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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 "SkIntersections.h"
9 #include "SkPathOpsCubic.h"
10 #include "SkPathOpsLine.h"
11 #include "SkPathOpsPoint.h"
12 #include "SkPathOpsQuad.h"
13 #include "SkPathOpsRect.h"
14 #include "SkReduceOrder.h"
15 #include "SkTSort.h"
16 
17 #if ONE_OFF_DEBUG
18 static const double tLimits1[2][2] = {{0.3, 0.4}, {0.8, 0.9}};
19 static const double tLimits2[2][2] = {{-0.8, -0.9}, {-0.8, -0.9}};
20 #endif
21 
22 #define DEBUG_QUAD_PART ONE_OFF_DEBUG && 1
23 #define DEBUG_QUAD_PART_SHOW_SIMPLE DEBUG_QUAD_PART && 0
24 #define SWAP_TOP_DEBUG 0
25 
26 static const int kCubicToQuadSubdivisionDepth = 8; // slots reserved for cubic to quads subdivision
27 
quadPart(const SkDCubic & cubic,double tStart,double tEnd,SkReduceOrder * reducer)28 static int quadPart(const SkDCubic& cubic, double tStart, double tEnd, SkReduceOrder* reducer) {
29     SkDCubic part = cubic.subDivide(tStart, tEnd);
30     SkDQuad quad = part.toQuad();
31     // FIXME: should reduceOrder be looser in this use case if quartic is going to blow up on an
32     // extremely shallow quadratic?
33     int order = reducer->reduce(quad);
34 #if DEBUG_QUAD_PART
35     SkDebugf("%s cubic=(%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)"
36             " t=(%1.9g,%1.9g)\n", __FUNCTION__, cubic[0].fX, cubic[0].fY,
37             cubic[1].fX, cubic[1].fY, cubic[2].fX, cubic[2].fY,
38             cubic[3].fX, cubic[3].fY, tStart, tEnd);
39     SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n"
40              "  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
41             part[0].fX, part[0].fY, part[1].fX, part[1].fY, part[2].fX, part[2].fY,
42             part[3].fX, part[3].fY, quad[0].fX, quad[0].fY,
43             quad[1].fX, quad[1].fY, quad[2].fX, quad[2].fY);
44 #if DEBUG_QUAD_PART_SHOW_SIMPLE
45     SkDebugf("%s simple=(%1.9g,%1.9g", __FUNCTION__, reducer->fQuad[0].fX, reducer->fQuad[0].fY);
46     if (order > 1) {
47         SkDebugf(" %1.9g,%1.9g", reducer->fQuad[1].fX, reducer->fQuad[1].fY);
48     }
49     if (order > 2) {
50         SkDebugf(" %1.9g,%1.9g", reducer->fQuad[2].fX, reducer->fQuad[2].fY);
51     }
52     SkDebugf(")\n");
53     SkASSERT(order < 4 && order > 0);
54 #endif
55 #endif
56     return order;
57 }
58 
intersectWithOrder(const SkDQuad & simple1,int order1,const SkDQuad & simple2,int order2,SkIntersections & i)59 static void intersectWithOrder(const SkDQuad& simple1, int order1, const SkDQuad& simple2,
60         int order2, SkIntersections& i) {
61     if (order1 == 3 && order2 == 3) {
62         i.intersect(simple1, simple2);
63     } else if (order1 <= 2 && order2 <= 2) {
64         i.intersect((const SkDLine&) simple1, (const SkDLine&) simple2);
65     } else if (order1 == 3 && order2 <= 2) {
66         i.intersect(simple1, (const SkDLine&) simple2);
67     } else {
68         SkASSERT(order1 <= 2 && order2 == 3);
69         i.intersect(simple2, (const SkDLine&) simple1);
70         i.swapPts();
71     }
72 }
73 
74 // this flavor centers potential intersections recursively. In contrast, '2' may inadvertently
75 // chase intersections near quadratic ends, requiring odd hacks to find them.
intersect(const SkDCubic & cubic1,double t1s,double t1e,const SkDCubic & cubic2,double t2s,double t2e,double precisionScale,SkIntersections & i)76 static void intersect(const SkDCubic& cubic1, double t1s, double t1e, const SkDCubic& cubic2,
77         double t2s, double t2e, double precisionScale, SkIntersections& i) {
78     i.upDepth();
79     SkDCubic c1 = cubic1.subDivide(t1s, t1e);
80     SkDCubic c2 = cubic2.subDivide(t2s, t2e);
81     SkSTArray<kCubicToQuadSubdivisionDepth, double, true> ts1;
82     // OPTIMIZE: if c1 == c2, call once (happens when detecting self-intersection)
83     c1.toQuadraticTs(c1.calcPrecision() * precisionScale, &ts1);
84     SkSTArray<kCubicToQuadSubdivisionDepth, double, true> ts2;
85     c2.toQuadraticTs(c2.calcPrecision() * precisionScale, &ts2);
86     double t1Start = t1s;
87     int ts1Count = ts1.count();
88     for (int i1 = 0; i1 <= ts1Count; ++i1) {
89         const double tEnd1 = i1 < ts1Count ? ts1[i1] : 1;
90         const double t1 = t1s + (t1e - t1s) * tEnd1;
91         SkReduceOrder s1;
92         int o1 = quadPart(cubic1, t1Start, t1, &s1);
93         double t2Start = t2s;
94         int ts2Count = ts2.count();
95         for (int i2 = 0; i2 <= ts2Count; ++i2) {
96             const double tEnd2 = i2 < ts2Count ? ts2[i2] : 1;
97             const double t2 = t2s + (t2e - t2s) * tEnd2;
98             if (&cubic1 == &cubic2 && t1Start >= t2Start) {
99                 t2Start = t2;
100                 continue;
101             }
102             SkReduceOrder s2;
103             int o2 = quadPart(cubic2, t2Start, t2, &s2);
104         #if ONE_OFF_DEBUG
105             char tab[] = "                  ";
106             if (tLimits1[0][0] >= t1Start && tLimits1[0][1] <= t1
107                     && tLimits1[1][0] >= t2Start && tLimits1[1][1] <= t2) {
108                 SkDebugf("%.*s %s t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)", i.depth()*2, tab,
109                         __FUNCTION__, t1Start, t1, t2Start, t2);
110                 SkIntersections xlocals;
111                 xlocals.allowNear(false);
112                 intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, xlocals);
113                 SkDebugf(" xlocals.fUsed=%d\n", xlocals.used());
114             }
115         #endif
116             SkIntersections locals;
117             locals.allowNear(false);
118             intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, locals);
119             int tCount = locals.used();
120             for (int tIdx = 0; tIdx < tCount; ++tIdx) {
121                 double to1 = t1Start + (t1 - t1Start) * locals[0][tIdx];
122                 double to2 = t2Start + (t2 - t2Start) * locals[1][tIdx];
123     // if the computed t is not sufficiently precise, iterate
124                 SkDPoint p1 = cubic1.ptAtT(to1);
125                 SkDPoint p2 = cubic2.ptAtT(to2);
126                 if (p1.approximatelyEqual(p2)) {
127     // FIXME: local edge may be coincident -- experiment with not propagating coincidence to caller
128 //                    SkASSERT(!locals.isCoincident(tIdx));
129                     if (&cubic1 != &cubic2 || !approximately_equal(to1, to2)) {
130                         if (i.swapped()) {  //  FIXME: insert should respect swap
131                             i.insert(to2, to1, p1);
132                         } else {
133                             i.insert(to1, to2, p1);
134                         }
135                     }
136                 } else {
137 /*for random cubics, 16 below catches 99.997% of the intersections. To test for the remaining 0.003%
138   look for nearly coincident curves. and check each 1/16th section.
139 */
140                     double offset = precisionScale / 16;  // FIXME: const is arbitrary: test, refine
141                     double c1Bottom = tIdx == 0 ? 0 :
142                             (t1Start + (t1 - t1Start) * locals[0][tIdx - 1] + to1) / 2;
143                     double c1Min = SkTMax(c1Bottom, to1 - offset);
144                     double c1Top = tIdx == tCount - 1 ? 1 :
145                             (t1Start + (t1 - t1Start) * locals[0][tIdx + 1] + to1) / 2;
146                     double c1Max = SkTMin(c1Top, to1 + offset);
147                     double c2Min = SkTMax(0., to2 - offset);
148                     double c2Max = SkTMin(1., to2 + offset);
149                 #if ONE_OFF_DEBUG
150                     SkDebugf("%.*s %s 1 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
151                             __FUNCTION__,
152                             c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
153                          && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
154                             to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
155                          && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
156                             c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
157                          && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
158                             to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
159                          && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
160                     SkDebugf("%.*s %s 1 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
161                             " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
162                             i.depth()*2, tab, __FUNCTION__, c1Bottom, c1Top, 0., 1.,
163                             to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
164                     SkDebugf("%.*s %s 1 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
165                             " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
166                             c1Max, c2Min, c2Max);
167                 #endif
168                     intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
169                 #if ONE_OFF_DEBUG
170                     SkDebugf("%.*s %s 1 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
171                             i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
172                 #endif
173                     if (tCount > 1) {
174                         c1Min = SkTMax(0., to1 - offset);
175                         c1Max = SkTMin(1., to1 + offset);
176                         double c2Bottom = tIdx == 0 ? to2 :
177                                 (t2Start + (t2 - t2Start) * locals[1][tIdx - 1] + to2) / 2;
178                         double c2Top = tIdx == tCount - 1 ? to2 :
179                                 (t2Start + (t2 - t2Start) * locals[1][tIdx + 1] + to2) / 2;
180                         if (c2Bottom > c2Top) {
181                             SkTSwap(c2Bottom, c2Top);
182                         }
183                         if (c2Bottom == to2) {
184                             c2Bottom = 0;
185                         }
186                         if (c2Top == to2) {
187                             c2Top = 1;
188                         }
189                         c2Min = SkTMax(c2Bottom, to2 - offset);
190                         c2Max = SkTMin(c2Top, to2 + offset);
191                     #if ONE_OFF_DEBUG
192                         SkDebugf("%.*s %s 2 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
193                             __FUNCTION__,
194                             c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
195                          && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
196                             to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
197                          && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
198                             c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
199                          && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
200                             to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
201                          && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
202                         SkDebugf("%.*s %s 2 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
203                                 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
204                                 i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top,
205                                 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
206                         SkDebugf("%.*s %s 2 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
207                                 " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
208                                 c1Max, c2Min, c2Max);
209                     #endif
210                         intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
211                 #if ONE_OFF_DEBUG
212                     SkDebugf("%.*s %s 2 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
213                             i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
214                 #endif
215                         c1Min = SkTMax(c1Bottom, to1 - offset);
216                         c1Max = SkTMin(c1Top, to1 + offset);
217                     #if ONE_OFF_DEBUG
218                         SkDebugf("%.*s %s 3 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
219                         __FUNCTION__,
220                             c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
221                          && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
222                             to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
223                          && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
224                             c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
225                          && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
226                             to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
227                          && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
228                         SkDebugf("%.*s %s 3 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
229                                 " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
230                                 i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top,
231                                 to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
232                         SkDebugf("%.*s %s 3 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
233                                 " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
234                                 c1Max, c2Min, c2Max);
235                     #endif
236                         intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
237                 #if ONE_OFF_DEBUG
238                     SkDebugf("%.*s %s 3 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
239                             i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
240                 #endif
241                     }
242           //          intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
243                     // FIXME: if no intersection is found, either quadratics intersected where
244                     // cubics did not, or the intersection was missed. In the former case, expect
245                     // the quadratics to be nearly parallel at the point of intersection, and check
246                     // for that.
247                 }
248             }
249             t2Start = t2;
250         }
251         t1Start = t1;
252     }
253     i.downDepth();
254 }
255 
256     // if two ends intersect, check middle for coincidence
cubicCheckCoincidence(const SkDCubic & c1,const SkDCubic & c2)257 bool SkIntersections::cubicCheckCoincidence(const SkDCubic& c1, const SkDCubic& c2) {
258     if (fUsed < 2) {
259         return false;
260     }
261     int last = fUsed - 1;
262     double tRange1 = fT[0][last] - fT[0][0];
263     double tRange2 = fT[1][last] - fT[1][0];
264     for (int index = 1; index < 5; ++index) {
265         double testT1 = fT[0][0] + tRange1 * index / 5;
266         double testT2 = fT[1][0] + tRange2 * index / 5;
267         SkDPoint testPt1 = c1.ptAtT(testT1);
268         SkDPoint testPt2 = c2.ptAtT(testT2);
269         if (!testPt1.approximatelyEqual(testPt2)) {
270             return false;
271         }
272     }
273     if (fUsed > 2) {
274         fPt[1] = fPt[last];
275         fT[0][1] = fT[0][last];
276         fT[1][1] = fT[1][last];
277         fUsed = 2;
278     }
279     fIsCoincident[0] = fIsCoincident[1] = 0x03;
280     return true;
281 }
282 
283 #define LINE_FRACTION 0.1
284 
285 // intersect the end of the cubic with the other. Try lines from the end to control and opposite
286 // end to determine range of t on opposite cubic.
cubicExactEnd(const SkDCubic & cubic1,bool start,const SkDCubic & cubic2)287 bool SkIntersections::cubicExactEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2) {
288     int t1Index = start ? 0 : 3;
289     double testT = (double) !start;
290     bool swap = swapped();
291     // quad/quad at this point checks to see if exact matches have already been found
292     // cubic/cubic can't reject so easily since cubics can intersect same point more than once
293     SkDLine tmpLine;
294     tmpLine[0] = tmpLine[1] = cubic2[t1Index];
295     tmpLine[1].fX += cubic2[2 - start].fY - cubic2[t1Index].fY;
296     tmpLine[1].fY -= cubic2[2 - start].fX - cubic2[t1Index].fX;
297     SkIntersections impTs;
298     impTs.allowNear(false);
299     impTs.intersectRay(cubic1, tmpLine);
300     for (int index = 0; index < impTs.used(); ++index) {
301         SkDPoint realPt = impTs.pt(index);
302         if (!tmpLine[0].approximatelyEqual(realPt)) {
303             continue;
304         }
305         if (swap) {
306             cubicInsert(testT, impTs[0][index], tmpLine[0], cubic2, cubic1);
307         } else {
308             cubicInsert(impTs[0][index], testT, tmpLine[0], cubic1, cubic2);
309         }
310         return true;
311     }
312     return false;
313 }
314 
315 
cubicInsert(double one,double two,const SkDPoint & pt,const SkDCubic & cubic1,const SkDCubic & cubic2)316 void SkIntersections::cubicInsert(double one, double two, const SkDPoint& pt,
317         const SkDCubic& cubic1, const SkDCubic& cubic2) {
318     for (int index = 0; index < fUsed; ++index) {
319         if (fT[0][index] == one) {
320             double oldTwo = fT[1][index];
321             if (oldTwo == two) {
322                 return;
323             }
324             SkDPoint mid = cubic2.ptAtT((oldTwo + two) / 2);
325             if (mid.approximatelyEqual(fPt[index])) {
326                 return;
327             }
328         }
329         if (fT[1][index] == two) {
330             SkDPoint mid = cubic1.ptAtT((fT[0][index] + two) / 2);
331             if (mid.approximatelyEqual(fPt[index])) {
332                 return;
333             }
334         }
335     }
336     insert(one, two, pt);
337 }
338 
cubicNearEnd(const SkDCubic & cubic1,bool start,const SkDCubic & cubic2,const SkDRect & bounds2)339 void SkIntersections::cubicNearEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2,
340                          const SkDRect& bounds2) {
341     SkDLine line;
342     int t1Index = start ? 0 : 3;
343     double testT = (double) !start;
344    // don't bother if the two cubics are connnected
345     static const int kPointsInCubic = 4; // FIXME: move to DCubic, replace '4' with this
346     static const int kMaxLineCubicIntersections = 3;
347     SkSTArray<(kMaxLineCubicIntersections - 1) * kMaxLineCubicIntersections, double, true> tVals;
348     line[0] = cubic1[t1Index];
349     // this variant looks for intersections with the end point and lines parallel to other points
350     for (int index = 0; index < kPointsInCubic; ++index) {
351         if (index == t1Index) {
352             continue;
353         }
354         SkDVector dxy1 = cubic1[index] - line[0];
355         dxy1 /= SkDCubic::gPrecisionUnit;
356         line[1] = line[0] + dxy1;
357         SkDRect lineBounds;
358         lineBounds.setBounds(line);
359         if (!bounds2.intersects(&lineBounds)) {
360             continue;
361         }
362         SkIntersections local;
363         if (!local.intersect(cubic2, line)) {
364             continue;
365         }
366         for (int idx2 = 0; idx2 < local.used(); ++idx2) {
367             double foundT = local[0][idx2];
368             if (approximately_less_than_zero(foundT)
369                     || approximately_greater_than_one(foundT)) {
370                 continue;
371             }
372             if (local.pt(idx2).approximatelyEqual(line[0])) {
373                 if (swapped()) {  // FIXME: insert should respect swap
374                     insert(foundT, testT, line[0]);
375                 } else {
376                     insert(testT, foundT, line[0]);
377                 }
378             } else {
379                 tVals.push_back(foundT);
380             }
381         }
382     }
383     if (tVals.count() == 0) {
384         return;
385     }
386     SkTQSort<double>(tVals.begin(), tVals.end() - 1);
387     double tMin1 = start ? 0 : 1 - LINE_FRACTION;
388     double tMax1 = start ? LINE_FRACTION : 1;
389     int tIdx = 0;
390     do {
391         int tLast = tIdx;
392         while (tLast + 1 < tVals.count() && roughly_equal(tVals[tLast + 1], tVals[tIdx])) {
393             ++tLast;
394         }
395         double tMin2 = SkTMax(tVals[tIdx] - LINE_FRACTION, 0.0);
396         double tMax2 = SkTMin(tVals[tLast] + LINE_FRACTION, 1.0);
397         int lastUsed = used();
398         if (start ? tMax1 < tMin2 : tMax2 < tMin1) {
399             ::intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, *this);
400         }
401         if (lastUsed == used()) {
402             tMin2 = SkTMax(tVals[tIdx] - (1.0 / SkDCubic::gPrecisionUnit), 0.0);
403             tMax2 = SkTMin(tVals[tLast] + (1.0 / SkDCubic::gPrecisionUnit), 1.0);
404             if (start ? tMax1 < tMin2 : tMax2 < tMin1) {
405                 ::intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, *this);
406             }
407         }
408         tIdx = tLast + 1;
409     } while (tIdx < tVals.count());
410     return;
411 }
412 
413 const double CLOSE_ENOUGH = 0.001;
414 
closeStart(const SkDCubic & cubic,int cubicIndex,SkIntersections & i,SkDPoint & pt)415 static bool closeStart(const SkDCubic& cubic, int cubicIndex, SkIntersections& i, SkDPoint& pt) {
416     if (i[cubicIndex][0] != 0 || i[cubicIndex][1] > CLOSE_ENOUGH) {
417         return false;
418     }
419     pt = cubic.ptAtT((i[cubicIndex][0] + i[cubicIndex][1]) / 2);
420     return true;
421 }
422 
closeEnd(const SkDCubic & cubic,int cubicIndex,SkIntersections & i,SkDPoint & pt)423 static bool closeEnd(const SkDCubic& cubic, int cubicIndex, SkIntersections& i, SkDPoint& pt) {
424     int last = i.used() - 1;
425     if (i[cubicIndex][last] != 1 || i[cubicIndex][last - 1] < 1 - CLOSE_ENOUGH) {
426         return false;
427     }
428     pt = cubic.ptAtT((i[cubicIndex][last] + i[cubicIndex][last - 1]) / 2);
429     return true;
430 }
431 
only_end_pts_in_common(const SkDCubic & c1,const SkDCubic & c2)432 static bool only_end_pts_in_common(const SkDCubic& c1, const SkDCubic& c2) {
433 // the idea here is to see at minimum do a quick reject by rotating all points
434 // to either side of the line formed by connecting the endpoints
435 // if the opposite curves points are on the line or on the other side, the
436 // curves at most intersect at the endpoints
437     for (int oddMan = 0; oddMan < 4; ++oddMan) {
438         const SkDPoint* endPt[3];
439         for (int opp = 1; opp < 4; ++opp) {
440             int end = oddMan ^ opp;  // choose a value not equal to oddMan
441             endPt[opp - 1] = &c1[end];
442         }
443         for (int triTest = 0; triTest < 3; ++triTest) {
444             double origX = endPt[triTest]->fX;
445             double origY = endPt[triTest]->fY;
446             int oppTest = triTest + 1;
447             if (3 == oppTest) {
448                 oppTest = 0;
449             }
450             double adj = endPt[oppTest]->fX - origX;
451             double opp = endPt[oppTest]->fY - origY;
452             if (adj == 0 && opp == 0) {  // if the other point equals the test point, ignore it
453                 continue;
454             }
455             double sign = (c1[oddMan].fY - origY) * adj - (c1[oddMan].fX - origX) * opp;
456             if (approximately_zero(sign)) {
457                 goto tryNextHalfPlane;
458             }
459             for (int n = 0; n < 4; ++n) {
460                 double test = (c2[n].fY - origY) * adj - (c2[n].fX - origX) * opp;
461                 if (test * sign > 0 && !precisely_zero(test)) {
462                     goto tryNextHalfPlane;
463                 }
464             }
465         }
466         return true;
467 tryNextHalfPlane:
468         ;
469     }
470     return false;
471 }
472 
intersect(const SkDCubic & c1,const SkDCubic & c2)473 int SkIntersections::intersect(const SkDCubic& c1, const SkDCubic& c2) {
474     if (fMax == 0) {
475         fMax = 9;
476     }
477     bool selfIntersect = &c1 == &c2;
478     if (selfIntersect) {
479         if (c1[0].approximatelyEqual(c1[3])) {
480             insert(0, 1, c1[0]);
481             return fUsed;
482         }
483     } else {
484         // OPTIMIZATION: set exact end bits here to avoid cubic exact end later
485         for (int i1 = 0; i1 < 4; i1 += 3) {
486             for (int i2 = 0; i2 < 4; i2 += 3) {
487                 if (c1[i1].approximatelyEqual(c2[i2])) {
488                     insert(i1 >> 1, i2 >> 1, c1[i1]);
489                 }
490             }
491         }
492     }
493     SkASSERT(fUsed < 4);
494     if (!selfIntersect) {
495         if (only_end_pts_in_common(c1, c2)) {
496             return fUsed;
497         }
498         if (only_end_pts_in_common(c2, c1)) {
499             return fUsed;
500         }
501     }
502     // quad/quad does linear test here -- cubic does not
503     // cubics which are really lines should have been detected in reduce step earlier
504     int exactEndBits = 0;
505     if (selfIntersect) {
506         if (fUsed) {
507             return fUsed;
508         }
509     } else {
510         exactEndBits |= cubicExactEnd(c1, false, c2) << 0;
511         exactEndBits |= cubicExactEnd(c1, true, c2) << 1;
512         swap();
513         exactEndBits |= cubicExactEnd(c2, false, c1) << 2;
514         exactEndBits |= cubicExactEnd(c2, true, c1) << 3;
515         swap();
516     }
517     if (cubicCheckCoincidence(c1, c2)) {
518         SkASSERT(!selfIntersect);
519         return fUsed;
520     }
521     // FIXME: pass in cached bounds from caller
522     SkDRect c2Bounds;
523     c2Bounds.setBounds(c2);
524     if (!(exactEndBits & 4)) {
525         cubicNearEnd(c1, false, c2, c2Bounds);
526     }
527     if (!(exactEndBits & 8)) {
528         if (selfIntersect && fUsed) {
529             return fUsed;
530         }
531         cubicNearEnd(c1, true, c2, c2Bounds);
532         if (selfIntersect && fUsed && ((approximately_less_than_zero(fT[0][0])
533                     && approximately_less_than_zero(fT[1][0]))
534                     || (approximately_greater_than_one(fT[0][0])
535                     && approximately_greater_than_one(fT[1][0])))) {
536             SkASSERT(fUsed == 1);
537             fUsed = 0;
538             return fUsed;
539         }
540     }
541     if (!selfIntersect) {
542         SkDRect c1Bounds;
543         c1Bounds.setBounds(c1);  // OPTIMIZE use setRawBounds ?
544         swap();
545         if (!(exactEndBits & 1)) {
546             cubicNearEnd(c2, false, c1, c1Bounds);
547         }
548         if (!(exactEndBits & 2)) {
549             cubicNearEnd(c2, true, c1, c1Bounds);
550         }
551         swap();
552     }
553     if (cubicCheckCoincidence(c1, c2)) {
554         SkASSERT(!selfIntersect);
555         return fUsed;
556     }
557     SkIntersections i;
558     i.fAllowNear = false;
559     i.fMax = 9;
560     ::intersect(c1, 0, 1, c2, 0, 1, 1, i);
561     int compCount = i.used();
562     if (compCount) {
563         int exactCount = used();
564         if (exactCount == 0) {
565             *this = i;
566         } else {
567             // at least one is exact or near, and at least one was computed. Eliminate duplicates
568             for (int exIdx = 0; exIdx < exactCount; ++exIdx) {
569                 for (int cpIdx = 0; cpIdx < compCount; ) {
570                     if (fT[0][0] == i[0][0] && fT[1][0] == i[1][0]) {
571                         i.removeOne(cpIdx);
572                         --compCount;
573                         continue;
574                     }
575                     double tAvg = (fT[0][exIdx] + i[0][cpIdx]) / 2;
576                     SkDPoint pt = c1.ptAtT(tAvg);
577                     if (!pt.approximatelyEqual(fPt[exIdx])) {
578                         ++cpIdx;
579                         continue;
580                     }
581                     tAvg = (fT[1][exIdx] + i[1][cpIdx]) / 2;
582                     pt = c2.ptAtT(tAvg);
583                     if (!pt.approximatelyEqual(fPt[exIdx])) {
584                         ++cpIdx;
585                         continue;
586                     }
587                     i.removeOne(cpIdx);
588                     --compCount;
589                 }
590             }
591             // if mid t evaluates to nearly the same point, skip the t
592             for (int cpIdx = 0; cpIdx < compCount - 1; ) {
593                 double tAvg = (fT[0][cpIdx] + i[0][cpIdx + 1]) / 2;
594                 SkDPoint pt = c1.ptAtT(tAvg);
595                 if (!pt.approximatelyEqual(fPt[cpIdx])) {
596                     ++cpIdx;
597                     continue;
598                 }
599                 tAvg = (fT[1][cpIdx] + i[1][cpIdx + 1]) / 2;
600                 pt = c2.ptAtT(tAvg);
601                 if (!pt.approximatelyEqual(fPt[cpIdx])) {
602                     ++cpIdx;
603                     continue;
604                 }
605                 i.removeOne(cpIdx);
606                 --compCount;
607             }
608             // in addition to adding below missing function, think about how to say
609             append(i);
610         }
611     }
612     // If an end point and a second point very close to the end is returned, the second
613     // point may have been detected because the approximate quads
614     // intersected at the end and close to it. Verify that the second point is valid.
615     if (fUsed <= 1) {
616         return fUsed;
617     }
618     SkDPoint pt[2];
619     if (closeStart(c1, 0, *this, pt[0]) && closeStart(c2, 1, *this, pt[1])
620             && pt[0].approximatelyEqual(pt[1])) {
621         removeOne(1);
622     }
623     if (closeEnd(c1, 0, *this, pt[0]) && closeEnd(c2, 1, *this, pt[1])
624             && pt[0].approximatelyEqual(pt[1])) {
625         removeOne(used() - 2);
626     }
627     // vet the pairs of t values to see if the mid value is also on the curve. If so, mark
628     // the span as coincident
629     if (fUsed >= 2 && !coincidentUsed()) {
630         int last = fUsed - 1;
631         int match = 0;
632         for (int index = 0; index < last; ++index) {
633             double mid1 = (fT[0][index] + fT[0][index + 1]) / 2;
634             double mid2 = (fT[1][index] + fT[1][index + 1]) / 2;
635             pt[0] = c1.ptAtT(mid1);
636             pt[1] = c2.ptAtT(mid2);
637             if (pt[0].approximatelyEqual(pt[1])) {
638                 match |= 1 << index;
639             }
640         }
641         if (match) {
642 #if DEBUG_CONCIDENT
643             if (((match + 1) & match) != 0) {
644                 SkDebugf("%s coincident hole\n", __FUNCTION__);
645             }
646 #endif
647             // for now, assume that everything from start to finish is coincident
648             if (fUsed > 2) {
649                   fPt[1] = fPt[last];
650                   fT[0][1] = fT[0][last];
651                   fT[1][1] = fT[1][last];
652                   fIsCoincident[0] = 0x03;
653                   fIsCoincident[1] = 0x03;
654                   fUsed = 2;
655             }
656         }
657     }
658     return fUsed;
659 }
660 
661 // Up promote the quad to a cubic.
662 // OPTIMIZATION If this is a common use case, optimize by duplicating
663 // the intersect 3 loop to avoid the promotion  / demotion code
intersect(const SkDCubic & cubic,const SkDQuad & quad)664 int SkIntersections::intersect(const SkDCubic& cubic, const SkDQuad& quad) {
665     fMax = 6;
666     SkDCubic up = quad.toCubic();
667     (void) intersect(cubic, up);
668     return used();
669 }
670 
671 /* http://www.ag.jku.at/compass/compasssample.pdf
672 ( Self-Intersection Problems and Approximate Implicitization by Jan B. Thomassen
673 Centre of Mathematics for Applications, University of Oslo http://www.cma.uio.no janbth@math.uio.no
674 SINTEF Applied Mathematics http://www.sintef.no )
675 describes a method to find the self intersection of a cubic by taking the gradient of the implicit
676 form dotted with the normal, and solving for the roots. My math foo is too poor to implement this.*/
677 
intersect(const SkDCubic & c)678 int SkIntersections::intersect(const SkDCubic& c) {
679     fMax = 1;
680     // check to see if x or y end points are the extrema. Are other quick rejects possible?
681     if (c.endsAreExtremaInXOrY()) {
682         return false;
683     }
684     // OPTIMIZATION: could quick reject if neither end point tangent ray intersected the line
685     // segment formed by the opposite end point to the control point
686     (void) intersect(c, c);
687     if (used() > 0) {
688         if (approximately_equal_double(fT[0][0], fT[1][0])) {
689             fUsed = 0;
690         } else {
691             SkASSERT(used() == 1);
692             if (fT[0][0] > fT[1][0]) {
693                 swapPts();
694             }
695         }
696     }
697     return used();
698 }
699