, dst, tValues, roots); 471 if (dst && roots > 0) { 472 // we do some cleanup to ensure our Y extrema are flat 473 flatten_double_cubic_extrema(&dst[0].fY); 474 if (roots == 2) { 475 flatten_double_cubic_extrema(&dst[3].fY); 476 } 477 } 478 return roots; 479} 480 481int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 482 SkScalar tValues[2]; 483 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 484 src[3].fX, tValues); 485 486 SkChopCubicAt(src, dst, tValues, roots); 487 if (dst && roots > 0) { 488 // we do some cleanup to ensure our Y extrema are flat 489 flatten_double_cubic_extrema(&dst[0].fX); 490 if (roots == 2) { 491 flatten_double_cubic_extrema(&dst[3].fX); 492 } 493 } 494 return roots; 495} 496 497/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 498 499 Inflection means that curvature is zero. 500 Curvature is [F' x F''] / [F'^3] 501 So we solve F'x X F''y - F'y X F''y == 0 502 After some canceling of the cubic term, we get 503 A = b - a 504 B = c - 2b + a 505 C = d - 3c + 3b - a 506 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 507*/ 508int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 509 SkScalar Ax = src[1].fX - src[0].fX; 510 SkScalar Ay = src[1].fY - src[0].fY; 511 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 512 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 513 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 514 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 515 516 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 517 Ax*Cy - Ay*Cx, 518 Ax*By - Ay*Bx, 519 tValues); 520} 521 522int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 523 SkScalar tValues[2]; 524 int count = SkFindCubicInflections(src, tValues); 525 526 if (dst) { 527 if (count == 0) { 528 memcpy(dst, src, 4 * sizeof(SkPoint)); 529 } else { 530 SkChopCubicAt(src, dst, tValues, count); 531 } 532 } 533 return count + 1; 534} 535 536// Assumes the third component of points is 1. 537// Calcs p0 . (p1 x p2) 538static double calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 539 const double xComp = (double) p0.fX * ((double) p1.fY - (double) p2.fY); 540 const double yComp = (double) p0.fY * ((double) p2.fX - (double) p1.fX); 541 const double wComp = (double) p1.fX * (double) p2.fY - (double) p1.fY * (double) p2.fX; 542 return (xComp + yComp + wComp); 543} 544 545// Calc coefficients of I(s,t) where roots of I are inflection points of curve 546// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 547// d0 = a1 - 2*a2+3*a3 548// d1 = -a2 + 3*a3 549// d2 = 3*a3 550// a1 = p0 . (p3 x p2) 551// a2 = p1 . (p0 x p3) 552// a3 = p2 . (p1 x p0) 553// Places the values of d1, d2, d3 in array d passed in 554static void calc_cubic_inflection_func(const SkPoint p[4], double d[4]) { 555 const double a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 556 const double a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 557 const double a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 558 559 d[3] = 3 * a3; 560 d[2] = d[3] - a2; 561 d[1] = d[2] - a2 + a1; 562 d[0] = 0; 563} 564 565static void normalize_t_s(double t[], double s[], int count) { 566 // Keep the exponents at or below zero to avoid overflow down the road. 567 for (int i = 0; i < count; ++i) { 568 SkASSERT(0 != s[i]); // classify_cubic should not call this method when s[i] is 0 or NaN. 569 570 uint64_t bitsT, bitsS; 571 memcpy(&bitsT, &t[i], sizeof(double)); 572 memcpy(&bitsS, &s[i], sizeof(double)); 573 574 uint64_t maxExponent = SkTMax(bitsT & 0x7ff0000000000000, bitsS & 0x7ff0000000000000); 575 576#ifdef SK_DEBUG 577 uint64_t maxExponentValue = maxExponent >> 52; 578 // Ensure max(absT,absS) is NOT in denormalized form. SkClassifyCubic is given fp32 points, 579 // and does not call this method when s==0, so this should never happen. 580 SkASSERT(0 != maxExponentValue); 581 // Ensure 1/max(absT,absS) will NOT be in denormalized form. SkClassifyCubic is given fp32 582 // points, so this should never happen. 583 SkASSERT(2046 != maxExponentValue); 584#endif 585 586 // Pick a normalizer that scales the larger exponent to 1 (aka 1023 in biased form), but 587 // does NOT change the mantissa (thus preserving accuracy). 588 double normalizer; 589 uint64_t normalizerExponent = (uint64_t(1023 * 2) << 52) - maxExponent; 590 memcpy(&normalizer, &normalizerExponent, sizeof(double)); 591 592 t[i] *= normalizer; 593 s[i] *= normalizer; 594 } 595} 596 597static void sort_and_orient_t_s(double t[2], double s[2]) { 598 // This copysign/abs business orients the implicit function so positive values are always on the 599 // "left" side of the curve. 600 t[1] = -copysign(t[1], t[1] * s[1]); 601 s[1] = -fabs(s[1]); 602 603 // Ensure t[0]/s[0] <= t[1]/s[1] (s[1] is negative from above). 604 if (copysign(s[1], s[0]) * t[0] > -fabs(s[0]) * t[1]) { 605 std::swap(t[0], t[1]); 606 std::swap(s[0], s[1]); 607 } 608} 609 610// See "Resolution Independent Curve Rendering using Programmable Graphics Hardware" 611// https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf 612// discr(I) = 3*d2^2 - 4*d1*d3 613// Classification: 614// d1 != 0, discr(I) > 0 Serpentine 615// d1 != 0, discr(I) < 0 Loop 616// d1 != 0, discr(I) = 0 Cusp (with inflection at infinity) 617// d1 = 0, d2 != 0 Cusp (with cusp at infinity) 618// d1 = d2 = 0, d3 != 0 Quadratic 619// d1 = d2 = d3 = 0 Line or Point 620static SkCubicType classify_cubic(const double d[4], double t[2], double s[2]) { 621 if (0 == d[1]) { 622 if (0 == d[2]) { 623 if (t && s) { 624 t[0] = t[1] = 1; 625 s[0] = s[1] = 0; // infinity 626 } 627 return 0 == d[3] ? SkCubicType::kLineOrPoint : SkCubicType::kQuadratic; 628 } 629 if (t && s) { 630 t[0] = d[3]; 631 s[0] = 3 * d[2]; 632 normalize_t_s(t, s, 1); 633 t[1] = 1; 634 s[1] = 0; // infinity 635 } 636 return SkCubicType::kCuspAtInfinity; 637 } 638 639 const double discr = 3 * d[2] * d[2] - 4 * d[1] * d[3]; 640 if (discr > 0) { 641 if (t && s) { 642 const double q = 3 * d[2] + copysign(sqrt(3 * discr), d[2]); 643 t[0] = q; 644 s[0] = 6 * d[1]; 645 t[1] = 2 * d[3]; 646 s[1] = q; 647 normalize_t_s(t, s, 2); 648 sort_and_orient_t_s(t, s); 649 } 650 return SkCubicType::kSerpentine; 651 } else if (discr < 0) { 652 if (t && s) { 653 const double q = d[2] + copysign(sqrt(-discr), d[2]); 654 t[0] = q; 655 s[0] = 2 * d[1]; 656 t[1] = 2 * (d[2] * d[2] - d[3] * d[1]); 657 s[1] = d[1] * q; 658 normalize_t_s(t, s, 2); 659 sort_and_orient_t_s(t, s); 660 } 661 return SkCubicType::kLoop; 662 } else { 663 if (t && s) { 664 t[0] = d[2]; 665 s[0] = 2 * d[1]; 666 normalize_t_s(t, s, 1); 667 t[1] = t[0]; 668 s[1] = s[0]; 669 sort_and_orient_t_s(t, s); 670 } 671 return SkCubicType::kLocalCusp; 672 } 673} 674 675SkCubicType SkClassifyCubic(const SkPoint src[4], double t[2], double s[2], double d[4]) { 676 double localD[4]; 677 double* dd = d ? d : localD; 678 calc_cubic_inflection_func(src, dd); 679 return classify_cubic(dd, t, s); 680} 681 682template <typename T> void bubble_sort(T array[], int count) { 683 for (int i = count - 1; i > 0; --i) 684 for (int j = i; j > 0; --j) 685 if (array[j] < array[j-1]) 686 { 687 T tmp(array[j]); 688 array[j] = array[j-1]; 689 array[j-1] = tmp; 690 } 691} 692 693/** 694 * Given an array and count, remove all pair-wise duplicates from the array, 695 * keeping the existing sorting, and return the new count 696 */ 697static int collaps_duplicates(SkScalar array[], int count) { 698 for (int n = count; n > 1; --n) { 699 if (array[0] == array[1]) { 700 for (int i = 1; i < n; ++i) { 701 array[i - 1] = array[i]; 702 } 703 count -= 1; 704 } else { 705 array += 1; 706 } 707 } 708 return count; 709} 710 711#ifdef SK_DEBUG 712 713#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 714 715static void test_collaps_duplicates() { 716 static bool gOnce; 717 if (gOnce) { return; } 718 gOnce = true; 719 const SkScalar src0[] = { 0 }; 720 const SkScalar src1[] = { 0, 0 }; 721 const SkScalar src2[] = { 0, 1 }; 722 const SkScalar src3[] = { 0, 0, 0 }; 723 const SkScalar src4[] = { 0, 0, 1 }; 724 const SkScalar src5[] = { 0, 1, 1 }; 725 const SkScalar src6[] = { 0, 1, 2 }; 726 const struct { 727 const SkScalar* fData; 728 int fCount; 729 int fCollapsedCount; 730 } data[] = { 731 { TEST_COLLAPS_ENTRY(src0), 1 }, 732 { TEST_COLLAPS_ENTRY(src1), 1 }, 733 { TEST_COLLAPS_ENTRY(src2), 2 }, 734 { TEST_COLLAPS_ENTRY(src3), 1 }, 735 { TEST_COLLAPS_ENTRY(src4), 2 }, 736 { TEST_COLLAPS_ENTRY(src5), 2 }, 737 { TEST_COLLAPS_ENTRY(src6), 3 }, 738 }; 739 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 740 SkScalar dst[3]; 741 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 742 int count = collaps_duplicates(dst, data[i].fCount); 743 SkASSERT(data[i].fCollapsedCount == count); 744 for (int j = 1; j < count; ++j) { 745 SkASSERT(dst[j-1] < dst[j]); 746 } 747 } 748} 749#endif 750 751static SkScalar SkScalarCubeRoot(SkScalar x) { 752 return SkScalarPow(x, 0.3333333f); 753} 754 755/* Solve coeff(t) == 0, returning the number of roots that 756 lie withing 0 < t < 1. 757 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 758 759 Eliminates repeated roots (so that all tValues are distinct, and are always 760 in increasing order. 761*/ 762static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 763 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 764 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 765 } 766 767 SkScalar a, b, c, Q, R; 768 769 { 770 SkASSERT(coeff[0] != 0); 771 772 SkScalar inva = SkScalarInvert(coeff[0]); 773 a = coeff[1] * inva; 774 b = coeff[2] * inva; 775 c = coeff[3] * inva; 776 } 777 Q = (a*a - b*3) / 9; 778 R = (2*a*a*a - 9*a*b + 27*c) / 54; 779 780 SkScalar Q3 = Q * Q * Q; 781 SkScalar R2MinusQ3 = R * R - Q3; 782 SkScalar adiv3 = a / 3; 783 784 SkScalar* roots = tValues; 785 SkScalar r; 786 787 if (R2MinusQ3 < 0) { // we have 3 real roots 788 // the divide/root can, due to finite precisions, be slightly outside of -1...1 789 SkScalar theta = SkScalarACos(SkScalarPin(R / SkScalarSqrt(Q3), -1, 1)); 790 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 791 792 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 793 if (is_unit_interval(r)) { 794 *roots++ = r; 795 } 796 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 797 if (is_unit_interval(r)) { 798 *roots++ = r; 799 } 800 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 801 if (is_unit_interval(r)) { 802 *roots++ = r; 803 } 804 SkDEBUGCODE(test_collaps_duplicates();) 805 806 // now sort the roots 807 int count = (int)(roots - tValues); 808 SkASSERT((unsigned)count <= 3); 809 bubble_sort(tValues, count); 810 count = collaps_duplicates(tValues, count); 811 roots = tValues + count; // so we compute the proper count below 812 } else { // we have 1 real root 813 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 814 A = SkScalarCubeRoot(A); 815 if (R > 0) { 816 A = -A; 817 } 818 if (A != 0) { 819 A += Q / A; 820 } 821 r = A - adiv3; 822 if (is_unit_interval(r)) { 823 *roots++ = r; 824 } 825 } 826 827 return (int)(roots - tValues); 828} 829 830/* Looking for F' dot F'' == 0 831 832 A = b - a 833 B = c - 2b + a 834 C = d - 3c + 3b - a 835 836 F' = 3Ct^2 + 6Bt + 3A 837 F'' = 6Ct + 6B 838 839 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 840*/ 841static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 842 SkScalar a = src[2] - src[0]; 843 SkScalar b = src[4] - 2 * src[2] + src[0]; 844 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 845 846 coeff[0] = c * c; 847 coeff[1] = 3 * b * c; 848 coeff[2] = 2 * b * b + c * a; 849 coeff[3] = a * b; 850} 851 852/* Looking for F' dot F'' == 0 853 854 A = b - a 855 B = c - 2b + a 856 C = d - 3c + 3b - a 857 858 F' = 3Ct^2 + 6Bt + 3A 859 F'' = 6Ct + 6B 860 861 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 862*/ 863int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 864 SkScalar coeffX[4], coeffY[4]; 865 int i; 866 867 formulate_F1DotF2(&src[0].fX, coeffX); 868 formulate_F1DotF2(&src[0].fY, coeffY); 869 870 for (i = 0; i < 4; i++) { 871 coeffX[i] += coeffY[i]; 872 } 873 874 SkScalar t[3]; 875 int count = solve_cubic_poly(coeffX, t); 876 int maxCount = 0; 877 878 // now remove extrema where the curvature is zero (mins) 879 // !!!! need a test for this !!!! 880 for (i = 0; i < count; i++) { 881 // if (not_min_curvature()) 882 if (t[i] > 0 && t[i] < SK_Scalar1) { 883 tValues[maxCount++] = t[i]; 884 } 885 } 886 return maxCount; 887} 888 889int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 890 SkScalar tValues[3]) { 891 SkScalar t_storage[3]; 892 893 if (tValues == nullptr) { 894 tValues = t_storage; 895 } 896 897 int count = SkFindCubicMaxCurvature(src, tValues); 898 899 if (dst) { 900 if (count == 0) { 901 memcpy(dst, src, 4 * sizeof(SkPoint)); 902 } else { 903 SkChopCubicAt(src, dst, tValues, count); 904 } 905 } 906 return count + 1; 907} 908 909#include "../pathops/SkPathOpsCubic.h" 910 911typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; 912 913static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], 914 InterceptProc method) { 915 SkDCubic cubic; 916 double roots[3]; 917 int count = (cubic.set(src).*method)(intercept, roots); 918 if (count > 0) { 919 SkDCubicPair pair = cubic.chopAt(roots[0]); 920 for (int i = 0; i < 7; ++i) { 921 dst[i] = pair.pts[i].asSkPoint(); 922 } 923 return true; 924 } 925 return false; 926} 927 928bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { 929 return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); 930} 931 932bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { 933 return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); 934} 935 936/////////////////////////////////////////////////////////////////////////////// 937// 938// NURB representation for conics. Helpful explanations at: 939// 940// http://citeseerx.ist.psu.edu/viewdoc/ 941// download?doi=10.1.1.44.5740&rep=rep1&type=ps 942// and 943// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 944// 945// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 946// ------------------------------------------ 947// ((1 - t)^2 + t^2 + 2 (1 - t) t w) 948// 949// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 950// ------------------------------------------------ 951// {t^2 (2 - 2 w), t (-2 + 2 w), 1} 952// 953 954// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 955// 956// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 957// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 958// t^0 : -2 P0 w + 2 P1 w 959// 960// We disregard magnitude, so we can freely ignore the denominator of F', and 961// divide the numerator by 2 962// 963// coeff[0] for t^2 964// coeff[1] for t^1 965// coeff[2] for t^0 966// 967static void conic_deriv_coeff(const SkScalar src[], 968 SkScalar w, 969 SkScalar coeff[3]) { 970 const SkScalar P20 = src[4] - src[0]; 971 const SkScalar P10 = src[2] - src[0]; 972 const SkScalar wP10 = w * P10; 973 coeff[0] = w * P20 - P20; 974 coeff[1] = P20 - 2 * wP10; 975 coeff[2] = wP10; 976} 977 978static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 979 SkScalar coeff[3]; 980 conic_deriv_coeff(src, w, coeff); 981 982 SkScalar tValues[2]; 983 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 984 SkASSERT(0 == roots || 1 == roots); 985 986 if (1 == roots) { 987 *t = tValues[0]; 988 return true; 989 } 990 return false; 991} 992 993// We only interpolate one dimension at a time (the first, at +0, +3, +6). 994static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 995 SkScalar ab = SkScalarInterp(src[0], src[3], t); 996 SkScalar bc = SkScalarInterp(src[3], src[6], t); 997 dst[0] = ab; 998 dst[3] = SkScalarInterp(ab, bc, t); 999 dst[6] = bc; 1000} 1001 1002static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkPoint3 dst[3]) { 1003 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1004 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1005 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1006} 1007 1008static SkPoint project_down(const SkPoint3& src) { 1009 return {src.fX / src.fZ, src.fY / src.fZ}; 1010} 1011 1012// return false if infinity or NaN is generated; caller must check 1013bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1014 SkPoint3 tmp[3], tmp2[3]; 1015 1016 ratquad_mapTo3D(fPts, fW, tmp); 1017 1018 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1019 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1020 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1021 1022 dst[0].fPts[0] = fPts[0]; 1023 dst[0].fPts[1] = project_down(tmp2[0]); 1024 dst[0].fPts[2] = project_down(tmp2[1]); dst[1].fPts[0] = dst[0].fPts[2]; 1025 dst[1].fPts[1] = project_down(tmp2[2]); 1026 dst[1].fPts[2] = fPts[2]; 1027 1028 // to put in "standard form", where w0 and w2 are both 1, we compute the 1029 // new w1 as sqrt(w1*w1/w0*w2) 1030 // or 1031 // w1 /= sqrt(w0*w2) 1032 // 1033 // However, in our case, we know that for dst[0]: 1034 // w0 == 1, and for dst[1], w2 == 1 1035 // 1036 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1037 dst[0].fW = tmp2[0].fZ / root; 1038 dst[1].fW = tmp2[2].fZ / root; 1039 SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7); 1040 SkASSERT(0 == offsetof(SkConic, fPts[0].fX)); 1041 return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2); 1042} 1043 1044void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { 1045 if (0 == t1 || 1 == t2) { 1046 if (0 == t1 && 1 == t2) { 1047 *dst = *this; 1048 return; 1049 } else { 1050 SkConic pair[2]; 1051 if (this->chopAt(t1 ? t1 : t2, pair)) { 1052 *dst = pair[SkToBool(t1)]; 1053 return; 1054 } 1055 } 1056 } 1057 SkConicCoeff coeff(*this); 1058 Sk2s tt1(t1); 1059 Sk2s aXY = coeff.fNumer.eval(tt1); 1060 Sk2s aZZ = coeff.fDenom.eval(tt1); 1061 Sk2s midTT((t1 + t2) / 2); 1062 Sk2s dXY = coeff.fNumer.eval(midTT); 1063 Sk2s dZZ = coeff.fDenom.eval(midTT); 1064 Sk2s tt2(t2); 1065 Sk2s cXY = coeff.fNumer.eval(tt2); 1066 Sk2s cZZ = coeff.fDenom.eval(tt2); 1067 Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); 1068 Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); 1069 dst->fPts[0] = to_point(aXY / aZZ); 1070 dst->fPts[1] = to_point(bXY / bZZ); 1071 dst->fPts[2] = to_point(cXY / cZZ); 1072 Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); 1073 dst->fW = ww[0]; 1074} 1075 1076SkPoint SkConic::evalAt(SkScalar t) const { 1077 return to_point(SkConicCoeff(*this).eval(t)); 1078} 1079 1080SkVector SkConic::evalTangentAt(SkScalar t) const { 1081 // The derivative equation returns a zero tangent vector when t is 0 or 1, 1082 // and the control point is equal to the end point. 1083 // In this case, use the conic endpoints to compute the tangent. 1084 if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { 1085 return fPts[2] - fPts[0]; 1086 } 1087 Sk2s p0 = from_point(fPts[0]); 1088 Sk2s p1 = from_point(fPts[1]); 1089 Sk2s p2 = from_point(fPts[2]); 1090 Sk2s ww(fW); 1091 1092 Sk2s p20 = p2 - p0; 1093 Sk2s p10 = p1 - p0; 1094 1095 Sk2s C = ww * p10; 1096 Sk2s A = ww * p20 - p20; 1097 Sk2s B = p20 - C - C; 1098 1099 return to_vector(SkQuadCoeff(A, B, C).eval(t)); 1100} 1101 1102void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1103 SkASSERT(t >= 0 && t <= SK_Scalar1); 1104 1105 if (pt) { 1106 *pt = this->evalAt(t); 1107 } 1108 if (tangent) { 1109 *tangent = this->evalTangentAt(t); 1110 } 1111} 1112 1113static SkScalar subdivide_w_value(SkScalar w) { 1114 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1115} 1116 1117void SkConic::chop(SkConic * SK_RESTRICT dst) const { 1118 Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); 1119 SkScalar newW = subdivide_w_value(fW); 1120 1121 Sk2s p0 = from_point(fPts[0]); 1122 Sk2s p1 = from_point(fPts[1]); 1123 Sk2s p2 = from_point(fPts[2]); 1124 Sk2s ww(fW); 1125 1126 Sk2s wp1 = ww * p1; 1127 Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f); 1128 1129 dst[0].fPts[0] = fPts[0]; 1130 dst[0].fPts[1] = to_point((p0 + wp1) * scale); 1131 dst[0].fPts[2] = dst[1].fPts[0] = to_point(m); 1132 dst[1].fPts[1] = to_point((wp1 + p2) * scale); 1133 dst[1].fPts[2] = fPts[2]; 1134 1135 dst[0].fW = dst[1].fW = newW; 1136} 1137 1138/* 1139 * "High order approximation of conic sections by quadratic splines" 1140 * by Michael Floater, 1993 1141 */ 1142#define AS_QUAD_ERROR_SETUP \ 1143 SkScalar a = fW - 1; \ 1144 SkScalar k = a / (4 * (2 + a)); \ 1145 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1146 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1147 1148void SkConic::computeAsQuadError(SkVector* err) const { 1149 AS_QUAD_ERROR_SETUP 1150 err->set(x, y); 1151} 1152 1153bool SkConic::asQuadTol(SkScalar tol) const { 1154 AS_QUAD_ERROR_SETUP 1155 return (x * x + y * y) <= tol * tol; 1156} 1157 1158// Limit the number of suggested quads to approximate a conic 1159#define kMaxConicToQuadPOW2 5 1160 1161int SkConic::computeQuadPOW2(SkScalar tol) const { 1162 if (tol < 0 || !SkScalarIsFinite(tol) || !SkPointPriv::AreFinite(fPts, 3)) { 1163 return 0; 1164 } 1165 1166 AS_QUAD_ERROR_SETUP 1167 1168 SkScalar error = SkScalarSqrt(x * x + y * y); 1169 int pow2; 1170 for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { 1171 if (error <= tol) { 1172 break; 1173 } 1174 error *= 0.25f; 1175 } 1176 // float version -- using ceil gives the same results as the above. 1177 if (false) { 1178 SkScalar err = SkScalarSqrt(x * x + y * y); 1179 if (err <= tol) { 1180 return 0; 1181 } 1182 SkScalar tol2 = tol * tol; 1183 if (tol2 == 0) { 1184 return kMaxConicToQuadPOW2; 1185 } 1186 SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; 1187 int altPow2 = SkScalarCeilToInt(fpow2); 1188 if (altPow2 != pow2) { 1189 SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); 1190 } 1191 pow2 = altPow2; 1192 } 1193 return pow2; 1194} 1195 1196// This was originally developed and tested for pathops: see SkOpTypes.h 1197// returns true if (a <= b <= c) || (a >= b >= c) 1198static bool between(SkScalar a, SkScalar b, SkScalar c) { 1199 return (a - b) * (c - b) <= 0; 1200} 1201 1202static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1203 SkASSERT(level >= 0); 1204 1205 if (0 == level) { 1206 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1207 return pts + 2; 1208 } else { 1209 SkConic dst[2]; 1210 src.chop(dst); 1211 const SkScalar startY = src.fPts[0].fY; 1212 const SkScalar endY = src.fPts[2].fY; 1213 if (between(startY, src.fPts[1].fY, endY)) { 1214 // If the input is monotonic and the output is not, the scan converter hangs. 1215 // Ensure that the chopped conics maintain their y-order. 1216 SkScalar midY = dst[0].fPts[2].fY; 1217 if (!between(startY, midY, endY)) { 1218 // If the computed midpoint is outside the ends, move it to the closer one. 1219 SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY; 1220 dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY; 1221 } 1222 if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) { 1223 // If the 1st control is not between the start and end, put it at the start. 1224 // This also reduces the quad to a line. 1225 dst[0].fPts[1].fY = startY; 1226 } 1227 if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) { 1228 // If the 2nd control is not between the start and end, put it at the end. 1229 // This also reduces the quad to a line. 1230 dst[1].fPts[1].fY = endY; 1231 } 1232 // Verify that all five points are in order. 1233 SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)); 1234 SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY)); 1235 SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY)); 1236 } 1237 --level; 1238 pts = subdivide(dst[0], pts, level); 1239 return subdivide(dst[1], pts, level); 1240 } 1241} 1242 1243int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1244 SkASSERT(pow2 >= 0); 1245 *pts = fPts[0]; 1246 SkDEBUGCODE(SkPoint* endPts); 1247 if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ... 1248 SkConic dst[2]; 1249 this->chop(dst); 1250 // check to see if the first chop generates a pair of lines 1251 if (SkPointPriv::EqualsWithinTolerance(dst[0].fPts[1], dst[0].fPts[2]) && 1252 SkPointPriv::EqualsWithinTolerance(dst[1].fPts[0], dst[1].fPts[1])) { 1253 pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines 1254 pts[4] = dst[1].fPts[2]; 1255 pow2 = 1; 1256 SkDEBUGCODE(endPts = &pts[5]); 1257 goto commonFinitePtCheck; 1258 } 1259 } 1260 SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2); 1261commonFinitePtCheck: 1262 const int quadCount = 1 << pow2; 1263 const int ptCount = 2 * quadCount + 1; 1264 SkASSERT(endPts - pts == ptCount); 1265 if (!SkPointPriv::AreFinite(pts, ptCount)) { 1266 // if we generated a non-finite, pin ourselves to the middle of the hull, 1267 // as our first and last are already on the first/last pts of the hull. 1268 for (int i = 1; i < ptCount - 1; ++i) { 1269 pts[i] = fPts[1]; 1270 } 1271 } 1272 return 1 << pow2; 1273} 1274 1275bool SkConic::findXExtrema(SkScalar* t) const { 1276 return conic_find_extrema(&fPts[0].fX, fW, t); 1277} 1278 1279bool SkConic::findYExtrema(SkScalar* t) const { 1280 return conic_find_extrema(&fPts[0].fY, fW, t); 1281} 1282 1283bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1284 SkScalar t; 1285 if (this->findXExtrema(&t)) { 1286 if (!this->chopAt(t, dst)) { 1287 // if chop can't return finite values, don't chop 1288 return false; 1289 } 1290 // now clean-up the middle, since we know t was meant to be at 1291 // an X-extrema 1292 SkScalar value = dst[0].fPts[2].fX; 1293 dst[0].fPts[1].fX = value; 1294 dst[1].fPts[0].fX = value; 1295 dst[1].fPts[1].fX = value; 1296 return true; 1297 } 1298 return false; 1299} 1300 1301bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1302 SkScalar t; 1303 if (this->findYExtrema(&t)) { 1304 if (!this->chopAt(t, dst)) { 1305 // if chop can't return finite values, don't chop 1306 return false; 1307 } 1308 // now clean-up the middle, since we know t was meant to be at 1309 // an Y-extrema 1310 SkScalar value = dst[0].fPts[2].fY; 1311 dst[0].fPts[1].fY = value; 1312 dst[1].fPts[0].fY = value; 1313 dst[1].fPts[1].fY = value; 1314 return true; 1315 } 1316 return false; 1317} 1318 1319void SkConic::computeTightBounds(SkRect* bounds) const { 1320 SkPoint pts[4]; 1321 pts[0] = fPts[0]; 1322 pts[1] = fPts[2]; 1323 int count = 2; 1324 1325 SkScalar t; 1326 if (this->findXExtrema(&t)) { 1327 this->evalAt(t, &pts[count++]); 1328 } 1329 if (this->findYExtrema(&t)) { 1330 this->evalAt(t, &pts[count++]); 1331 } 1332 bounds->set(pts, count); 1333} 1334 1335void SkConic::computeFastBounds(SkRect* bounds) const { 1336 bounds->set(fPts, 3); 1337} 1338 1339#if 0 // unimplemented 1340bool SkConic::findMaxCurvature(SkScalar* t) const { 1341 // TODO: Implement me 1342 return false; 1343} 1344#endif 1345 1346SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, 1347 const SkMatrix& matrix) { 1348 if (!matrix.hasPerspective()) { 1349 return w; 1350 } 1351 1352 SkPoint3 src[3], dst[3]; 1353 1354 ratquad_mapTo3D(pts, w, src); 1355 1356 matrix.mapHomogeneousPoints(dst, src, 3); 1357 1358 // w' = sqrt(w1*w1/w0*w2) 1359 SkScalar w0 = dst[0].fZ; 1360 SkScalar w1 = dst[1].fZ; 1361 SkScalar w2 = dst[2].fZ; 1362 w = SkScalarSqrt((w1 * w1) / (w0 * w2)); 1363 return w; 1364} 1365 1366int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, 1367 const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { 1368 // rotate by x,y so that uStart is (1.0) 1369 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1370 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1371 1372 SkScalar absY = SkScalarAbs(y); 1373 1374 // check for (effectively) coincident vectors 1375 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1376 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1377 if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) || 1378 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1379 return 0; 1380 } 1381 1382 if (dir == kCCW_SkRotationDirection) { 1383 y = -y; 1384 } 1385 1386 // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? 1387 // 0 == [0 .. 90) 1388 // 1 == [90 ..180) 1389 // 2 == [180..270) 1390 // 3 == [270..360) 1391 // 1392 int quadrant = 0; 1393 if (0 == y) { 1394 quadrant = 2; // 180 1395 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1396 } else if (0 == x) { 1397 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1398 quadrant = y > 0 ? 1 : 3; // 90 : 270 1399 } else { 1400 if (y < 0) { 1401 quadrant += 2; 1402 } 1403 if ((x < 0) != (y < 0)) { 1404 quadrant += 1; 1405 } 1406 } 1407 1408 const SkPoint quadrantPts[] = { 1409 { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } 1410 }; 1411 const SkScalar quadrantWeight = SK_ScalarRoot2Over2; 1412 1413 int conicCount = quadrant; 1414 for (int i = 0; i < conicCount; ++i) { 1415 dst[i].set(&quadrantPts[i * 2], quadrantWeight); 1416 } 1417 1418 // Now compute any remaing (sub-90-degree) arc for the last conic 1419 const SkPoint finalP = { x, y }; 1420 const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector 1421 const SkScalar dot = SkVector::DotProduct(lastQ, finalP); 1422 SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); 1423 1424 if (dot < 1) { 1425 SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; 1426 // compute the bisector vector, and then rescale to be the off-curve point. 1427 // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get 1428 // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. 1429 // This is nice, since our computed weight is cos(theta/2) as well! 1430 // 1431 const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); 1432 offCurve.setLength(SkScalarInvert(cosThetaOver2)); 1433 if (!SkPointPriv::EqualsWithinTolerance(lastQ, offCurve)) { 1434 dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); 1435 conicCount += 1; 1436 } 1437 } 1438 1439 // now handle counter-clockwise and the initial unitStart rotation 1440 SkMatrix matrix; 1441 matrix.setSinCos(uStart.fY, uStart.fX); 1442 if (dir == kCCW_SkRotationDirection) { 1443 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1444 } 1445 if (userMatrix) { 1446 matrix.postConcat(*userMatrix); 1447 } 1448 for (int i = 0; i < conicCount; ++i) { 1449 matrix.mapPoints(dst[i].fPts, 3); 1450 } 1451 return conicCount; 1452} 1453