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 "SkPathOpsCurve.h"
9 #include "SkPathOpsLine.h"
10 #include "SkPathOpsQuad.h"
11
12 /*
13 Find the interection of a line and quadratic by solving for valid t values.
14
15 From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve
16
17 "A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three
18 control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where
19 A, B and C are points and t goes from zero to one.
20
21 This will give you two equations:
22
23 x = a(1 - t)^2 + b(1 - t)t + ct^2
24 y = d(1 - t)^2 + e(1 - t)t + ft^2
25
26 If you add for instance the line equation (y = kx + m) to that, you'll end up
27 with three equations and three unknowns (x, y and t)."
28
29 Similar to above, the quadratic is represented as
30 x = a(1-t)^2 + 2b(1-t)t + ct^2
31 y = d(1-t)^2 + 2e(1-t)t + ft^2
32 and the line as
33 y = g*x + h
34
35 Using Mathematica, solve for the values of t where the quadratic intersects the
36 line:
37
38 (in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x,
39 d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x]
40 (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 +
41 g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2)
42 (in) Solve[t1 == 0, t]
43 (out) {
44 {t -> (-2 d + 2 e + 2 a g - 2 b g -
45 Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 -
46 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) /
47 (2 (-d + 2 e - f + a g - 2 b g + c g))
48 },
49 {t -> (-2 d + 2 e + 2 a g - 2 b g +
50 Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 -
51 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) /
52 (2 (-d + 2 e - f + a g - 2 b g + c g))
53 }
54 }
55
56 Using the results above (when the line tends towards horizontal)
57 A = (-(d - 2*e + f) + g*(a - 2*b + c) )
58 B = 2*( (d - e ) - g*(a - b ) )
59 C = (-(d ) + g*(a ) + h )
60
61 If g goes to infinity, we can rewrite the line in terms of x.
62 x = g'*y + h'
63
64 And solve accordingly in Mathematica:
65
66 (in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h',
67 d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y]
68 (out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 -
69 g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2)
70 (in) Solve[t2 == 0, t]
71 (out) {
72 {t -> (2 a - 2 b - 2 d g' + 2 e g' -
73 Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 -
74 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) /
75 (2 (a - 2 b + c - d g' + 2 e g' - f g'))
76 },
77 {t -> (2 a - 2 b - 2 d g' + 2 e g' +
78 Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 -
79 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/
80 (2 (a - 2 b + c - d g' + 2 e g' - f g'))
81 }
82 }
83
84 Thus, if the slope of the line tends towards vertical, we use:
85 A = ( (a - 2*b + c) - g'*(d - 2*e + f) )
86 B = 2*(-(a - b ) + g'*(d - e ) )
87 C = ( (a ) - g'*(d ) - h' )
88 */
89
90 class LineQuadraticIntersections {
91 public:
92 enum PinTPoint {
93 kPointUninitialized,
94 kPointInitialized
95 };
96
LineQuadraticIntersections(const SkDQuad & q,const SkDLine & l,SkIntersections * i)97 LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i)
98 : fQuad(q)
99 , fLine(&l)
100 , fIntersections(i)
101 , fAllowNear(true) {
102 i->setMax(5); // allow short partial coincidence plus discrete intersections
103 }
104
LineQuadraticIntersections(const SkDQuad & q)105 LineQuadraticIntersections(const SkDQuad& q)
106 : fQuad(q)
107 SkDEBUGPARAMS(fLine(nullptr))
108 SkDEBUGPARAMS(fIntersections(nullptr))
109 SkDEBUGPARAMS(fAllowNear(false)) {
110 }
111
allowNear(bool allow)112 void allowNear(bool allow) {
113 fAllowNear = allow;
114 }
115
checkCoincident()116 void checkCoincident() {
117 int last = fIntersections->used() - 1;
118 for (int index = 0; index < last; ) {
119 double quadMidT = ((*fIntersections)[0][index] + (*fIntersections)[0][index + 1]) / 2;
120 SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
121 double t = fLine->nearPoint(quadMidPt, nullptr);
122 if (t < 0) {
123 ++index;
124 continue;
125 }
126 if (fIntersections->isCoincident(index)) {
127 fIntersections->removeOne(index);
128 --last;
129 } else if (fIntersections->isCoincident(index + 1)) {
130 fIntersections->removeOne(index + 1);
131 --last;
132 } else {
133 fIntersections->setCoincident(index++);
134 }
135 fIntersections->setCoincident(index);
136 }
137 }
138
intersectRay(double roots[2])139 int intersectRay(double roots[2]) {
140 /*
141 solve by rotating line+quad so line is horizontal, then finding the roots
142 set up matrix to rotate quad to x-axis
143 |cos(a) -sin(a)|
144 |sin(a) cos(a)|
145 note that cos(a) = A(djacent) / Hypoteneuse
146 sin(a) = O(pposite) / Hypoteneuse
147 since we are computing Ts, we can ignore hypoteneuse, the scale factor:
148 | A -O |
149 | O A |
150 A = line[1].fX - line[0].fX (adjacent side of the right triangle)
151 O = line[1].fY - line[0].fY (opposite side of the right triangle)
152 for each of the three points (e.g. n = 0 to 2)
153 quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
154 */
155 double adj = (*fLine)[1].fX - (*fLine)[0].fX;
156 double opp = (*fLine)[1].fY - (*fLine)[0].fY;
157 double r[3];
158 for (int n = 0; n < 3; ++n) {
159 r[n] = (fQuad[n].fY - (*fLine)[0].fY) * adj - (fQuad[n].fX - (*fLine)[0].fX) * opp;
160 }
161 double A = r[2];
162 double B = r[1];
163 double C = r[0];
164 A += C - 2 * B; // A = a - 2*b + c
165 B -= C; // B = -(b - c)
166 return SkDQuad::RootsValidT(A, 2 * B, C, roots);
167 }
168
intersect()169 int intersect() {
170 addExactEndPoints();
171 if (fAllowNear) {
172 addNearEndPoints();
173 }
174 double rootVals[2];
175 int roots = intersectRay(rootVals);
176 for (int index = 0; index < roots; ++index) {
177 double quadT = rootVals[index];
178 double lineT = findLineT(quadT);
179 SkDPoint pt;
180 if (pinTs(&quadT, &lineT, &pt, kPointUninitialized) && uniqueAnswer(quadT, pt)) {
181 fIntersections->insert(quadT, lineT, pt);
182 }
183 }
184 checkCoincident();
185 return fIntersections->used();
186 }
187
horizontalIntersect(double axisIntercept,double roots[2])188 int horizontalIntersect(double axisIntercept, double roots[2]) {
189 double D = fQuad[2].fY; // f
190 double E = fQuad[1].fY; // e
191 double F = fQuad[0].fY; // d
192 D += F - 2 * E; // D = d - 2*e + f
193 E -= F; // E = -(d - e)
194 F -= axisIntercept;
195 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
196 }
197
horizontalIntersect(double axisIntercept,double left,double right,bool flipped)198 int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
199 addExactHorizontalEndPoints(left, right, axisIntercept);
200 if (fAllowNear) {
201 addNearHorizontalEndPoints(left, right, axisIntercept);
202 }
203 double rootVals[2];
204 int roots = horizontalIntersect(axisIntercept, rootVals);
205 for (int index = 0; index < roots; ++index) {
206 double quadT = rootVals[index];
207 SkDPoint pt = fQuad.ptAtT(quadT);
208 double lineT = (pt.fX - left) / (right - left);
209 if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
210 fIntersections->insert(quadT, lineT, pt);
211 }
212 }
213 if (flipped) {
214 fIntersections->flip();
215 }
216 checkCoincident();
217 return fIntersections->used();
218 }
219
uniqueAnswer(double quadT,const SkDPoint & pt)220 bool uniqueAnswer(double quadT, const SkDPoint& pt) {
221 for (int inner = 0; inner < fIntersections->used(); ++inner) {
222 if (fIntersections->pt(inner) != pt) {
223 continue;
224 }
225 double existingQuadT = (*fIntersections)[0][inner];
226 if (quadT == existingQuadT) {
227 return false;
228 }
229 // check if midway on quad is also same point. If so, discard this
230 double quadMidT = (existingQuadT + quadT) / 2;
231 SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
232 if (quadMidPt.approximatelyEqual(pt)) {
233 return false;
234 }
235 }
236 #if ONE_OFF_DEBUG
237 SkDPoint qPt = fQuad.ptAtT(quadT);
238 SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY,
239 qPt.fX, qPt.fY);
240 #endif
241 return true;
242 }
243
verticalIntersect(double axisIntercept,double roots[2])244 int verticalIntersect(double axisIntercept, double roots[2]) {
245 double D = fQuad[2].fX; // f
246 double E = fQuad[1].fX; // e
247 double F = fQuad[0].fX; // d
248 D += F - 2 * E; // D = d - 2*e + f
249 E -= F; // E = -(d - e)
250 F -= axisIntercept;
251 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
252 }
253
verticalIntersect(double axisIntercept,double top,double bottom,bool flipped)254 int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
255 addExactVerticalEndPoints(top, bottom, axisIntercept);
256 if (fAllowNear) {
257 addNearVerticalEndPoints(top, bottom, axisIntercept);
258 }
259 double rootVals[2];
260 int roots = verticalIntersect(axisIntercept, rootVals);
261 for (int index = 0; index < roots; ++index) {
262 double quadT = rootVals[index];
263 SkDPoint pt = fQuad.ptAtT(quadT);
264 double lineT = (pt.fY - top) / (bottom - top);
265 if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
266 fIntersections->insert(quadT, lineT, pt);
267 }
268 }
269 if (flipped) {
270 fIntersections->flip();
271 }
272 checkCoincident();
273 return fIntersections->used();
274 }
275
276 protected:
277 // add endpoints first to get zero and one t values exactly
addExactEndPoints()278 void addExactEndPoints() {
279 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
280 double lineT = fLine->exactPoint(fQuad[qIndex]);
281 if (lineT < 0) {
282 continue;
283 }
284 double quadT = (double) (qIndex >> 1);
285 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
286 }
287 }
288
addNearEndPoints()289 void addNearEndPoints() {
290 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
291 double quadT = (double) (qIndex >> 1);
292 if (fIntersections->hasT(quadT)) {
293 continue;
294 }
295 double lineT = fLine->nearPoint(fQuad[qIndex], nullptr);
296 if (lineT < 0) {
297 continue;
298 }
299 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
300 }
301 this->addLineNearEndPoints();
302 }
303
addLineNearEndPoints()304 void addLineNearEndPoints() {
305 for (int lIndex = 0; lIndex < 2; ++lIndex) {
306 double lineT = (double) lIndex;
307 if (fIntersections->hasOppT(lineT)) {
308 continue;
309 }
310 double quadT = ((SkDCurve*) &fQuad)->nearPoint(SkPath::kQuad_Verb,
311 (*fLine)[lIndex], (*fLine)[!lIndex]);
312 if (quadT < 0) {
313 continue;
314 }
315 fIntersections->insert(quadT, lineT, (*fLine)[lIndex]);
316 }
317 }
318
addExactHorizontalEndPoints(double left,double right,double y)319 void addExactHorizontalEndPoints(double left, double right, double y) {
320 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
321 double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
322 if (lineT < 0) {
323 continue;
324 }
325 double quadT = (double) (qIndex >> 1);
326 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
327 }
328 }
329
addNearHorizontalEndPoints(double left,double right,double y)330 void addNearHorizontalEndPoints(double left, double right, double y) {
331 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
332 double quadT = (double) (qIndex >> 1);
333 if (fIntersections->hasT(quadT)) {
334 continue;
335 }
336 double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
337 if (lineT < 0) {
338 continue;
339 }
340 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
341 }
342 this->addLineNearEndPoints();
343 }
344
addExactVerticalEndPoints(double top,double bottom,double x)345 void addExactVerticalEndPoints(double top, double bottom, double x) {
346 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
347 double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
348 if (lineT < 0) {
349 continue;
350 }
351 double quadT = (double) (qIndex >> 1);
352 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
353 }
354 }
355
addNearVerticalEndPoints(double top,double bottom,double x)356 void addNearVerticalEndPoints(double top, double bottom, double x) {
357 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
358 double quadT = (double) (qIndex >> 1);
359 if (fIntersections->hasT(quadT)) {
360 continue;
361 }
362 double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
363 if (lineT < 0) {
364 continue;
365 }
366 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
367 }
368 this->addLineNearEndPoints();
369 }
370
findLineT(double t)371 double findLineT(double t) {
372 SkDPoint xy = fQuad.ptAtT(t);
373 double dx = (*fLine)[1].fX - (*fLine)[0].fX;
374 double dy = (*fLine)[1].fY - (*fLine)[0].fY;
375 if (fabs(dx) > fabs(dy)) {
376 return (xy.fX - (*fLine)[0].fX) / dx;
377 }
378 return (xy.fY - (*fLine)[0].fY) / dy;
379 }
380
pinTs(double * quadT,double * lineT,SkDPoint * pt,PinTPoint ptSet)381 bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
382 if (!approximately_one_or_less_double(*lineT)) {
383 return false;
384 }
385 if (!approximately_zero_or_more_double(*lineT)) {
386 return false;
387 }
388 double qT = *quadT = SkPinT(*quadT);
389 double lT = *lineT = SkPinT(*lineT);
390 if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
391 *pt = (*fLine).ptAtT(lT);
392 } else if (ptSet == kPointUninitialized) {
393 *pt = fQuad.ptAtT(qT);
394 }
395 SkPoint gridPt = pt->asSkPoint();
396 if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[0].asSkPoint())) {
397 *pt = (*fLine)[0];
398 *lineT = 0;
399 } else if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[1].asSkPoint())) {
400 *pt = (*fLine)[1];
401 *lineT = 1;
402 }
403 if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) {
404 return false;
405 }
406 if (gridPt == fQuad[0].asSkPoint()) {
407 *pt = fQuad[0];
408 *quadT = 0;
409 } else if (gridPt == fQuad[2].asSkPoint()) {
410 *pt = fQuad[2];
411 *quadT = 1;
412 }
413 return true;
414 }
415
416 private:
417 const SkDQuad& fQuad;
418 const SkDLine* fLine;
419 SkIntersections* fIntersections;
420 bool fAllowNear;
421 };
422
horizontal(const SkDQuad & quad,double left,double right,double y,bool flipped)423 int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
424 bool flipped) {
425 SkDLine line = {{{ left, y }, { right, y }}};
426 LineQuadraticIntersections q(quad, line, this);
427 return q.horizontalIntersect(y, left, right, flipped);
428 }
429
vertical(const SkDQuad & quad,double top,double bottom,double x,bool flipped)430 int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
431 bool flipped) {
432 SkDLine line = {{{ x, top }, { x, bottom }}};
433 LineQuadraticIntersections q(quad, line, this);
434 return q.verticalIntersect(x, top, bottom, flipped);
435 }
436
intersect(const SkDQuad & quad,const SkDLine & line)437 int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
438 LineQuadraticIntersections q(quad, line, this);
439 q.allowNear(fAllowNear);
440 return q.intersect();
441 }
442
intersectRay(const SkDQuad & quad,const SkDLine & line)443 int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
444 LineQuadraticIntersections q(quad, line, this);
445 fUsed = q.intersectRay(fT[0]);
446 for (int index = 0; index < fUsed; ++index) {
447 fPt[index] = quad.ptAtT(fT[0][index]);
448 }
449 return fUsed;
450 }
451
HorizontalIntercept(const SkDQuad & quad,SkScalar y,double * roots)452 int SkIntersections::HorizontalIntercept(const SkDQuad& quad, SkScalar y, double* roots) {
453 LineQuadraticIntersections q(quad);
454 return q.horizontalIntersect(y, roots);
455 }
456
VerticalIntercept(const SkDQuad & quad,SkScalar x,double * roots)457 int SkIntersections::VerticalIntercept(const SkDQuad& quad, SkScalar x, double* roots) {
458 LineQuadraticIntersections q(quad);
459 return q.verticalIntersect(x, roots);
460 }
461
462 // SkDQuad accessors to Intersection utilities
463
horizontalIntersect(double yIntercept,double roots[2]) const464 int SkDQuad::horizontalIntersect(double yIntercept, double roots[2]) const {
465 return SkIntersections::HorizontalIntercept(*this, yIntercept, roots);
466 }
467
verticalIntersect(double xIntercept,double roots[2]) const468 int SkDQuad::verticalIntersect(double xIntercept, double roots[2]) const {
469 return SkIntersections::VerticalIntercept(*this, xIntercept, roots);
470 }
471