1// Another approach is to start with the implicit form of one curve and solve 2// (seek implicit coefficients in QuadraticParameter.cpp 3// by substituting in the parametric form of the other. 4// The downside of this approach is that early rejects are difficult to come by. 5// http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html#step 6 7 8#include "SkDQuadImplicit.h" 9#include "SkIntersections.h" 10#include "SkPathOpsLine.h" 11#include "SkQuarticRoot.h" 12#include "SkTArray.h" 13#include "SkTSort.h" 14 15/* given the implicit form 0 = Ax^2 + Bxy + Cy^2 + Dx + Ey + F 16 * and given x = at^2 + bt + c (the parameterized form) 17 * y = dt^2 + et + f 18 * then 19 * 0 = A(at^2+bt+c)(at^2+bt+c)+B(at^2+bt+c)(dt^2+et+f)+C(dt^2+et+f)(dt^2+et+f)+D(at^2+bt+c)+E(dt^2+et+f)+F 20 */ 21 22static int findRoots(const SkDQuadImplicit& i, const SkDQuad& quad, double roots[4], 23 bool oneHint, bool flip, int firstCubicRoot) { 24 SkDQuad flipped; 25 const SkDQuad& q = flip ? (flipped = quad.flip()) : quad; 26 double a, b, c; 27 SkDQuad::SetABC(&q[0].fX, &a, &b, &c); 28 double d, e, f; 29 SkDQuad::SetABC(&q[0].fY, &d, &e, &f); 30 const double t4 = i.x2() * a * a 31 + i.xy() * a * d 32 + i.y2() * d * d; 33 const double t3 = 2 * i.x2() * a * b 34 + i.xy() * (a * e + b * d) 35 + 2 * i.y2() * d * e; 36 const double t2 = i.x2() * (b * b + 2 * a * c) 37 + i.xy() * (c * d + b * e + a * f) 38 + i.y2() * (e * e + 2 * d * f) 39 + i.x() * a 40 + i.y() * d; 41 const double t1 = 2 * i.x2() * b * c 42 + i.xy() * (c * e + b * f) 43 + 2 * i.y2() * e * f 44 + i.x() * b 45 + i.y() * e; 46 const double t0 = i.x2() * c * c 47 + i.xy() * c * f 48 + i.y2() * f * f 49 + i.x() * c 50 + i.y() * f 51 + i.c(); 52 int rootCount = SkReducedQuarticRoots(t4, t3, t2, t1, t0, oneHint, roots); 53 if (rootCount < 0) { 54 rootCount = SkQuarticRootsReal(firstCubicRoot, t4, t3, t2, t1, t0, roots); 55 } 56 if (flip) { 57 for (int index = 0; index < rootCount; ++index) { 58 roots[index] = 1 - roots[index]; 59 } 60 } 61 return rootCount; 62} 63 64static int addValidRoots(const double roots[4], const int count, double valid[4]) { 65 int result = 0; 66 int index; 67 for (index = 0; index < count; ++index) { 68 if (!approximately_zero_or_more(roots[index]) || !approximately_one_or_less(roots[index])) { 69 continue; 70 } 71 double t = 1 - roots[index]; 72 if (approximately_less_than_zero(t)) { 73 t = 0; 74 } else if (approximately_greater_than_one(t)) { 75 t = 1; 76 } 77 valid[result++] = t; 78 } 79 return result; 80} 81 82static bool only_end_pts_in_common(const SkDQuad& q1, const SkDQuad& q2) { 83// the idea here is to see at minimum do a quick reject by rotating all points 84// to either side of the line formed by connecting the endpoints 85// if the opposite curves points are on the line or on the other side, the 86// curves at most intersect at the endpoints 87 for (int oddMan = 0; oddMan < 3; ++oddMan) { 88 const SkDPoint* endPt[2]; 89 for (int opp = 1; opp < 3; ++opp) { 90 int end = oddMan ^ opp; // choose a value not equal to oddMan 91 if (3 == end) { // and correct so that largest value is 1 or 2 92 end = opp; 93 } 94 endPt[opp - 1] = &q1[end]; 95 } 96 double origX = endPt[0]->fX; 97 double origY = endPt[0]->fY; 98 double adj = endPt[1]->fX - origX; 99 double opp = endPt[1]->fY - origY; 100 double sign = (q1[oddMan].fY - origY) * adj - (q1[oddMan].fX - origX) * opp; 101 if (approximately_zero(sign)) { 102 goto tryNextHalfPlane; 103 } 104 for (int n = 0; n < 3; ++n) { 105 double test = (q2[n].fY - origY) * adj - (q2[n].fX - origX) * opp; 106 if (test * sign > 0 && !precisely_zero(test)) { 107 goto tryNextHalfPlane; 108 } 109 } 110 return true; 111tryNextHalfPlane: 112 ; 113 } 114 return false; 115} 116 117// returns false if there's more than one intercept or the intercept doesn't match the point 118// returns true if the intercept was successfully added or if the 119// original quads need to be subdivided 120static bool add_intercept(const SkDQuad& q1, const SkDQuad& q2, double tMin, double tMax, 121 SkIntersections* i, bool* subDivide) { 122 double tMid = (tMin + tMax) / 2; 123 SkDPoint mid = q2.ptAtT(tMid); 124 SkDLine line; 125 line[0] = line[1] = mid; 126 SkDVector dxdy = q2.dxdyAtT(tMid); 127 line[0] -= dxdy; 128 line[1] += dxdy; 129 SkIntersections rootTs; 130 rootTs.allowNear(false); 131 int roots = rootTs.intersect(q1, line); 132 if (roots == 0) { 133 if (subDivide) { 134 *subDivide = true; 135 } 136 return true; 137 } 138 if (roots == 2) { 139 return false; 140 } 141 SkDPoint pt2 = q1.ptAtT(rootTs[0][0]); 142 if (!pt2.approximatelyEqual(mid)) { 143 return false; 144 } 145 i->insertSwap(rootTs[0][0], tMid, pt2); 146 return true; 147} 148 149static bool is_linear_inner(const SkDQuad& q1, double t1s, double t1e, const SkDQuad& q2, 150 double t2s, double t2e, SkIntersections* i, bool* subDivide) { 151 SkDQuad hull = q1.subDivide(t1s, t1e); 152 SkDLine line = {{hull[2], hull[0]}}; 153 const SkDLine* testLines[] = { &line, (const SkDLine*) &hull[0], (const SkDLine*) &hull[1] }; 154 const size_t kTestCount = SK_ARRAY_COUNT(testLines); 155 SkSTArray<kTestCount * 2, double, true> tsFound; 156 for (size_t index = 0; index < kTestCount; ++index) { 157 SkIntersections rootTs; 158 rootTs.allowNear(false); 159 int roots = rootTs.intersect(q2, *testLines[index]); 160 for (int idx2 = 0; idx2 < roots; ++idx2) { 161 double t = rootTs[0][idx2]; 162#ifdef SK_DEBUG 163 SkDPoint qPt = q2.ptAtT(t); 164 SkDPoint lPt = testLines[index]->ptAtT(rootTs[1][idx2]); 165 SkASSERT(qPt.approximatelyPEqual(lPt)); 166#endif 167 if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) { 168 continue; 169 } 170 tsFound.push_back(rootTs[0][idx2]); 171 } 172 } 173 int tCount = tsFound.count(); 174 if (tCount <= 0) { 175 return true; 176 } 177 double tMin, tMax; 178 if (tCount == 1) { 179 tMin = tMax = tsFound[0]; 180 } else { 181 SkASSERT(tCount > 1); 182 SkTQSort<double>(tsFound.begin(), tsFound.end() - 1); 183 tMin = tsFound[0]; 184 tMax = tsFound[tsFound.count() - 1]; 185 } 186 SkDPoint end = q2.ptAtT(t2s); 187 bool startInTriangle = hull.pointInHull(end); 188 if (startInTriangle) { 189 tMin = t2s; 190 } 191 end = q2.ptAtT(t2e); 192 bool endInTriangle = hull.pointInHull(end); 193 if (endInTriangle) { 194 tMax = t2e; 195 } 196 int split = 0; 197 SkDVector dxy1, dxy2; 198 if (tMin != tMax || tCount > 2) { 199 dxy2 = q2.dxdyAtT(tMin); 200 for (int index = 1; index < tCount; ++index) { 201 dxy1 = dxy2; 202 dxy2 = q2.dxdyAtT(tsFound[index]); 203 double dot = dxy1.dot(dxy2); 204 if (dot < 0) { 205 split = index - 1; 206 break; 207 } 208 } 209 } 210 if (split == 0) { // there's one point 211 if (add_intercept(q1, q2, tMin, tMax, i, subDivide)) { 212 return true; 213 } 214 i->swap(); 215 return is_linear_inner(q2, tMin, tMax, q1, t1s, t1e, i, subDivide); 216 } 217 // At this point, we have two ranges of t values -- treat each separately at the split 218 bool result; 219 if (add_intercept(q1, q2, tMin, tsFound[split - 1], i, subDivide)) { 220 result = true; 221 } else { 222 i->swap(); 223 result = is_linear_inner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i, subDivide); 224 } 225 if (add_intercept(q1, q2, tsFound[split], tMax, i, subDivide)) { 226 result = true; 227 } else { 228 i->swap(); 229 result |= is_linear_inner(q2, tsFound[split], tMax, q1, t1s, t1e, i, subDivide); 230 } 231 return result; 232} 233 234static double flat_measure(const SkDQuad& q) { 235 SkDVector mid = q[1] - q[0]; 236 SkDVector dxy = q[2] - q[0]; 237 double length = dxy.length(); // OPTIMIZE: get rid of sqrt 238 return fabs(mid.cross(dxy) / length); 239} 240 241// FIXME ? should this measure both and then use the quad that is the flattest as the line? 242static bool is_linear(const SkDQuad& q1, const SkDQuad& q2, SkIntersections* i) { 243 double measure = flat_measure(q1); 244 // OPTIMIZE: (get rid of sqrt) use approximately_zero 245 if (!approximately_zero_sqrt(measure)) { 246 return false; 247 } 248 return is_linear_inner(q1, 0, 1, q2, 0, 1, i, NULL); 249} 250 251// FIXME: if flat measure is sufficiently large, then probably the quartic solution failed 252// avoid imprecision incurred with chopAt 253static void relaxed_is_linear(const SkDQuad* q1, double s1, double e1, const SkDQuad* q2, 254 double s2, double e2, SkIntersections* i) { 255 double m1 = flat_measure(*q1); 256 double m2 = flat_measure(*q2); 257 i->reset(); 258 const SkDQuad* rounder, *flatter; 259 double sf, midf, ef, sr, er; 260 if (m2 < m1) { 261 rounder = q1; 262 sr = s1; 263 er = e1; 264 flatter = q2; 265 sf = s2; 266 midf = (s2 + e2) / 2; 267 ef = e2; 268 } else { 269 rounder = q2; 270 sr = s2; 271 er = e2; 272 flatter = q1; 273 sf = s1; 274 midf = (s1 + e1) / 2; 275 ef = e1; 276 } 277 bool subDivide = false; 278 is_linear_inner(*flatter, sf, ef, *rounder, sr, er, i, &subDivide); 279 if (subDivide) { 280 relaxed_is_linear(flatter, sf, midf, rounder, sr, er, i); 281 relaxed_is_linear(flatter, midf, ef, rounder, sr, er, i); 282 } 283 if (m2 < m1) { 284 i->swapPts(); 285 } 286} 287 288// each time through the loop, this computes values it had from the last loop 289// if i == j == 1, the center values are still good 290// otherwise, for i != 1 or j != 1, four of the values are still good 291// and if i == 1 ^ j == 1, an additional value is good 292static bool binary_search(const SkDQuad& quad1, const SkDQuad& quad2, double* t1Seed, 293 double* t2Seed, SkDPoint* pt) { 294 double tStep = ROUGH_EPSILON; 295 SkDPoint t1[3], t2[3]; 296 int calcMask = ~0; 297 do { 298 if (calcMask & (1 << 1)) t1[1] = quad1.ptAtT(*t1Seed); 299 if (calcMask & (1 << 4)) t2[1] = quad2.ptAtT(*t2Seed); 300 if (t1[1].approximatelyEqual(t2[1])) { 301 *pt = t1[1]; 302 #if ONE_OFF_DEBUG 303 SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) == (%1.9g,%1.9g)\n", __FUNCTION__, 304 t1Seed, t2Seed, t1[1].fX, t1[1].fY, t2[1].fX, t2[1].fY); 305 #endif 306 return true; 307 } 308 if (calcMask & (1 << 0)) t1[0] = quad1.ptAtT(*t1Seed - tStep); 309 if (calcMask & (1 << 2)) t1[2] = quad1.ptAtT(*t1Seed + tStep); 310 if (calcMask & (1 << 3)) t2[0] = quad2.ptAtT(*t2Seed - tStep); 311 if (calcMask & (1 << 5)) t2[2] = quad2.ptAtT(*t2Seed + tStep); 312 double dist[3][3]; 313 // OPTIMIZE: using calcMask value permits skipping some distance calcuations 314 // if prior loop's results are moved to correct slot for reuse 315 dist[1][1] = t1[1].distanceSquared(t2[1]); 316 int best_i = 1, best_j = 1; 317 for (int i = 0; i < 3; ++i) { 318 for (int j = 0; j < 3; ++j) { 319 if (i == 1 && j == 1) { 320 continue; 321 } 322 dist[i][j] = t1[i].distanceSquared(t2[j]); 323 if (dist[best_i][best_j] > dist[i][j]) { 324 best_i = i; 325 best_j = j; 326 } 327 } 328 } 329 if (best_i == 1 && best_j == 1) { 330 tStep /= 2; 331 if (tStep < FLT_EPSILON_HALF) { 332 break; 333 } 334 calcMask = (1 << 0) | (1 << 2) | (1 << 3) | (1 << 5); 335 continue; 336 } 337 if (best_i == 0) { 338 *t1Seed -= tStep; 339 t1[2] = t1[1]; 340 t1[1] = t1[0]; 341 calcMask = 1 << 0; 342 } else if (best_i == 2) { 343 *t1Seed += tStep; 344 t1[0] = t1[1]; 345 t1[1] = t1[2]; 346 calcMask = 1 << 2; 347 } else { 348 calcMask = 0; 349 } 350 if (best_j == 0) { 351 *t2Seed -= tStep; 352 t2[2] = t2[1]; 353 t2[1] = t2[0]; 354 calcMask |= 1 << 3; 355 } else if (best_j == 2) { 356 *t2Seed += tStep; 357 t2[0] = t2[1]; 358 t2[1] = t2[2]; 359 calcMask |= 1 << 5; 360 } 361 } while (true); 362#if ONE_OFF_DEBUG 363 SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) != (%1.9g,%1.9g) %s\n", __FUNCTION__, 364 t1Seed, t2Seed, t1[1].fX, t1[1].fY, t1[2].fX, t1[2].fY); 365#endif 366 return false; 367} 368 369static void lookNearEnd(const SkDQuad& q1, const SkDQuad& q2, int testT, 370 const SkIntersections& orig, bool swap, SkIntersections* i) { 371 if (orig.used() == 1 && orig[!swap][0] == testT) { 372 return; 373 } 374 if (orig.used() == 2 && orig[!swap][1] == testT) { 375 return; 376 } 377 SkDLine tmpLine; 378 int testTIndex = testT << 1; 379 tmpLine[0] = tmpLine[1] = q2[testTIndex]; 380 tmpLine[1].fX += q2[1].fY - q2[testTIndex].fY; 381 tmpLine[1].fY -= q2[1].fX - q2[testTIndex].fX; 382 SkIntersections impTs; 383 impTs.intersectRay(q1, tmpLine); 384 for (int index = 0; index < impTs.used(); ++index) { 385 SkDPoint realPt = impTs.pt(index); 386 if (!tmpLine[0].approximatelyEqual(realPt)) { 387 continue; 388 } 389 if (swap) { 390 i->insert(testT, impTs[0][index], tmpLine[0]); 391 } else { 392 i->insert(impTs[0][index], testT, tmpLine[0]); 393 } 394 } 395} 396 397int SkIntersections::intersect(const SkDQuad& q1, const SkDQuad& q2) { 398 fMax = 4; 399 // if the quads share an end point, check to see if they overlap 400 for (int i1 = 0; i1 < 3; i1 += 2) { 401 for (int i2 = 0; i2 < 3; i2 += 2) { 402 if (q1[i1].asSkPoint() == q2[i2].asSkPoint()) { 403 insert(i1 >> 1, i2 >> 1, q1[i1]); 404 } 405 } 406 } 407 SkASSERT(fUsed < 3); 408 if (only_end_pts_in_common(q1, q2)) { 409 return fUsed; 410 } 411 if (only_end_pts_in_common(q2, q1)) { 412 return fUsed; 413 } 414 // see if either quad is really a line 415 // FIXME: figure out why reduce step didn't find this earlier 416 if (is_linear(q1, q2, this)) { 417 return fUsed; 418 } 419 SkIntersections swapped; 420 swapped.setMax(fMax); 421 if (is_linear(q2, q1, &swapped)) { 422 swapped.swapPts(); 423 set(swapped); 424 return fUsed; 425 } 426 SkIntersections copyI(*this); 427 lookNearEnd(q1, q2, 0, *this, false, ©I); 428 lookNearEnd(q1, q2, 1, *this, false, ©I); 429 lookNearEnd(q2, q1, 0, *this, true, ©I); 430 lookNearEnd(q2, q1, 1, *this, true, ©I); 431 int innerEqual = 0; 432 if (copyI.fUsed >= 2) { 433 SkASSERT(copyI.fUsed <= 4); 434 double width = copyI[0][1] - copyI[0][0]; 435 int midEnd = 1; 436 for (int index = 2; index < copyI.fUsed; ++index) { 437 double testWidth = copyI[0][index] - copyI[0][index - 1]; 438 if (testWidth <= width) { 439 continue; 440 } 441 midEnd = index; 442 } 443 for (int index = 0; index < 2; ++index) { 444 double testT = (copyI[0][midEnd] * (index + 1) 445 + copyI[0][midEnd - 1] * (2 - index)) / 3; 446 SkDPoint testPt1 = q1.ptAtT(testT); 447 testT = (copyI[1][midEnd] * (index + 1) + copyI[1][midEnd - 1] * (2 - index)) / 3; 448 SkDPoint testPt2 = q2.ptAtT(testT); 449 innerEqual += testPt1.approximatelyEqual(testPt2); 450 } 451 } 452 bool expectCoincident = copyI.fUsed >= 2 && innerEqual == 2; 453 if (expectCoincident) { 454 reset(); 455 insertCoincident(copyI[0][0], copyI[1][0], copyI.fPt[0]); 456 int last = copyI.fUsed - 1; 457 insertCoincident(copyI[0][last], copyI[1][last], copyI.fPt[last]); 458 return fUsed; 459 } 460 SkDQuadImplicit i1(q1); 461 SkDQuadImplicit i2(q2); 462 int index; 463 bool flip1 = q1[2] == q2[0]; 464 bool flip2 = q1[0] == q2[2]; 465 bool useCubic = q1[0] == q2[0]; 466 double roots1[4]; 467 int rootCount = findRoots(i2, q1, roots1, useCubic, flip1, 0); 468 // OPTIMIZATION: could short circuit here if all roots are < 0 or > 1 469 double roots1Copy[4]; 470 int r1Count = addValidRoots(roots1, rootCount, roots1Copy); 471 SkDPoint pts1[4]; 472 for (index = 0; index < r1Count; ++index) { 473 pts1[index] = q1.ptAtT(roots1Copy[index]); 474 } 475 double roots2[4]; 476 int rootCount2 = findRoots(i1, q2, roots2, useCubic, flip2, 0); 477 double roots2Copy[4]; 478 int r2Count = addValidRoots(roots2, rootCount2, roots2Copy); 479 SkDPoint pts2[4]; 480 for (index = 0; index < r2Count; ++index) { 481 pts2[index] = q2.ptAtT(roots2Copy[index]); 482 } 483 if (r1Count == r2Count && r1Count <= 1) { 484 if (r1Count == 1 && used() == 0) { 485 if (pts1[0].approximatelyEqual(pts2[0])) { 486 insert(roots1Copy[0], roots2Copy[0], pts1[0]); 487 } else if (pts1[0].moreRoughlyEqual(pts2[0])) { 488 // experiment: try to find intersection by chasing t 489 if (binary_search(q1, q2, roots1Copy, roots2Copy, pts1)) { 490 insert(roots1Copy[0], roots2Copy[0], pts1[0]); 491 } 492 } 493 } 494 return fUsed; 495 } 496 int closest[4]; 497 double dist[4]; 498 bool foundSomething = false; 499 for (index = 0; index < r1Count; ++index) { 500 dist[index] = DBL_MAX; 501 closest[index] = -1; 502 for (int ndex2 = 0; ndex2 < r2Count; ++ndex2) { 503 if (!pts2[ndex2].approximatelyEqual(pts1[index])) { 504 continue; 505 } 506 double dx = pts2[ndex2].fX - pts1[index].fX; 507 double dy = pts2[ndex2].fY - pts1[index].fY; 508 double distance = dx * dx + dy * dy; 509 if (dist[index] <= distance) { 510 continue; 511 } 512 for (int outer = 0; outer < index; ++outer) { 513 if (closest[outer] != ndex2) { 514 continue; 515 } 516 if (dist[outer] < distance) { 517 goto next; 518 } 519 closest[outer] = -1; 520 } 521 dist[index] = distance; 522 closest[index] = ndex2; 523 foundSomething = true; 524 next: 525 ; 526 } 527 } 528 if (r1Count && r2Count && !foundSomething) { 529 relaxed_is_linear(&q1, 0, 1, &q2, 0, 1, this); 530 return fUsed; 531 } 532 int used = 0; 533 do { 534 double lowest = DBL_MAX; 535 int lowestIndex = -1; 536 for (index = 0; index < r1Count; ++index) { 537 if (closest[index] < 0) { 538 continue; 539 } 540 if (roots1Copy[index] < lowest) { 541 lowestIndex = index; 542 lowest = roots1Copy[index]; 543 } 544 } 545 if (lowestIndex < 0) { 546 break; 547 } 548 insert(roots1Copy[lowestIndex], roots2Copy[closest[lowestIndex]], 549 pts1[lowestIndex]); 550 closest[lowestIndex] = -1; 551 } while (++used < r1Count); 552 return fUsed; 553} 554