1 /* 2 * Copyright 2012 Google Inc. 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 "CubicUtilities.h" 9 #include "CurveIntersection.h" 10 #include "Intersections.h" 11 #include "IntersectionUtilities.h" 12 #include "LineIntersection.h" 13 #include "LineUtilities.h" 14 #include "QuadraticUtilities.h" 15 #include "TSearch.h" 16 17 #if 0 18 #undef ONE_OFF_DEBUG 19 #define ONE_OFF_DEBUG 0 20 #endif 21 22 #if ONE_OFF_DEBUG 23 static const double tLimits1[2][2] = {{0.36, 0.37}, {0.63, 0.64}}; 24 static const double tLimits2[2][2] = {{-0.865211397, -0.865215212}, {-0.865207696, -0.865208078}}; 25 #endif 26 27 #define DEBUG_QUAD_PART 0 28 #define SWAP_TOP_DEBUG 0 29 30 static int quadPart(const Cubic& cubic, double tStart, double tEnd, Quadratic& simple) { 31 Cubic part; 32 sub_divide(cubic, tStart, tEnd, part); 33 Quadratic quad; 34 demote_cubic_to_quad(part, quad); 35 // FIXME: should reduceOrder be looser in this use case if quartic is going to blow up on an 36 // extremely shallow quadratic? 37 int order = reduceOrder(quad, simple, kReduceOrder_TreatAsFill); 38 #if DEBUG_QUAD_PART 39 SkDebugf("%s cubic=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g) t=(%1.17g,%1.17g)\n", 40 __FUNCTION__, cubic[0].x, cubic[0].y, cubic[1].x, cubic[1].y, cubic[2].x, cubic[2].y, 41 cubic[3].x, cubic[3].y, tStart, tEnd); 42 SkDebugf("%s part=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)" 43 " quad=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)\n", __FUNCTION__, part[0].x, part[0].y, 44 part[1].x, part[1].y, part[2].x, part[2].y, part[3].x, part[3].y, quad[0].x, quad[0].y, 45 quad[1].x, quad[1].y, quad[2].x, quad[2].y); 46 SkDebugf("%s simple=(%1.17g,%1.17g", __FUNCTION__, simple[0].x, simple[0].y); 47 if (order > 1) { 48 SkDebugf(" %1.17g,%1.17g", simple[1].x, simple[1].y); 49 } 50 if (order > 2) { 51 SkDebugf(" %1.17g,%1.17g", simple[2].x, simple[2].y); 52 } 53 SkDebugf(")\n"); 54 SkASSERT(order < 4 && order > 0); 55 #endif 56 return order; 57 } 58 59 static void intersectWithOrder(const Quadratic& simple1, int order1, const Quadratic& simple2, 60 int order2, Intersections& i) { 61 if (order1 == 3 && order2 == 3) { 62 intersect2(simple1, simple2, i); 63 } else if (order1 <= 2 && order2 <= 2) { 64 intersect((const _Line&) simple1, (const _Line&) simple2, i); 65 } else if (order1 == 3 && order2 <= 2) { 66 intersect(simple1, (const _Line&) simple2, i); 67 } else { 68 SkASSERT(order1 <= 2 && order2 == 3); 69 intersect(simple2, (const _Line&) simple1, i); 70 for (int s = 0; s < i.fUsed; ++s) { 71 SkTSwap(i.fT[0][s], i.fT[1][s]); 72 } 73 } 74 } 75 76 // this flavor centers potential intersections recursively. In contrast, '2' may inadvertently 77 // chase intersections near quadratic ends, requiring odd hacks to find them. 78 static bool intersect3(const Cubic& cubic1, double t1s, double t1e, const Cubic& cubic2, 79 double t2s, double t2e, double precisionScale, Intersections& i) { 80 i.upDepth(); 81 bool result = false; 82 Cubic c1, c2; 83 sub_divide(cubic1, t1s, t1e, c1); 84 sub_divide(cubic2, t2s, t2e, c2); 85 SkTDArray<double> ts1; 86 // OPTIMIZE: if c1 == c2, call once (happens when detecting self-intersection) 87 cubic_to_quadratics(c1, calcPrecision(c1) * precisionScale, ts1); 88 SkTDArray<double> ts2; 89 cubic_to_quadratics(c2, calcPrecision(c2) * precisionScale, ts2); 90 double t1Start = t1s; 91 int ts1Count = ts1.count(); 92 for (int i1 = 0; i1 <= ts1Count; ++i1) { 93 const double tEnd1 = i1 < ts1Count ? ts1[i1] : 1; 94 const double t1 = t1s + (t1e - t1s) * tEnd1; 95 Quadratic s1; 96 int o1 = quadPart(cubic1, t1Start, t1, s1); 97 double t2Start = t2s; 98 int ts2Count = ts2.count(); 99 for (int i2 = 0; i2 <= ts2Count; ++i2) { 100 const double tEnd2 = i2 < ts2Count ? ts2[i2] : 1; 101 const double t2 = t2s + (t2e - t2s) * tEnd2; 102 if (cubic1 == cubic2 && t1Start >= t2Start) { 103 t2Start = t2; 104 continue; 105 } 106 Quadratic s2; 107 int o2 = quadPart(cubic2, t2Start, t2, s2); 108 #if ONE_OFF_DEBUG 109 char tab[] = " "; 110 if (tLimits1[0][0] >= t1Start && tLimits1[0][1] <= t1 111 && tLimits1[1][0] >= t2Start && tLimits1[1][1] <= t2) { 112 Cubic cSub1, cSub2; 113 sub_divide(cubic1, t1Start, t1, cSub1); 114 sub_divide(cubic2, t2Start, t2, cSub2); 115 SkDebugf("%.*s %s t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)", i.depth()*2, tab, __FUNCTION__, 116 t1Start, t1, t2Start, t2); 117 Intersections xlocals; 118 intersectWithOrder(s1, o1, s2, o2, xlocals); 119 SkDebugf(" xlocals.fUsed=%d\n", xlocals.used()); 120 } 121 #endif 122 Intersections locals; 123 intersectWithOrder(s1, o1, s2, o2, locals); 124 double coStart[2] = { -1 }; 125 _Point coPoint; 126 int tCount = locals.used(); 127 for (int tIdx = 0; tIdx < tCount; ++tIdx) { 128 double to1 = t1Start + (t1 - t1Start) * locals.fT[0][tIdx]; 129 double to2 = t2Start + (t2 - t2Start) * locals.fT[1][tIdx]; 130 // if the computed t is not sufficiently precise, iterate 131 _Point p1 = xy_at_t(cubic1, to1); 132 _Point p2 = xy_at_t(cubic2, to2); 133 if (p1.approximatelyEqual(p2)) { 134 if (locals.fIsCoincident[0] & 1 << tIdx) { 135 if (coStart[0] < 0) { 136 coStart[0] = to1; 137 coStart[1] = to2; 138 coPoint = p1; 139 } else { 140 i.insertCoincidentPair(coStart[0], to1, coStart[1], to2, coPoint, p1); 141 coStart[0] = -1; 142 } 143 result = true; 144 } else if (cubic1 != cubic2 || !approximately_equal(to1, to2)) { 145 if (i.swapped()) { // FIXME: insert should respect swap 146 i.insert(to2, to1, p1); 147 } else { 148 i.insert(to1, to2, p1); 149 } 150 result = true; 151 } 152 } else { 153 double offset = precisionScale / 16; // FIME: const is arbitrary -- test & refine 154 #if 1 155 double c1Bottom = tIdx == 0 ? 0 : 156 (t1Start + (t1 - t1Start) * locals.fT[0][tIdx - 1] + to1) / 2; 157 double c1Min = SkTMax(c1Bottom, to1 - offset); 158 double c1Top = tIdx == tCount - 1 ? 1 : 159 (t1Start + (t1 - t1Start) * locals.fT[0][tIdx + 1] + to1) / 2; 160 double c1Max = SkTMin(c1Top, to1 + offset); 161 double c2Min = SkTMax(0., to2 - offset); 162 double c2Max = SkTMin(1., to2 + offset); 163 #if ONE_OFF_DEBUG 164 SkDebugf("%.*s %s 1 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab, __FUNCTION__, 165 c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max 166 && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max, 167 to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset 168 && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset, 169 c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max 170 && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max, 171 to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset 172 && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset); 173 SkDebugf("%.*s %s 1 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g" 174 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n", 175 i.depth()*2, tab, __FUNCTION__, c1Bottom, c1Top, 0., 1., 176 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset); 177 SkDebugf("%.*s %s 1 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g" 178 " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min, c1Max, c2Min, c2Max); 179 #endif 180 intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); 181 #if ONE_OFF_DEBUG 182 SkDebugf("%.*s %s 1 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__, i.used(), 183 i.used() > 0 ? i.fT[0][i.used() - 1] : -1); 184 #endif 185 if (tCount > 1) { 186 c1Min = SkTMax(0., to1 - offset); 187 c1Max = SkTMin(1., to1 + offset); 188 double c2Bottom = tIdx == 0 ? to2 : 189 (t2Start + (t2 - t2Start) * locals.fT[1][tIdx - 1] + to2) / 2; 190 double c2Top = tIdx == tCount - 1 ? to2 : 191 (t2Start + (t2 - t2Start) * locals.fT[1][tIdx + 1] + to2) / 2; 192 if (c2Bottom > c2Top) { 193 SkTSwap(c2Bottom, c2Top); 194 } 195 if (c2Bottom == to2) { 196 c2Bottom = 0; 197 } 198 if (c2Top == to2) { 199 c2Top = 1; 200 } 201 c2Min = SkTMax(c2Bottom, to2 - offset); 202 c2Max = SkTMin(c2Top, to2 + offset); 203 #if ONE_OFF_DEBUG 204 SkDebugf("%.*s %s 2 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab, __FUNCTION__, 205 c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max 206 && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max, 207 to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset 208 && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset, 209 c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max 210 && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max, 211 to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset 212 && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset); 213 SkDebugf("%.*s %s 2 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g" 214 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n", 215 i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top, 216 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset); 217 SkDebugf("%.*s %s 2 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g" 218 " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min, c1Max, c2Min, c2Max); 219 #endif 220 intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); 221 #if ONE_OFF_DEBUG 222 SkDebugf("%.*s %s 2 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__, i.used(), 223 i.used() > 0 ? i.fT[0][i.used() - 1] : -1); 224 #endif 225 c1Min = SkTMax(c1Bottom, to1 - offset); 226 c1Max = SkTMin(c1Top, to1 + offset); 227 #if ONE_OFF_DEBUG 228 SkDebugf("%.*s %s 3 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab, __FUNCTION__, 229 c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max 230 && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max, 231 to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset 232 && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset, 233 c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max 234 && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max, 235 to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset 236 && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset); 237 SkDebugf("%.*s %s 3 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g" 238 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n", 239 i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top, 240 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset); 241 SkDebugf("%.*s %s 3 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g" 242 " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min, c1Max, c2Min, c2Max); 243 #endif 244 intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); 245 #if ONE_OFF_DEBUG 246 SkDebugf("%.*s %s 3 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__, i.used(), 247 i.used() > 0 ? i.fT[0][i.used() - 1] : -1); 248 #endif 249 } 250 #else 251 double c1Bottom = tIdx == 0 ? 0 : 252 (t1Start + (t1 - t1Start) * locals.fT[0][tIdx - 1] + to1) / 2; 253 double c1Min = SkTMax(c1Bottom, to1 - offset); 254 double c1Top = tIdx == tCount - 1 ? 1 : 255 (t1Start + (t1 - t1Start) * locals.fT[0][tIdx + 1] + to1) / 2; 256 double c1Max = SkTMin(c1Top, to1 + offset); 257 double c2Bottom = tIdx == 0 ? to2 : 258 (t2Start + (t2 - t2Start) * locals.fT[1][tIdx - 1] + to2) / 2; 259 double c2Top = tIdx == tCount - 1 ? to2 : 260 (t2Start + (t2 - t2Start) * locals.fT[1][tIdx + 1] + to2) / 2; 261 if (c2Bottom > c2Top) { 262 SkTSwap(c2Bottom, c2Top); 263 } 264 if (c2Bottom == to2) { 265 c2Bottom = 0; 266 } 267 if (c2Top == to2) { 268 c2Top = 1; 269 } 270 double c2Min = SkTMax(c2Bottom, to2 - offset); 271 double c2Max = SkTMin(c2Top, to2 + offset); 272 #if ONE_OFF_DEBUG 273 SkDebugf("%s contains1=%d/%d contains2=%d/%d\n", __FUNCTION__, 274 c1Min <= 0.210357794 && 0.210357794 <= c1Max 275 && c2Min <= 0.223476406 && 0.223476406 <= c2Max, 276 to1 - offset <= 0.210357794 && 0.210357794 <= to1 + offset 277 && to2 - offset <= 0.223476406 && 0.223476406 <= to2 + offset, 278 c1Min <= 0.211324707 && 0.211324707 <= c1Max 279 && c2Min <= 0.211327209 && 0.211327209 <= c2Max, 280 to1 - offset <= 0.211324707 && 0.211324707 <= to1 + offset 281 && to2 - offset <= 0.211327209 && 0.211327209 <= to2 + offset); 282 SkDebugf("%s c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g" 283 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n", 284 __FUNCTION__, c1Bottom, c1Top, c2Bottom, c2Top, 285 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset); 286 SkDebugf("%s to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g" 287 " c2Max=%1.9g\n", __FUNCTION__, to1, to2, c1Min, c1Max, c2Min, c2Max); 288 #endif 289 #endif 290 intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); 291 // TODO: if no intersection is found, either quadratics intersected where 292 // cubics did not, or the intersection was missed. In the former case, expect 293 // the quadratics to be nearly parallel at the point of intersection, and check 294 // for that. 295 } 296 } 297 SkASSERT(coStart[0] == -1); 298 t2Start = t2; 299 } 300 t1Start = t1; 301 } 302 i.downDepth(); 303 return result; 304 } 305 306 #if 0 307 #define LINE_FRACTION (1.0 / gPrecisionUnit) 308 #else 309 #define LINE_FRACTION 0.1 310 #endif 311 312 // intersect the end of the cubic with the other. Try lines from the end to control and opposite 313 // end to determine range of t on opposite cubic. 314 static bool intersectEnd(const Cubic& cubic1, bool start, const Cubic& cubic2, const _Rect& bounds2, 315 Intersections& i) { 316 // bool selfIntersect = cubic1 == cubic2; 317 _Line line; 318 int t1Index = start ? 0 : 3; 319 line[0] = cubic1[t1Index]; 320 // don't bother if the two cubics are connnected 321 #if 0 322 if (!selfIntersect && (line[0].approximatelyEqual(cubic2[0]) 323 || line[0].approximatelyEqual(cubic2[3]))) { 324 return false; 325 } 326 #endif 327 bool result = false; 328 SkTDArray<double> tVals; // OPTIMIZE: replace with hard-sized array 329 for (int index = 0; index < 4; ++index) { 330 if (index == t1Index) { 331 continue; 332 } 333 _Vector dxy1 = cubic1[index] - line[0]; 334 dxy1 /= gPrecisionUnit; 335 line[1] = line[0] + dxy1; 336 _Rect lineBounds; 337 lineBounds.setBounds(line); 338 if (!bounds2.intersects(lineBounds)) { 339 continue; 340 } 341 Intersections local; 342 if (!intersect(cubic2, line, local)) { 343 continue; 344 } 345 for (int idx2 = 0; idx2 < local.used(); ++idx2) { 346 double foundT = local.fT[0][idx2]; 347 if (approximately_less_than_zero(foundT) 348 || approximately_greater_than_one(foundT)) { 349 continue; 350 } 351 if (local.fPt[idx2].approximatelyEqual(line[0])) { 352 if (i.swapped()) { // FIXME: insert should respect swap 353 i.insert(foundT, start ? 0 : 1, line[0]); 354 } else { 355 i.insert(start ? 0 : 1, foundT, line[0]); 356 } 357 result = true; 358 } else { 359 *tVals.append() = local.fT[0][idx2]; 360 } 361 } 362 } 363 if (tVals.count() == 0) { 364 return result; 365 } 366 QSort<double>(tVals.begin(), tVals.end() - 1); 367 double tMin1 = start ? 0 : 1 - LINE_FRACTION; 368 double tMax1 = start ? LINE_FRACTION : 1; 369 int tIdx = 0; 370 do { 371 int tLast = tIdx; 372 while (tLast + 1 < tVals.count() && roughly_equal(tVals[tLast + 1], tVals[tIdx])) { 373 ++tLast; 374 } 375 double tMin2 = SkTMax(tVals[tIdx] - LINE_FRACTION, 0.0); 376 double tMax2 = SkTMin(tVals[tLast] + LINE_FRACTION, 1.0); 377 int lastUsed = i.used(); 378 result |= intersect3(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i); 379 if (lastUsed == i.used()) { 380 tMin2 = SkTMax(tVals[tIdx] - (1.0 / gPrecisionUnit), 0.0); 381 tMax2 = SkTMin(tVals[tLast] + (1.0 / gPrecisionUnit), 1.0); 382 result |= intersect3(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i); 383 } 384 tIdx = tLast + 1; 385 } while (tIdx < tVals.count()); 386 return result; 387 } 388 389 const double CLOSE_ENOUGH = 0.001; 390 391 static bool closeStart(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) { 392 if (i.fT[cubicIndex][0] != 0 || i.fT[cubicIndex][1] > CLOSE_ENOUGH) { 393 return false; 394 } 395 pt = xy_at_t(cubic, (i.fT[cubicIndex][0] + i.fT[cubicIndex][1]) / 2); 396 return true; 397 } 398 399 static bool closeEnd(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) { 400 int last = i.used() - 1; 401 if (i.fT[cubicIndex][last] != 1 || i.fT[cubicIndex][last - 1] < 1 - CLOSE_ENOUGH) { 402 return false; 403 } 404 pt = xy_at_t(cubic, (i.fT[cubicIndex][last] + i.fT[cubicIndex][last - 1]) / 2); 405 return true; 406 } 407 408 bool intersect3(const Cubic& c1, const Cubic& c2, Intersections& i) { 409 bool result = intersect3(c1, 0, 1, c2, 0, 1, 1, i); 410 // FIXME: pass in cached bounds from caller 411 _Rect c1Bounds, c2Bounds; 412 c1Bounds.setBounds(c1); // OPTIMIZE use setRawBounds ? 413 c2Bounds.setBounds(c2); 414 result |= intersectEnd(c1, false, c2, c2Bounds, i); 415 result |= intersectEnd(c1, true, c2, c2Bounds, i); 416 bool selfIntersect = c1 == c2; 417 if (!selfIntersect) { 418 i.swap(); 419 result |= intersectEnd(c2, false, c1, c1Bounds, i); 420 result |= intersectEnd(c2, true, c1, c1Bounds, i); 421 i.swap(); 422 } 423 // If an end point and a second point very close to the end is returned, the second 424 // point may have been detected because the approximate quads 425 // intersected at the end and close to it. Verify that the second point is valid. 426 if (i.used() <= 1 || i.coincidentUsed()) { 427 return result; 428 } 429 _Point pt[2]; 430 if (closeStart(c1, 0, i, pt[0]) && closeStart(c2, 1, i, pt[1]) 431 && pt[0].approximatelyEqual(pt[1])) { 432 i.removeOne(1); 433 } 434 if (closeEnd(c1, 0, i, pt[0]) && closeEnd(c2, 1, i, pt[1]) 435 && pt[0].approximatelyEqual(pt[1])) { 436 i.removeOne(i.used() - 2); 437 } 438 return result; 439 } 440 441 // Up promote the quad to a cubic. 442 // OPTIMIZATION If this is a common use case, optimize by duplicating 443 // the intersect 3 loop to avoid the promotion / demotion code 444 int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& i) { 445 Cubic up; 446 toCubic(quad, up); 447 (void) intersect3(cubic, up, i); 448 return i.used(); 449 } 450 451 /* http://www.ag.jku.at/compass/compasssample.pdf 452 ( Self-Intersection Problems and Approximate Implicitization by Jan B. Thomassen 453 Centre of Mathematics for Applications, University of Oslo http://www.cma.uio.no janbth (at) math.uio.no 454 SINTEF Applied Mathematics http://www.sintef.no ) 455 describes a method to find the self intersection of a cubic by taking the gradient of the implicit 456 form dotted with the normal, and solving for the roots. My math foo is too poor to implement this.*/ 457 458 int intersect(const Cubic& c, Intersections& i) { 459 // check to see if x or y end points are the extrema. Are other quick rejects possible? 460 if (ends_are_extrema_in_x_or_y(c)) { 461 return false; 462 } 463 (void) intersect3(c, c, i); 464 if (i.used() > 0) { 465 SkASSERT(i.used() == 1); 466 if (i.fT[0][0] > i.fT[1][0]) { 467 SkTSwap(i.fT[0][0], i.fT[1][0]); 468 } 469 } 470 return i.used(); 471 } 472
