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
allowNear(bool allow)104 void allowNear(bool allow) {
105 fAllowNear = allow;
106 }
107
intersectRay(double roots[2])108 int intersectRay(double roots[2]) {
109 /*
110 solve by rotating line+quad so line is horizontal, then finding the roots
111 set up matrix to rotate quad to x-axis
112 |cos(a) -sin(a)|
113 |sin(a) cos(a)|
114 note that cos(a) = A(djacent) / Hypoteneuse
115 sin(a) = O(pposite) / Hypoteneuse
116 since we are computing Ts, we can ignore hypoteneuse, the scale factor:
117 | A -O |
118 | O A |
119 A = line[1].fX - line[0].fX (adjacent side of the right triangle)
120 O = line[1].fY - line[0].fY (opposite side of the right triangle)
121 for each of the three points (e.g. n = 0 to 2)
122 quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
123 */
124 double adj = fLine[1].fX - fLine[0].fX;
125 double opp = fLine[1].fY - fLine[0].fY;
126 double r[3];
127 for (int n = 0; n < 3; ++n) {
128 r[n] = (fQuad[n].fY - fLine[0].fY) * adj - (fQuad[n].fX - fLine[0].fX) * opp;
129 }
130 double A = r[2];
131 double B = r[1];
132 double C = r[0];
133 A += C - 2 * B; // A = a - 2*b + c
134 B -= C; // B = -(b - c)
135 return SkDQuad::RootsValidT(A, 2 * B, C, roots);
136 }
137
intersect()138 int intersect() {
139 addExactEndPoints();
140 if (fAllowNear) {
141 addNearEndPoints();
142 }
143 if (fIntersections->used() == 2) {
144 // FIXME : need sharable code that turns spans into coincident if middle point is on
145 } else {
146 double rootVals[2];
147 int roots = intersectRay(rootVals);
148 for (int index = 0; index < roots; ++index) {
149 double quadT = rootVals[index];
150 double lineT = findLineT(quadT);
151 SkDPoint pt;
152 if (pinTs(&quadT, &lineT, &pt, kPointUninitialized)) {
153 fIntersections->insert(quadT, lineT, pt);
154 }
155 }
156 }
157 return fIntersections->used();
158 }
159
horizontalIntersect(double axisIntercept,double roots[2])160 int horizontalIntersect(double axisIntercept, double roots[2]) {
161 double D = fQuad[2].fY; // f
162 double E = fQuad[1].fY; // e
163 double F = fQuad[0].fY; // d
164 D += F - 2 * E; // D = d - 2*e + f
165 E -= F; // E = -(d - e)
166 F -= axisIntercept;
167 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
168 }
169
horizontalIntersect(double axisIntercept,double left,double right,bool flipped)170 int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
171 addExactHorizontalEndPoints(left, right, axisIntercept);
172 if (fAllowNear) {
173 addNearHorizontalEndPoints(left, right, axisIntercept);
174 }
175 double rootVals[2];
176 int roots = horizontalIntersect(axisIntercept, rootVals);
177 for (int index = 0; index < roots; ++index) {
178 double quadT = rootVals[index];
179 SkDPoint pt = fQuad.ptAtT(quadT);
180 double lineT = (pt.fX - left) / (right - left);
181 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
182 fIntersections->insert(quadT, lineT, pt);
183 }
184 }
185 if (flipped) {
186 fIntersections->flip();
187 }
188 return fIntersections->used();
189 }
190
verticalIntersect(double axisIntercept,double roots[2])191 int verticalIntersect(double axisIntercept, double roots[2]) {
192 double D = fQuad[2].fX; // f
193 double E = fQuad[1].fX; // e
194 double F = fQuad[0].fX; // d
195 D += F - 2 * E; // D = d - 2*e + f
196 E -= F; // E = -(d - e)
197 F -= axisIntercept;
198 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
199 }
200
verticalIntersect(double axisIntercept,double top,double bottom,bool flipped)201 int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
202 addExactVerticalEndPoints(top, bottom, axisIntercept);
203 if (fAllowNear) {
204 addNearVerticalEndPoints(top, bottom, axisIntercept);
205 }
206 double rootVals[2];
207 int roots = verticalIntersect(axisIntercept, rootVals);
208 for (int index = 0; index < roots; ++index) {
209 double quadT = rootVals[index];
210 SkDPoint pt = fQuad.ptAtT(quadT);
211 double lineT = (pt.fY - top) / (bottom - top);
212 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
213 fIntersections->insert(quadT, lineT, pt);
214 }
215 }
216 if (flipped) {
217 fIntersections->flip();
218 }
219 return fIntersections->used();
220 }
221
222 protected:
223 // add endpoints first to get zero and one t values exactly
addExactEndPoints()224 void addExactEndPoints() {
225 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
226 double lineT = fLine.exactPoint(fQuad[qIndex]);
227 if (lineT < 0) {
228 continue;
229 }
230 double quadT = (double) (qIndex >> 1);
231 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
232 }
233 }
234
addNearEndPoints()235 void addNearEndPoints() {
236 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
237 double quadT = (double) (qIndex >> 1);
238 if (fIntersections->hasT(quadT)) {
239 continue;
240 }
241 double lineT = fLine.nearPoint(fQuad[qIndex], NULL);
242 if (lineT < 0) {
243 continue;
244 }
245 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
246 }
247 // FIXME: see if line end is nearly on quad
248 }
249
addExactHorizontalEndPoints(double left,double right,double y)250 void addExactHorizontalEndPoints(double left, double right, double y) {
251 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
252 double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
253 if (lineT < 0) {
254 continue;
255 }
256 double quadT = (double) (qIndex >> 1);
257 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
258 }
259 }
260
addNearHorizontalEndPoints(double left,double right,double y)261 void addNearHorizontalEndPoints(double left, double right, double y) {
262 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
263 double quadT = (double) (qIndex >> 1);
264 if (fIntersections->hasT(quadT)) {
265 continue;
266 }
267 double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
268 if (lineT < 0) {
269 continue;
270 }
271 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
272 }
273 // FIXME: see if line end is nearly on quad
274 }
275
addExactVerticalEndPoints(double top,double bottom,double x)276 void addExactVerticalEndPoints(double top, double bottom, double x) {
277 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
278 double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
279 if (lineT < 0) {
280 continue;
281 }
282 double quadT = (double) (qIndex >> 1);
283 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
284 }
285 }
286
addNearVerticalEndPoints(double top,double bottom,double x)287 void addNearVerticalEndPoints(double top, double bottom, double x) {
288 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
289 double quadT = (double) (qIndex >> 1);
290 if (fIntersections->hasT(quadT)) {
291 continue;
292 }
293 double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
294 if (lineT < 0) {
295 continue;
296 }
297 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
298 }
299 // FIXME: see if line end is nearly on quad
300 }
301
findLineT(double t)302 double findLineT(double t) {
303 SkDPoint xy = fQuad.ptAtT(t);
304 double dx = fLine[1].fX - fLine[0].fX;
305 double dy = fLine[1].fY - fLine[0].fY;
306 if (fabs(dx) > fabs(dy)) {
307 return (xy.fX - fLine[0].fX) / dx;
308 }
309 return (xy.fY - fLine[0].fY) / dy;
310 }
311
pinTs(double * quadT,double * lineT,SkDPoint * pt,PinTPoint ptSet)312 bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
313 if (!approximately_one_or_less_double(*lineT)) {
314 return false;
315 }
316 if (!approximately_zero_or_more_double(*lineT)) {
317 return false;
318 }
319 double qT = *quadT = SkPinT(*quadT);
320 double lT = *lineT = SkPinT(*lineT);
321 if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
322 *pt = fLine.ptAtT(lT);
323 } else if (ptSet == kPointUninitialized) {
324 *pt = fQuad.ptAtT(qT);
325 }
326 SkPoint gridPt = pt->asSkPoint();
327 if (SkDPoint::ApproximatelyEqual(gridPt, fLine[0].asSkPoint())) {
328 *pt = fLine[0];
329 *lineT = 0;
330 } else if (SkDPoint::ApproximatelyEqual(gridPt, fLine[1].asSkPoint())) {
331 *pt = fLine[1];
332 *lineT = 1;
333 }
334 if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) {
335 return false;
336 }
337 if (gridPt == fQuad[0].asSkPoint()) {
338 *pt = fQuad[0];
339 *quadT = 0;
340 } else if (gridPt == fQuad[2].asSkPoint()) {
341 *pt = fQuad[2];
342 *quadT = 1;
343 }
344 return true;
345 }
346
347 private:
348 const SkDQuad& fQuad;
349 const SkDLine& fLine;
350 SkIntersections* fIntersections;
351 bool fAllowNear;
352 };
353
horizontal(const SkDQuad & quad,double left,double right,double y,bool flipped)354 int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
355 bool flipped) {
356 SkDLine line = {{{ left, y }, { right, y }}};
357 LineQuadraticIntersections q(quad, line, this);
358 return q.horizontalIntersect(y, left, right, flipped);
359 }
360
vertical(const SkDQuad & quad,double top,double bottom,double x,bool flipped)361 int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
362 bool flipped) {
363 SkDLine line = {{{ x, top }, { x, bottom }}};
364 LineQuadraticIntersections q(quad, line, this);
365 return q.verticalIntersect(x, top, bottom, flipped);
366 }
367
intersect(const SkDQuad & quad,const SkDLine & line)368 int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
369 LineQuadraticIntersections q(quad, line, this);
370 q.allowNear(fAllowNear);
371 return q.intersect();
372 }
373
intersectRay(const SkDQuad & quad,const SkDLine & line)374 int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
375 LineQuadraticIntersections q(quad, line, this);
376 fUsed = q.intersectRay(fT[0]);
377 for (int index = 0; index < fUsed; ++index) {
378 fPt[index] = quad.ptAtT(fT[0][index]);
379 }
380 return fUsed;
381 }
382