• Home
  • Line#
  • Scopes#
  • Navigate#
  • Raw
  • Download
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 #include "SkIntersections.h"
8 #include "SkPathOpsLine.h"
9 #include "SkPathOpsQuad.h"
10 
11 /*
12 Find the interection of a line and quadratic by solving for valid t values.
13 
14 From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve
15 
16 "A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three
17 control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where
18 A, B and C are points and t goes from zero to one.
19 
20 This will give you two equations:
21 
22   x = a(1 - t)^2 + b(1 - t)t + ct^2
23   y = d(1 - t)^2 + e(1 - t)t + ft^2
24 
25 If you add for instance the line equation (y = kx + m) to that, you'll end up
26 with three equations and three unknowns (x, y and t)."
27 
28 Similar to above, the quadratic is represented as
29   x = a(1-t)^2 + 2b(1-t)t + ct^2
30   y = d(1-t)^2 + 2e(1-t)t + ft^2
31 and the line as
32   y = g*x + h
33 
34 Using Mathematica, solve for the values of t where the quadratic intersects the
35 line:
36 
37   (in)  t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x,
38                        d*(1 - t)^2 + 2*e*(1 - t)*t  + f*t^2 - g*x - h, x]
39   (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 +
40          g  (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2)
41   (in)  Solve[t1 == 0, t]
42   (out) {
43     {t -> (-2 d + 2 e +   2 a g - 2 b g    -
44       Sqrt[(2 d - 2 e -   2 a g + 2 b g)^2 -
45           4 (-d + 2 e - f + a g - 2 b g    + c g) (-d + a g + h)]) /
46          (2 (-d + 2 e - f + a g - 2 b g    + c g))
47          },
48     {t -> (-2 d + 2 e +   2 a g - 2 b g    +
49       Sqrt[(2 d - 2 e -   2 a g + 2 b g)^2 -
50           4 (-d + 2 e - f + a g - 2 b g    + c g) (-d + a g + h)]) /
51          (2 (-d + 2 e - f + a g - 2 b g    + c g))
52          }
53         }
54 
55 Using the results above (when the line tends towards horizontal)
56        A =   (-(d - 2*e + f) + g*(a - 2*b + c)     )
57        B = 2*( (d -   e    ) - g*(a -   b    )     )
58        C =   (-(d          ) + g*(a          ) + h )
59 
60 If g goes to infinity, we can rewrite the line in terms of x.
61   x = g'*y + h'
62 
63 And solve accordingly in Mathematica:
64 
65   (in)  t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h',
66                        d*(1 - t)^2 + 2*e*(1 - t)*t  + f*t^2 - y, y]
67   (out)  a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 -
68          g'  (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2)
69   (in)  Solve[t2 == 0, t]
70   (out) {
71     {t -> (2 a - 2 b -   2 d g' + 2 e g'    -
72     Sqrt[(-2 a + 2 b +   2 d g' - 2 e g')^2 -
73           4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) /
74          (2 (a - 2 b + c - d g' + 2 e g' - f g'))
75          },
76     {t -> (2 a - 2 b -   2 d g' + 2 e g'    +
77     Sqrt[(-2 a + 2 b +   2 d g' - 2 e g')^2 -
78           4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/
79          (2 (a - 2 b + c - d g' + 2 e g' - f g'))
80          }
81         }
82 
83 Thus, if the slope of the line tends towards vertical, we use:
84        A =   ( (a - 2*b + c) - g'*(d  - 2*e + f)      )
85        B = 2*(-(a -   b    ) + g'*(d  -   e    )      )
86        C =   ( (a          ) - g'*(d           ) - h' )
87  */
88 
89 class LineQuadraticIntersections {
90 public:
91     enum PinTPoint {
92         kPointUninitialized,
93         kPointInitialized
94     };
95 
LineQuadraticIntersections(const SkDQuad & q,const SkDLine & l,SkIntersections * i)96     LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i)
97         : fQuad(q)
98         , fLine(&l)
99         , fIntersections(i)
100         , fAllowNear(true) {
101         i->setMax(3);  // allow short partial coincidence plus discrete intersection
102     }
103 
LineQuadraticIntersections(const SkDQuad & q)104     LineQuadraticIntersections(const SkDQuad& q)
105         : fQuad(q)
106         SkDEBUGPARAMS(fLine(nullptr))
107         SkDEBUGPARAMS(fIntersections(nullptr))
108         SkDEBUGPARAMS(fAllowNear(false)) {
109     }
110 
allowNear(bool allow)111     void allowNear(bool allow) {
112         fAllowNear = allow;
113     }
114 
checkCoincident()115     void checkCoincident() {
116         int last = fIntersections->used() - 1;
117         for (int index = 0; index < last; ) {
118             double quadMidT = ((*fIntersections)[0][index] + (*fIntersections)[0][index + 1]) / 2;
119             SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
120             double t = fLine->nearPoint(quadMidPt, nullptr);
121             if (t < 0) {
122                 ++index;
123                 continue;
124             }
125             if (fIntersections->isCoincident(index)) {
126                 fIntersections->removeOne(index);
127                 --last;
128             } else if (fIntersections->isCoincident(index + 1)) {
129                 fIntersections->removeOne(index + 1);
130                 --last;
131             } else {
132                 fIntersections->setCoincident(index++);
133             }
134             fIntersections->setCoincident(index);
135         }
136     }
137 
intersectRay(double roots[2])138     int intersectRay(double roots[2]) {
139     /*
140         solve by rotating line+quad so line is horizontal, then finding the roots
141         set up matrix to rotate quad to x-axis
142         |cos(a) -sin(a)|
143         |sin(a)  cos(a)|
144         note that cos(a) = A(djacent) / Hypoteneuse
145                   sin(a) = O(pposite) / Hypoteneuse
146         since we are computing Ts, we can ignore hypoteneuse, the scale factor:
147         |  A     -O    |
148         |  O      A    |
149         A = line[1].fX - line[0].fX (adjacent side of the right triangle)
150         O = line[1].fY - line[0].fY (opposite side of the right triangle)
151         for each of the three points (e.g. n = 0 to 2)
152         quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
153     */
154         double adj = (*fLine)[1].fX - (*fLine)[0].fX;
155         double opp = (*fLine)[1].fY - (*fLine)[0].fY;
156         double r[3];
157         for (int n = 0; n < 3; ++n) {
158             r[n] = (fQuad[n].fY - (*fLine)[0].fY) * adj - (fQuad[n].fX - (*fLine)[0].fX) * opp;
159         }
160         double A = r[2];
161         double B = r[1];
162         double C = r[0];
163         A += C - 2 * B;  // A = a - 2*b + c
164         B -= C;  // B = -(b - c)
165         return SkDQuad::RootsValidT(A, 2 * B, C, roots);
166     }
167 
intersect()168     int intersect() {
169         addExactEndPoints();
170         if (fAllowNear) {
171             addNearEndPoints();
172         }
173         double rootVals[2];
174         int roots = intersectRay(rootVals);
175         for (int index = 0; index < roots; ++index) {
176             double quadT = rootVals[index];
177             double lineT = findLineT(quadT);
178             SkDPoint pt;
179             if (pinTs(&quadT, &lineT, &pt, kPointUninitialized) && uniqueAnswer(quadT, pt)) {
180                 fIntersections->insert(quadT, lineT, pt);
181             }
182         }
183         checkCoincident();
184         return fIntersections->used();
185     }
186 
horizontalIntersect(double axisIntercept,double roots[2])187     int horizontalIntersect(double axisIntercept, double roots[2]) {
188         double D = fQuad[2].fY;  // f
189         double E = fQuad[1].fY;  // e
190         double F = fQuad[0].fY;  // d
191         D += F - 2 * E;         // D = d - 2*e + f
192         E -= F;                 // E = -(d - e)
193         F -= axisIntercept;
194         return SkDQuad::RootsValidT(D, 2 * E, F, roots);
195     }
196 
horizontalIntersect(double axisIntercept,double left,double right,bool flipped)197     int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
198         addExactHorizontalEndPoints(left, right, axisIntercept);
199         if (fAllowNear) {
200             addNearHorizontalEndPoints(left, right, axisIntercept);
201         }
202         double rootVals[2];
203         int roots = horizontalIntersect(axisIntercept, rootVals);
204         for (int index = 0; index < roots; ++index) {
205             double quadT = rootVals[index];
206             SkDPoint pt = fQuad.ptAtT(quadT);
207             double lineT = (pt.fX - left) / (right - left);
208             if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
209                 fIntersections->insert(quadT, lineT, pt);
210             }
211         }
212         if (flipped) {
213             fIntersections->flip();
214         }
215         checkCoincident();
216         return fIntersections->used();
217     }
218 
uniqueAnswer(double quadT,const SkDPoint & pt)219     bool uniqueAnswer(double quadT, const SkDPoint& pt) {
220         for (int inner = 0; inner < fIntersections->used(); ++inner) {
221             if (fIntersections->pt(inner) != pt) {
222                 continue;
223             }
224             double existingQuadT = (*fIntersections)[0][inner];
225             if (quadT == existingQuadT) {
226                 return false;
227             }
228             // check if midway on quad is also same point. If so, discard this
229             double quadMidT = (existingQuadT + quadT) / 2;
230             SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
231             if (quadMidPt.approximatelyEqual(pt)) {
232                 return false;
233             }
234         }
235 #if ONE_OFF_DEBUG
236         SkDPoint qPt = fQuad.ptAtT(quadT);
237         SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY,
238                 qPt.fX, qPt.fY);
239 #endif
240         return true;
241     }
242 
verticalIntersect(double axisIntercept,double roots[2])243     int verticalIntersect(double axisIntercept, double roots[2]) {
244         double D = fQuad[2].fX;  // f
245         double E = fQuad[1].fX;  // e
246         double F = fQuad[0].fX;  // d
247         D += F - 2 * E;         // D = d - 2*e + f
248         E -= F;                 // E = -(d - e)
249         F -= axisIntercept;
250         return SkDQuad::RootsValidT(D, 2 * E, F, roots);
251     }
252 
verticalIntersect(double axisIntercept,double top,double bottom,bool flipped)253     int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
254         addExactVerticalEndPoints(top, bottom, axisIntercept);
255         if (fAllowNear) {
256             addNearVerticalEndPoints(top, bottom, axisIntercept);
257         }
258         double rootVals[2];
259         int roots = verticalIntersect(axisIntercept, rootVals);
260         for (int index = 0; index < roots; ++index) {
261             double quadT = rootVals[index];
262             SkDPoint pt = fQuad.ptAtT(quadT);
263             double lineT = (pt.fY - top) / (bottom - top);
264             if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
265                 fIntersections->insert(quadT, lineT, pt);
266             }
267         }
268         if (flipped) {
269             fIntersections->flip();
270         }
271         checkCoincident();
272         return fIntersections->used();
273     }
274 
275 protected:
276     // add endpoints first to get zero and one t values exactly
addExactEndPoints()277     void addExactEndPoints() {
278         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
279             double lineT = fLine->exactPoint(fQuad[qIndex]);
280             if (lineT < 0) {
281                 continue;
282             }
283             double quadT = (double) (qIndex >> 1);
284             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
285         }
286     }
287 
addNearEndPoints()288     void addNearEndPoints() {
289         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
290             double quadT = (double) (qIndex >> 1);
291             if (fIntersections->hasT(quadT)) {
292                 continue;
293             }
294             double lineT = fLine->nearPoint(fQuad[qIndex], nullptr);
295             if (lineT < 0) {
296                 continue;
297             }
298             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
299         }
300         // FIXME: see if line end is nearly on quad
301     }
302 
addExactHorizontalEndPoints(double left,double right,double y)303     void addExactHorizontalEndPoints(double left, double right, double y) {
304         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
305             double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
306             if (lineT < 0) {
307                 continue;
308             }
309             double quadT = (double) (qIndex >> 1);
310             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
311         }
312     }
313 
addNearHorizontalEndPoints(double left,double right,double y)314     void addNearHorizontalEndPoints(double left, double right, double y) {
315         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
316             double quadT = (double) (qIndex >> 1);
317             if (fIntersections->hasT(quadT)) {
318                 continue;
319             }
320             double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
321             if (lineT < 0) {
322                 continue;
323             }
324             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
325         }
326         // FIXME: see if line end is nearly on quad
327     }
328 
addExactVerticalEndPoints(double top,double bottom,double x)329     void addExactVerticalEndPoints(double top, double bottom, double x) {
330         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
331             double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
332             if (lineT < 0) {
333                 continue;
334             }
335             double quadT = (double) (qIndex >> 1);
336             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
337         }
338     }
339 
addNearVerticalEndPoints(double top,double bottom,double x)340     void addNearVerticalEndPoints(double top, double bottom, double x) {
341         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
342             double quadT = (double) (qIndex >> 1);
343             if (fIntersections->hasT(quadT)) {
344                 continue;
345             }
346             double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
347             if (lineT < 0) {
348                 continue;
349             }
350             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
351         }
352         // FIXME: see if line end is nearly on quad
353     }
354 
findLineT(double t)355     double findLineT(double t) {
356         SkDPoint xy = fQuad.ptAtT(t);
357         double dx = (*fLine)[1].fX - (*fLine)[0].fX;
358         double dy = (*fLine)[1].fY - (*fLine)[0].fY;
359         if (fabs(dx) > fabs(dy)) {
360             return (xy.fX - (*fLine)[0].fX) / dx;
361         }
362         return (xy.fY - (*fLine)[0].fY) / dy;
363     }
364 
pinTs(double * quadT,double * lineT,SkDPoint * pt,PinTPoint ptSet)365     bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
366         if (!approximately_one_or_less_double(*lineT)) {
367             return false;
368         }
369         if (!approximately_zero_or_more_double(*lineT)) {
370             return false;
371         }
372         double qT = *quadT = SkPinT(*quadT);
373         double lT = *lineT = SkPinT(*lineT);
374         if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
375             *pt = (*fLine).ptAtT(lT);
376         } else if (ptSet == kPointUninitialized) {
377             *pt = fQuad.ptAtT(qT);
378         }
379         SkPoint gridPt = pt->asSkPoint();
380         if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[0].asSkPoint())) {
381             *pt = (*fLine)[0];
382             *lineT = 0;
383         } else if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[1].asSkPoint())) {
384             *pt = (*fLine)[1];
385             *lineT = 1;
386         }
387         if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) {
388             return false;
389         }
390         if (gridPt == fQuad[0].asSkPoint()) {
391             *pt = fQuad[0];
392             *quadT = 0;
393         } else if (gridPt == fQuad[2].asSkPoint()) {
394             *pt = fQuad[2];
395             *quadT = 1;
396         }
397         return true;
398     }
399 
400 private:
401     const SkDQuad& fQuad;
402     const SkDLine* fLine;
403     SkIntersections* fIntersections;
404     bool fAllowNear;
405 };
406 
horizontal(const SkDQuad & quad,double left,double right,double y,bool flipped)407 int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
408                                 bool flipped) {
409     SkDLine line = {{{ left, y }, { right, y }}};
410     LineQuadraticIntersections q(quad, line, this);
411     return q.horizontalIntersect(y, left, right, flipped);
412 }
413 
vertical(const SkDQuad & quad,double top,double bottom,double x,bool flipped)414 int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
415                               bool flipped) {
416     SkDLine line = {{{ x, top }, { x, bottom }}};
417     LineQuadraticIntersections q(quad, line, this);
418     return q.verticalIntersect(x, top, bottom, flipped);
419 }
420 
intersect(const SkDQuad & quad,const SkDLine & line)421 int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
422     LineQuadraticIntersections q(quad, line, this);
423     q.allowNear(fAllowNear);
424     return q.intersect();
425 }
426 
intersectRay(const SkDQuad & quad,const SkDLine & line)427 int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
428     LineQuadraticIntersections q(quad, line, this);
429     fUsed = q.intersectRay(fT[0]);
430     for (int index = 0; index < fUsed; ++index) {
431         fPt[index] = quad.ptAtT(fT[0][index]);
432     }
433     return fUsed;
434 }
435 
HorizontalIntercept(const SkDQuad & quad,SkScalar y,double * roots)436 int SkIntersections::HorizontalIntercept(const SkDQuad& quad, SkScalar y, double* roots) {
437     LineQuadraticIntersections q(quad);
438     return q.horizontalIntersect(y, roots);
439 }
440 
VerticalIntercept(const SkDQuad & quad,SkScalar x,double * roots)441 int SkIntersections::VerticalIntercept(const SkDQuad& quad, SkScalar x, double* roots) {
442     LineQuadraticIntersections q(quad);
443     return q.verticalIntersect(x, roots);
444 }
445 
446 // SkDQuad accessors to Intersection utilities
447 
horizontalIntersect(double yIntercept,double roots[2]) const448 int SkDQuad::horizontalIntersect(double yIntercept, double roots[2]) const {
449     return SkIntersections::HorizontalIntercept(*this, yIntercept, roots);
450 }
451 
verticalIntersect(double xIntercept,double roots[2]) const452 int SkDQuad::verticalIntersect(double xIntercept, double roots[2]) const {
453     return SkIntersections::VerticalIntercept(*this, xIntercept, roots);
454 }
455