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comparison: gfx/skia/trunk/src/pathops/SkDQuadLineIntersection.cpp

gfx/skia/trunk/src/pathops/SkDQuadLineIntersection.cpp

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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 #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
90 class LineQuadraticIntersections {
91 public:
92 enum PinTPoint {
93 kPointUninitialized,
94 kPointInitialized
95 };
96
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(2);
103 }
104
105 void allowNear(bool allow) {
106 fAllowNear = allow;
107 }
108
109 int intersectRay(double roots[2]) {
110 /*
111 solve by rotating line+quad so line is horizontal, then finding the roots
112 set up matrix to rotate quad to x-axis
113 |cos(a) -sin(a)|
114 |sin(a) cos(a)|
115 note that cos(a) = A(djacent) / Hypoteneuse
116 sin(a) = O(pposite) / Hypoteneuse
117 since we are computing Ts, we can ignore hypoteneuse, the scale factor:
118 | A -O |
119 | O A |
120 A = line[1].fX - line[0].fX (adjacent side of the right triangle)
121 O = line[1].fY - line[0].fY (opposite side of the right triangle)
122 for each of the three points (e.g. n = 0 to 2)
123 quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
124 */
125 double adj = fLine[1].fX - fLine[0].fX;
126 double opp = fLine[1].fY - fLine[0].fY;
127 double r[3];
128 for (int n = 0; n < 3; ++n) {
129 r[n] = (fQuad[n].fY - fLine[0].fY) * adj - (fQuad[n].fX - fLine[0].fX) * opp;
130 }
131 double A = r[2];
132 double B = r[1];
133 double C = r[0];
134 A += C - 2 * B; // A = a - 2*b + c
135 B -= C; // B = -(b - c)
136 return SkDQuad::RootsValidT(A, 2 * B, C, roots);
137 }
138
139 int intersect() {
140 addExactEndPoints();
141 if (fAllowNear) {
142 addNearEndPoints();
143 }
144 if (fIntersections->used() == 2) {
145 // FIXME : need sharable code that turns spans into coincident if middle point is on
146 } else {
147 double rootVals[2];
148 int roots = intersectRay(rootVals);
149 for (int index = 0; index < roots; ++index) {
150 double quadT = rootVals[index];
151 double lineT = findLineT(quadT);
152 SkDPoint pt;
153 if (pinTs(&quadT, &lineT, &pt, kPointUninitialized)) {
154 fIntersections->insert(quadT, lineT, pt);
155 }
156 }
157 }
158 return fIntersections->used();
159 }
160
161 int horizontalIntersect(double axisIntercept, double roots[2]) {
162 double D = fQuad[2].fY; // f
163 double E = fQuad[1].fY; // e
164 double F = fQuad[0].fY; // d
165 D += F - 2 * E; // D = d - 2*e + f
166 E -= F; // E = -(d - e)
167 F -= axisIntercept;
168 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
169 }
170
171 int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
172 addExactHorizontalEndPoints(left, right, axisIntercept);
173 if (fAllowNear) {
174 addNearHorizontalEndPoints(left, right, axisIntercept);
175 }
176 double rootVals[2];
177 int roots = horizontalIntersect(axisIntercept, rootVals);
178 for (int index = 0; index < roots; ++index) {
179 double quadT = rootVals[index];
180 SkDPoint pt = fQuad.ptAtT(quadT);
181 double lineT = (pt.fX - left) / (right - left);
182 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
183 fIntersections->insert(quadT, lineT, pt);
184 }
185 }
186 if (flipped) {
187 fIntersections->flip();
188 }
189 return fIntersections->used();
190 }
191
192 int verticalIntersect(double axisIntercept, double roots[2]) {
193 double D = fQuad[2].fX; // f
194 double E = fQuad[1].fX; // e
195 double F = fQuad[0].fX; // d
196 D += F - 2 * E; // D = d - 2*e + f
197 E -= F; // E = -(d - e)
198 F -= axisIntercept;
199 return SkDQuad::RootsValidT(D, 2 * E, F, roots);
200 }
201
202 int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
203 addExactVerticalEndPoints(top, bottom, axisIntercept);
204 if (fAllowNear) {
205 addNearVerticalEndPoints(top, bottom, axisIntercept);
206 }
207 double rootVals[2];
208 int roots = verticalIntersect(axisIntercept, rootVals);
209 for (int index = 0; index < roots; ++index) {
210 double quadT = rootVals[index];
211 SkDPoint pt = fQuad.ptAtT(quadT);
212 double lineT = (pt.fY - top) / (bottom - top);
213 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
214 fIntersections->insert(quadT, lineT, pt);
215 }
216 }
217 if (flipped) {
218 fIntersections->flip();
219 }
220 return fIntersections->used();
221 }
222
223 protected:
224 // add endpoints first to get zero and one t values exactly
225 void addExactEndPoints() {
226 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
227 double lineT = fLine.exactPoint(fQuad[qIndex]);
228 if (lineT < 0) {
229 continue;
230 }
231 double quadT = (double) (qIndex >> 1);
232 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
233 }
234 }
235
236 void addNearEndPoints() {
237 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
238 double quadT = (double) (qIndex >> 1);
239 if (fIntersections->hasT(quadT)) {
240 continue;
241 }
242 double lineT = fLine.nearPoint(fQuad[qIndex]);
243 if (lineT < 0) {
244 continue;
245 }
246 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
247 }
248 // FIXME: see if line end is nearly on quad
249 }
250
251 void addExactHorizontalEndPoints(double left, double right, double y) {
252 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
253 double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
254 if (lineT < 0) {
255 continue;
256 }
257 double quadT = (double) (qIndex >> 1);
258 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
259 }
260 }
261
262 void addNearHorizontalEndPoints(double left, double right, double y) {
263 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
264 double quadT = (double) (qIndex >> 1);
265 if (fIntersections->hasT(quadT)) {
266 continue;
267 }
268 double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
269 if (lineT < 0) {
270 continue;
271 }
272 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
273 }
274 // FIXME: see if line end is nearly on quad
275 }
276
277 void addExactVerticalEndPoints(double top, double bottom, double x) {
278 for (int qIndex = 0; qIndex < 3; qIndex += 2) {
279 double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
280 if (lineT < 0) {
281 continue;
282 }
283 double quadT = (double) (qIndex >> 1);
284 fIntersections->insert(quadT, lineT, fQuad[qIndex]);
285 }
286 }
287
288 void addNearVerticalEndPoints(double top, double bottom, double x) {
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 = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
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
303 double findLineT(double t) {
304 SkDPoint xy = fQuad.ptAtT(t);
305 double dx = fLine[1].fX - fLine[0].fX;
306 double dy = fLine[1].fY - fLine[0].fY;
307 if (fabs(dx) > fabs(dy)) {
308 return (xy.fX - fLine[0].fX) / dx;
309 }
310 return (xy.fY - fLine[0].fY) / dy;
311 }
312
313 bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
314 if (!approximately_one_or_less(*lineT)) {
315 return false;
316 }
317 if (!approximately_zero_or_more(*lineT)) {
318 return false;
319 }
320 double qT = *quadT = SkPinT(*quadT);
321 double lT = *lineT = SkPinT(*lineT);
322 if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
323 *pt = fLine.ptAtT(lT);
324 } else if (ptSet == kPointUninitialized) {
325 *pt = fQuad.ptAtT(qT);
326 }
327 SkPoint gridPt = pt->asSkPoint();
328 if (gridPt == fLine[0].asSkPoint()) {
329 *lineT = 0;
330 } else if (gridPt == fLine[1].asSkPoint()) {
331 *lineT = 1;
332 }
333 if (gridPt == fQuad[0].asSkPoint()) {
334 *quadT = 0;
335 } else if (gridPt == fQuad[2].asSkPoint()) {
336 *quadT = 1;
337 }
338 return true;
339 }
340
341 private:
342 const SkDQuad& fQuad;
343 const SkDLine& fLine;
344 SkIntersections* fIntersections;
345 bool fAllowNear;
346 };
347
348 // utility for pairs of coincident quads
349 static double horizontalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
350 LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
351 static_cast<SkIntersections*>(0));
352 double rootVals[2];
353 int roots = q.horizontalIntersect(pt.fY, rootVals);
354 for (int index = 0; index < roots; ++index) {
355 double t = rootVals[index];
356 SkDPoint qPt = quad.ptAtT(t);
357 if (AlmostEqualUlps(qPt.fX, pt.fX)) {
358 return t;
359 }
360 }
361 return -1;
362 }
363
364 static double verticalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
365 LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
366 static_cast<SkIntersections*>(0));
367 double rootVals[2];
368 int roots = q.verticalIntersect(pt.fX, rootVals);
369 for (int index = 0; index < roots; ++index) {
370 double t = rootVals[index];
371 SkDPoint qPt = quad.ptAtT(t);
372 if (AlmostEqualUlps(qPt.fY, pt.fY)) {
373 return t;
374 }
375 }
376 return -1;
377 }
378
379 double SkIntersections::Axial(const SkDQuad& q1, const SkDPoint& p, bool vertical) {
380 if (vertical) {
381 return verticalIntersect(q1, p);
382 }
383 return horizontalIntersect(q1, p);
384 }
385
386 int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
387 bool flipped) {
388 SkDLine line = {{{ left, y }, { right, y }}};
389 LineQuadraticIntersections q(quad, line, this);
390 return q.horizontalIntersect(y, left, right, flipped);
391 }
392
393 int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
394 bool flipped) {
395 SkDLine line = {{{ x, top }, { x, bottom }}};
396 LineQuadraticIntersections q(quad, line, this);
397 return q.verticalIntersect(x, top, bottom, flipped);
398 }
399
400 int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
401 LineQuadraticIntersections q(quad, line, this);
402 q.allowNear(fAllowNear);
403 return q.intersect();
404 }
405
406 int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
407 LineQuadraticIntersections q(quad, line, this);
408 fUsed = q.intersectRay(fT[0]);
409 for (int index = 0; index < fUsed; ++index) {
410 fPt[index] = quad.ptAtT(fT[0][index]);
411 }
412 return fUsed;
413 }

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