Line data Source code
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 :
97 0 : LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i)
98 0 : : fQuad(q)
99 : , fLine(&l)
100 : , fIntersections(i)
101 0 : , fAllowNear(true) {
102 0 : i->setMax(5); // allow short partial coincidence plus discrete intersections
103 0 : }
104 :
105 0 : LineQuadraticIntersections(const SkDQuad& q)
106 0 : : fQuad(q)
107 : SkDEBUGPARAMS(fLine(nullptr))
108 : SkDEBUGPARAMS(fIntersections(nullptr))
109 0 : SkDEBUGPARAMS(fAllowNear(false)) {
110 0 : }
111 :
112 0 : void allowNear(bool allow) {
113 0 : fAllowNear = allow;
114 0 : }
115 :
116 0 : void checkCoincident() {
117 0 : int last = fIntersections->used() - 1;
118 0 : for (int index = 0; index < last; ) {
119 0 : double quadMidT = ((*fIntersections)[0][index] + (*fIntersections)[0][index + 1]) / 2;
120 0 : SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
121 0 : double t = fLine->nearPoint(quadMidPt, nullptr);
122 0 : if (t < 0) {
123 0 : ++index;
124 0 : continue;
125 : }
126 0 : if (fIntersections->isCoincident(index)) {
127 0 : fIntersections->removeOne(index);
128 0 : --last;
129 0 : } else if (fIntersections->isCoincident(index + 1)) {
130 0 : fIntersections->removeOne(index + 1);
131 0 : --last;
132 : } else {
133 0 : fIntersections->setCoincident(index++);
134 : }
135 0 : fIntersections->setCoincident(index);
136 : }
137 0 : }
138 :
139 0 : 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 0 : double adj = (*fLine)[1].fX - (*fLine)[0].fX;
156 0 : double opp = (*fLine)[1].fY - (*fLine)[0].fY;
157 : double r[3];
158 0 : for (int n = 0; n < 3; ++n) {
159 0 : r[n] = (fQuad[n].fY - (*fLine)[0].fY) * adj - (fQuad[n].fX - (*fLine)[0].fX) * opp;
160 : }
161 0 : double A = r[2];
162 0 : double B = r[1];
163 0 : double C = r[0];
164 0 : A += C - 2 * B; // A = a - 2*b + c
165 0 : B -= C; // B = -(b - c)
166 0 : return SkDQuad::RootsValidT(A, 2 * B, C, roots);
167 : }
168 :
169 0 : int intersect() {
170 0 : addExactEndPoints();
171 0 : if (fAllowNear) {
172 0 : addNearEndPoints();
173 : }
174 : double rootVals[2];
175 0 : int roots = intersectRay(rootVals);
176 0 : for (int index = 0; index < roots; ++index) {
177 0 : double quadT = rootVals[index];
178 0 : double lineT = findLineT(quadT);
179 : SkDPoint pt;
180 0 : if (pinTs(&quadT, &lineT, &pt, kPointUninitialized) && uniqueAnswer(quadT, pt)) {
181 0 : fIntersections->insert(quadT, lineT, pt);
182 : }
183 : }
184 0 : checkCoincident();
185 0 : return fIntersections->used();
186 : }
187 :
188 0 : int horizontalIntersect(double axisIntercept, double roots[2]) {
189 0 : double D = fQuad[2].fY; // f
190 0 : double E = fQuad[1].fY; // e
191 0 : double F = fQuad[0].fY; // d
192 0 : D += F - 2 * E; // D = d - 2*e + f
193 0 : E -= F; // E = -(d - e)
194 0 : F -= axisIntercept;
195 0 : return SkDQuad::RootsValidT(D, 2 * E, F, roots);
196 : }
197 :
198 0 : int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
199 0 : addExactHorizontalEndPoints(left, right, axisIntercept);
200 0 : if (fAllowNear) {
201 0 : addNearHorizontalEndPoints(left, right, axisIntercept);
202 : }
203 : double rootVals[2];
204 0 : int roots = horizontalIntersect(axisIntercept, rootVals);
205 0 : for (int index = 0; index < roots; ++index) {
206 0 : double quadT = rootVals[index];
207 0 : SkDPoint pt = fQuad.ptAtT(quadT);
208 0 : double lineT = (pt.fX - left) / (right - left);
209 0 : if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
210 0 : fIntersections->insert(quadT, lineT, pt);
211 : }
212 : }
213 0 : if (flipped) {
214 0 : fIntersections->flip();
215 : }
216 0 : checkCoincident();
217 0 : return fIntersections->used();
218 : }
219 :
220 0 : bool uniqueAnswer(double quadT, const SkDPoint& pt) {
221 0 : for (int inner = 0; inner < fIntersections->used(); ++inner) {
222 0 : if (fIntersections->pt(inner) != pt) {
223 0 : continue;
224 : }
225 0 : double existingQuadT = (*fIntersections)[0][inner];
226 0 : if (quadT == existingQuadT) {
227 0 : return false;
228 : }
229 : // check if midway on quad is also same point. If so, discard this
230 0 : double quadMidT = (existingQuadT + quadT) / 2;
231 0 : SkDPoint quadMidPt = fQuad.ptAtT(quadMidT);
232 0 : if (quadMidPt.approximatelyEqual(pt)) {
233 0 : 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 0 : return true;
242 : }
243 :
244 0 : int verticalIntersect(double axisIntercept, double roots[2]) {
245 0 : double D = fQuad[2].fX; // f
246 0 : double E = fQuad[1].fX; // e
247 0 : double F = fQuad[0].fX; // d
248 0 : D += F - 2 * E; // D = d - 2*e + f
249 0 : E -= F; // E = -(d - e)
250 0 : F -= axisIntercept;
251 0 : return SkDQuad::RootsValidT(D, 2 * E, F, roots);
252 : }
253 :
254 0 : int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
255 0 : addExactVerticalEndPoints(top, bottom, axisIntercept);
256 0 : if (fAllowNear) {
257 0 : addNearVerticalEndPoints(top, bottom, axisIntercept);
258 : }
259 : double rootVals[2];
260 0 : int roots = verticalIntersect(axisIntercept, rootVals);
261 0 : for (int index = 0; index < roots; ++index) {
262 0 : double quadT = rootVals[index];
263 0 : SkDPoint pt = fQuad.ptAtT(quadT);
264 0 : double lineT = (pt.fY - top) / (bottom - top);
265 0 : if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) {
266 0 : fIntersections->insert(quadT, lineT, pt);
267 : }
268 : }
269 0 : if (flipped) {
270 0 : fIntersections->flip();
271 : }
272 0 : checkCoincident();
273 0 : return fIntersections->used();
274 : }
275 :
276 : protected:
277 : // add endpoints first to get zero and one t values exactly
278 0 : void addExactEndPoints() {
279 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
280 0 : double lineT = fLine->exactPoint(fQuad[qIndex]);
281 0 : if (lineT < 0) {
282 0 : continue;
283 : }
284 0 : double quadT = (double) (qIndex >> 1);
285 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
286 : }
287 0 : }
288 :
289 0 : void addNearEndPoints() {
290 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
291 0 : double quadT = (double) (qIndex >> 1);
292 0 : if (fIntersections->hasT(quadT)) {
293 0 : continue;
294 : }
295 0 : double lineT = fLine->nearPoint(fQuad[qIndex], nullptr);
296 0 : if (lineT < 0) {
297 0 : continue;
298 : }
299 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
300 : }
301 0 : this->addLineNearEndPoints();
302 0 : }
303 :
304 0 : void addLineNearEndPoints() {
305 0 : for (int lIndex = 0; lIndex < 2; ++lIndex) {
306 0 : double lineT = (double) lIndex;
307 0 : if (fIntersections->hasOppT(lineT)) {
308 0 : continue;
309 : }
310 0 : double quadT = ((SkDCurve*) &fQuad)->nearPoint(SkPath::kQuad_Verb,
311 0 : (*fLine)[lIndex], (*fLine)[!lIndex]);
312 0 : if (quadT < 0) {
313 0 : continue;
314 : }
315 0 : fIntersections->insert(quadT, lineT, (*fLine)[lIndex]);
316 : }
317 0 : }
318 :
319 0 : void addExactHorizontalEndPoints(double left, double right, double y) {
320 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
321 0 : double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
322 0 : if (lineT < 0) {
323 0 : continue;
324 : }
325 0 : double quadT = (double) (qIndex >> 1);
326 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
327 : }
328 0 : }
329 :
330 0 : void addNearHorizontalEndPoints(double left, double right, double y) {
331 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
332 0 : double quadT = (double) (qIndex >> 1);
333 0 : if (fIntersections->hasT(quadT)) {
334 0 : continue;
335 : }
336 0 : double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
337 0 : if (lineT < 0) {
338 0 : continue;
339 : }
340 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
341 : }
342 0 : this->addLineNearEndPoints();
343 0 : }
344 :
345 0 : void addExactVerticalEndPoints(double top, double bottom, double x) {
346 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
347 0 : double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
348 0 : if (lineT < 0) {
349 0 : continue;
350 : }
351 0 : double quadT = (double) (qIndex >> 1);
352 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
353 : }
354 0 : }
355 :
356 0 : void addNearVerticalEndPoints(double top, double bottom, double x) {
357 0 : for (int qIndex = 0; qIndex < 3; qIndex += 2) {
358 0 : double quadT = (double) (qIndex >> 1);
359 0 : if (fIntersections->hasT(quadT)) {
360 0 : continue;
361 : }
362 0 : double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
363 0 : if (lineT < 0) {
364 0 : continue;
365 : }
366 0 : fIntersections->insert(quadT, lineT, fQuad[qIndex]);
367 : }
368 0 : this->addLineNearEndPoints();
369 0 : }
370 :
371 0 : double findLineT(double t) {
372 0 : SkDPoint xy = fQuad.ptAtT(t);
373 0 : double dx = (*fLine)[1].fX - (*fLine)[0].fX;
374 0 : double dy = (*fLine)[1].fY - (*fLine)[0].fY;
375 0 : if (fabs(dx) > fabs(dy)) {
376 0 : return (xy.fX - (*fLine)[0].fX) / dx;
377 : }
378 0 : return (xy.fY - (*fLine)[0].fY) / dy;
379 : }
380 :
381 0 : bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
382 0 : if (!approximately_one_or_less_double(*lineT)) {
383 0 : return false;
384 : }
385 0 : if (!approximately_zero_or_more_double(*lineT)) {
386 0 : return false;
387 : }
388 0 : double qT = *quadT = SkPinT(*quadT);
389 0 : double lT = *lineT = SkPinT(*lineT);
390 0 : if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
391 0 : *pt = (*fLine).ptAtT(lT);
392 0 : } else if (ptSet == kPointUninitialized) {
393 0 : *pt = fQuad.ptAtT(qT);
394 : }
395 0 : SkPoint gridPt = pt->asSkPoint();
396 0 : if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[0].asSkPoint())) {
397 0 : *pt = (*fLine)[0];
398 0 : *lineT = 0;
399 0 : } else if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[1].asSkPoint())) {
400 0 : *pt = (*fLine)[1];
401 0 : *lineT = 1;
402 : }
403 0 : if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) {
404 0 : return false;
405 : }
406 0 : if (gridPt == fQuad[0].asSkPoint()) {
407 0 : *pt = fQuad[0];
408 0 : *quadT = 0;
409 0 : } else if (gridPt == fQuad[2].asSkPoint()) {
410 0 : *pt = fQuad[2];
411 0 : *quadT = 1;
412 : }
413 0 : return true;
414 : }
415 :
416 : private:
417 : const SkDQuad& fQuad;
418 : const SkDLine* fLine;
419 : SkIntersections* fIntersections;
420 : bool fAllowNear;
421 : };
422 :
423 0 : int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
424 : bool flipped) {
425 0 : SkDLine line = {{{ left, y }, { right, y }}};
426 0 : LineQuadraticIntersections q(quad, line, this);
427 0 : return q.horizontalIntersect(y, left, right, flipped);
428 : }
429 :
430 0 : int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
431 : bool flipped) {
432 0 : SkDLine line = {{{ x, top }, { x, bottom }}};
433 0 : LineQuadraticIntersections q(quad, line, this);
434 0 : return q.verticalIntersect(x, top, bottom, flipped);
435 : }
436 :
437 0 : int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
438 0 : LineQuadraticIntersections q(quad, line, this);
439 0 : q.allowNear(fAllowNear);
440 0 : return q.intersect();
441 : }
442 :
443 0 : int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
444 0 : LineQuadraticIntersections q(quad, line, this);
445 0 : fUsed = q.intersectRay(fT[0]);
446 0 : for (int index = 0; index < fUsed; ++index) {
447 0 : fPt[index] = quad.ptAtT(fT[0][index]);
448 : }
449 0 : return fUsed;
450 : }
451 :
452 0 : int SkIntersections::HorizontalIntercept(const SkDQuad& quad, SkScalar y, double* roots) {
453 0 : LineQuadraticIntersections q(quad);
454 0 : return q.horizontalIntersect(y, roots);
455 : }
456 :
457 0 : int SkIntersections::VerticalIntercept(const SkDQuad& quad, SkScalar x, double* roots) {
458 0 : LineQuadraticIntersections q(quad);
459 0 : return q.verticalIntersect(x, roots);
460 : }
461 :
462 : // SkDQuad accessors to Intersection utilities
463 :
464 0 : int SkDQuad::horizontalIntersect(double yIntercept, double roots[2]) const {
465 0 : return SkIntersections::HorizontalIntercept(*this, yIntercept, roots);
466 : }
467 :
468 0 : int SkDQuad::verticalIntersect(double xIntercept, double roots[2]) const {
469 0 : return SkIntersections::VerticalIntercept(*this, xIntercept, roots);
470 : }
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