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/*
12Find the interection of a line and quadratic by solving for valid t values.
13
14From 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
17control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where
18A, B and C are points and t goes from zero to one.
19
20This 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
25If you add for instance the line equation (y = kx + m) to that, you'll end up
26with three equations and three unknowns (x, y and t)."
27
28Similar 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
31and the line as
32  y = g*x + h
33
34Using Mathematica, solve for the values of t where the quadratic intersects the
35line:
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
55Using 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
60If g goes to infinity, we can rewrite the line in terms of x.
61  x = g'*y + h'
62
63And 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
83Thus, 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
90class LineQuadraticIntersections {
91public:
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
223protected:
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
341private:
342    const SkDQuad& fQuad;
343    const SkDLine& fLine;
344    SkIntersections* fIntersections;
345    bool fAllowNear;
346};
347
348// utility for pairs of coincident quads
349static 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
364static 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
379double 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
386int 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
393int 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
400int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
401    LineQuadraticIntersections q(quad, line, this);
402    q.allowNear(fAllowNear);
403    return q.intersect();
404}
405
406int 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}
414