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 "SkPathOpsCubic.h"
9#include "SkPathOpsLine.h"
10
11/*
12Find the interection of a line and cubic by solving for valid t values.
13
14Analogous to line-quadratic intersection, solve line-cubic intersection by
15representing the cubic as:
16  x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3
17  y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3
18and the line as:
19  y = i*x + j  (if the line is more horizontal)
20or:
21  x = i*y + j  (if the line is more vertical)
22
23Then using Mathematica, solve for the values of t where the cubic intersects the
24line:
25
26  (in) Resultant[
27        a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x,
28        e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x]
29  (out) -e     +   j     +
30       3 e t   - 3 f t   -
31       3 e t^2 + 6 f t^2 - 3 g t^2 +
32         e t^3 - 3 f t^3 + 3 g t^3 - h t^3 +
33     i ( a     -
34       3 a t + 3 b t +
35       3 a t^2 - 6 b t^2 + 3 c t^2 -
36         a t^3 + 3 b t^3 - 3 c t^3 + d t^3 )
37
38if i goes to infinity, we can rewrite the line in terms of x. Mathematica:
39
40  (in) Resultant[
41        a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j,
42        e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y,       y]
43  (out)  a     -   j     -
44       3 a t   + 3 b t   +
45       3 a t^2 - 6 b t^2 + 3 c t^2 -
46         a t^3 + 3 b t^3 - 3 c t^3 + d t^3 -
47     i ( e     -
48       3 e t   + 3 f t   +
49       3 e t^2 - 6 f t^2 + 3 g t^2 -
50         e t^3 + 3 f t^3 - 3 g t^3 + h t^3 )
51
52Solving this with Mathematica produces an expression with hundreds of terms;
53instead, use Numeric Solutions recipe to solve the cubic.
54
55The near-horizontal case, in terms of:  Ax^3 + Bx^2 + Cx + D == 0
56    A =   (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d)     )
57    B = 3*(-( e - 2*f +   g    ) + i*( a - 2*b +   c    )     )
58    C = 3*(-(-e +   f          ) + i*(-a +   b          )     )
59    D =   (-( e                ) + i*( a                ) + j )
60
61The near-vertical case, in terms of:  Ax^3 + Bx^2 + Cx + D == 0
62    A =   ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h)     )
63    B = 3*( ( a - 2*b +   c    ) - i*( e - 2*f +   g    )     )
64    C = 3*( (-a +   b          ) - i*(-e +   f          )     )
65    D =   ( ( a                ) - i*( e                ) - j )
66
67For horizontal lines:
68(in) Resultant[
69      a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j,
70      e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y]
71(out)  e     -   j     -
72     3 e t   + 3 f t   +
73     3 e t^2 - 6 f t^2 + 3 g t^2 -
74       e t^3 + 3 f t^3 - 3 g t^3 + h t^3
75 */
76
77class LineCubicIntersections {
78public:
79    enum PinTPoint {
80        kPointUninitialized,
81        kPointInitialized
82    };
83
84    LineCubicIntersections(const SkDCubic& c, const SkDLine& l, SkIntersections* i)
85        : fCubic(c)
86        , fLine(l)
87        , fIntersections(i)
88        , fAllowNear(true) {
89        i->setMax(3);
90    }
91
92    void allowNear(bool allow) {
93        fAllowNear = allow;
94    }
95
96    // see parallel routine in line quadratic intersections
97    int intersectRay(double roots[3]) {
98        double adj = fLine[1].fX - fLine[0].fX;
99        double opp = fLine[1].fY - fLine[0].fY;
100        SkDCubic c;
101        for (int n = 0; n < 4; ++n) {
102            c[n].fX = (fCubic[n].fY - fLine[0].fY) * adj - (fCubic[n].fX - fLine[0].fX) * opp;
103        }
104        double A, B, C, D;
105        SkDCubic::Coefficients(&c[0].fX, &A, &B, &C, &D);
106        int count = SkDCubic::RootsValidT(A, B, C, D, roots);
107        for (int index = 0; index < count; ++index) {
108            SkDPoint calcPt = c.ptAtT(roots[index]);
109            if (!approximately_zero(calcPt.fX)) {
110                for (int n = 0; n < 4; ++n) {
111                    c[n].fY = (fCubic[n].fY - fLine[0].fY) * opp
112                            + (fCubic[n].fX - fLine[0].fX) * adj;
113                }
114                double extremeTs[6];
115                int extrema = SkDCubic::FindExtrema(c[0].fX, c[1].fX, c[2].fX, c[3].fX, extremeTs);
116                count = c.searchRoots(extremeTs, extrema, 0, SkDCubic::kXAxis, roots);
117                break;
118            }
119        }
120        return count;
121    }
122
123    int intersect() {
124        addExactEndPoints();
125        if (fAllowNear) {
126            addNearEndPoints();
127        }
128        double rootVals[3];
129        int roots = intersectRay(rootVals);
130        for (int index = 0; index < roots; ++index) {
131            double cubicT = rootVals[index];
132            double lineT = findLineT(cubicT);
133            SkDPoint pt;
134            if (pinTs(&cubicT, &lineT, &pt, kPointUninitialized)) {
135    #if ONE_OFF_DEBUG
136                SkDPoint cPt = fCubic.ptAtT(cubicT);
137                SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY,
138                        cPt.fX, cPt.fY);
139    #endif
140                for (int inner = 0; inner < fIntersections->used(); ++inner) {
141                    if (fIntersections->pt(inner) != pt) {
142                        continue;
143                    }
144                    double existingCubicT = (*fIntersections)[0][inner];
145                    if (cubicT == existingCubicT) {
146                        goto skipInsert;
147                    }
148                    // check if midway on cubic is also same point. If so, discard this
149                    double cubicMidT = (existingCubicT + cubicT) / 2;
150                    SkDPoint cubicMidPt = fCubic.ptAtT(cubicMidT);
151                    if (cubicMidPt.approximatelyEqual(pt)) {
152                        goto skipInsert;
153                    }
154                }
155                fIntersections->insert(cubicT, lineT, pt);
156        skipInsert:
157                ;
158            }
159        }
160        return fIntersections->used();
161    }
162
163    static int HorizontalIntersect(const SkDCubic& c, double axisIntercept, double roots[3]) {
164        double A, B, C, D;
165        SkDCubic::Coefficients(&c[0].fY, &A, &B, &C, &D);
166        D -= axisIntercept;
167        int count = SkDCubic::RootsValidT(A, B, C, D, roots);
168        for (int index = 0; index < count; ++index) {
169            SkDPoint calcPt = c.ptAtT(roots[index]);
170            if (!approximately_equal(calcPt.fY, axisIntercept)) {
171                double extremeTs[6];
172                int extrema = SkDCubic::FindExtrema(c[0].fY, c[1].fY, c[2].fY, c[3].fY, extremeTs);
173                count = c.searchRoots(extremeTs, extrema, axisIntercept, SkDCubic::kYAxis, roots);
174                break;
175            }
176        }
177        return count;
178    }
179
180    int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
181        addExactHorizontalEndPoints(left, right, axisIntercept);
182        if (fAllowNear) {
183            addNearHorizontalEndPoints(left, right, axisIntercept);
184        }
185        double roots[3];
186        int count = HorizontalIntersect(fCubic, axisIntercept, roots);
187        for (int index = 0; index < count; ++index) {
188            double cubicT = roots[index];
189            SkDPoint pt;
190            pt.fX = fCubic.ptAtT(cubicT).fX;
191            pt.fY = axisIntercept;
192            double lineT = (pt.fX - left) / (right - left);
193            if (pinTs(&cubicT, &lineT, &pt, kPointInitialized)) {
194                fIntersections->insert(cubicT, lineT, pt);
195            }
196        }
197        if (flipped) {
198            fIntersections->flip();
199        }
200        return fIntersections->used();
201    }
202
203    static int VerticalIntersect(const SkDCubic& c, double axisIntercept, double roots[3]) {
204        double A, B, C, D;
205        SkDCubic::Coefficients(&c[0].fX, &A, &B, &C, &D);
206        D -= axisIntercept;
207        int count = SkDCubic::RootsValidT(A, B, C, D, roots);
208        for (int index = 0; index < count; ++index) {
209            SkDPoint calcPt = c.ptAtT(roots[index]);
210            if (!approximately_equal(calcPt.fX, axisIntercept)) {
211                double extremeTs[6];
212                int extrema = SkDCubic::FindExtrema(c[0].fX, c[1].fX, c[2].fX, c[3].fX, extremeTs);
213                count = c.searchRoots(extremeTs, extrema, axisIntercept, SkDCubic::kXAxis, roots);
214                break;
215            }
216        }
217        return count;
218    }
219
220    int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
221        addExactVerticalEndPoints(top, bottom, axisIntercept);
222        if (fAllowNear) {
223            addNearVerticalEndPoints(top, bottom, axisIntercept);
224        }
225        double roots[3];
226        int count = VerticalIntersect(fCubic, axisIntercept, roots);
227        for (int index = 0; index < count; ++index) {
228            double cubicT = roots[index];
229            SkDPoint pt;
230            pt.fX = axisIntercept;
231            pt.fY = fCubic.ptAtT(cubicT).fY;
232            double lineT = (pt.fY - top) / (bottom - top);
233            if (pinTs(&cubicT, &lineT, &pt, kPointInitialized)) {
234                fIntersections->insert(cubicT, lineT, pt);
235            }
236        }
237        if (flipped) {
238            fIntersections->flip();
239        }
240        return fIntersections->used();
241    }
242
243    protected:
244
245    void addExactEndPoints() {
246        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
247            double lineT = fLine.exactPoint(fCubic[cIndex]);
248            if (lineT < 0) {
249                continue;
250            }
251            double cubicT = (double) (cIndex >> 1);
252            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
253        }
254    }
255
256    /* Note that this does not look for endpoints of the line that are near the cubic.
257       These points are found later when check ends looks for missing points */
258    void addNearEndPoints() {
259        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
260            double cubicT = (double) (cIndex >> 1);
261            if (fIntersections->hasT(cubicT)) {
262                continue;
263            }
264            double lineT = fLine.nearPoint(fCubic[cIndex], NULL);
265            if (lineT < 0) {
266                continue;
267            }
268            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
269        }
270    }
271
272    void addExactHorizontalEndPoints(double left, double right, double y) {
273        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
274            double lineT = SkDLine::ExactPointH(fCubic[cIndex], left, right, y);
275            if (lineT < 0) {
276                continue;
277            }
278            double cubicT = (double) (cIndex >> 1);
279            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
280        }
281    }
282
283    void addNearHorizontalEndPoints(double left, double right, double y) {
284        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
285            double cubicT = (double) (cIndex >> 1);
286            if (fIntersections->hasT(cubicT)) {
287                continue;
288            }
289            double lineT = SkDLine::NearPointH(fCubic[cIndex], left, right, y);
290            if (lineT < 0) {
291                continue;
292            }
293            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
294        }
295        // FIXME: see if line end is nearly on cubic
296    }
297
298    void addExactVerticalEndPoints(double top, double bottom, double x) {
299        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
300            double lineT = SkDLine::ExactPointV(fCubic[cIndex], top, bottom, x);
301            if (lineT < 0) {
302                continue;
303            }
304            double cubicT = (double) (cIndex >> 1);
305            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
306        }
307    }
308
309    void addNearVerticalEndPoints(double top, double bottom, double x) {
310        for (int cIndex = 0; cIndex < 4; cIndex += 3) {
311            double cubicT = (double) (cIndex >> 1);
312            if (fIntersections->hasT(cubicT)) {
313                continue;
314            }
315            double lineT = SkDLine::NearPointV(fCubic[cIndex], top, bottom, x);
316            if (lineT < 0) {
317                continue;
318            }
319            fIntersections->insert(cubicT, lineT, fCubic[cIndex]);
320        }
321        // FIXME: see if line end is nearly on cubic
322    }
323
324    double findLineT(double t) {
325        SkDPoint xy = fCubic.ptAtT(t);
326        double dx = fLine[1].fX - fLine[0].fX;
327        double dy = fLine[1].fY - fLine[0].fY;
328        if (fabs(dx) > fabs(dy)) {
329            return (xy.fX - fLine[0].fX) / dx;
330        }
331        return (xy.fY - fLine[0].fY) / dy;
332    }
333
334    bool pinTs(double* cubicT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
335        if (!approximately_one_or_less(*lineT)) {
336            return false;
337        }
338        if (!approximately_zero_or_more(*lineT)) {
339            return false;
340        }
341        double cT = *cubicT = SkPinT(*cubicT);
342        double lT = *lineT = SkPinT(*lineT);
343        SkDPoint lPt = fLine.ptAtT(lT);
344        SkDPoint cPt = fCubic.ptAtT(cT);
345        if (!lPt.moreRoughlyEqual(cPt)) {
346            return false;
347        }
348        // FIXME: if points are roughly equal but not approximately equal, need to do
349        // a binary search like quad/quad intersection to find more precise t values
350        if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && cT != 0 && cT != 1)) {
351            *pt = lPt;
352        } else if (ptSet == kPointUninitialized) {
353            *pt = cPt;
354        }
355        SkPoint gridPt = pt->asSkPoint();
356        if (gridPt == fLine[0].asSkPoint()) {
357            *lineT = 0;
358        } else if (gridPt == fLine[1].asSkPoint()) {
359            *lineT = 1;
360        }
361        if (gridPt == fCubic[0].asSkPoint() && approximately_equal(*cubicT, 0)) {
362            *cubicT = 0;
363        } else if (gridPt == fCubic[3].asSkPoint() && approximately_equal(*cubicT, 1)) {
364            *cubicT = 1;
365        }
366        return true;
367    }
368
369private:
370    const SkDCubic& fCubic;
371    const SkDLine& fLine;
372    SkIntersections* fIntersections;
373    bool fAllowNear;
374};
375
376int SkIntersections::horizontal(const SkDCubic& cubic, double left, double right, double y,
377        bool flipped) {
378    SkDLine line = {{{ left, y }, { right, y }}};
379    LineCubicIntersections c(cubic, line, this);
380    return c.horizontalIntersect(y, left, right, flipped);
381}
382
383int SkIntersections::vertical(const SkDCubic& cubic, double top, double bottom, double x,
384        bool flipped) {
385    SkDLine line = {{{ x, top }, { x, bottom }}};
386    LineCubicIntersections c(cubic, line, this);
387    return c.verticalIntersect(x, top, bottom, flipped);
388}
389
390int SkIntersections::intersect(const SkDCubic& cubic, const SkDLine& line) {
391    LineCubicIntersections c(cubic, line, this);
392    c.allowNear(fAllowNear);
393    return c.intersect();
394}
395
396int SkIntersections::intersectRay(const SkDCubic& cubic, const SkDLine& line) {
397    LineCubicIntersections c(cubic, line, this);
398    fUsed = c.intersectRay(fT[0]);
399    for (int index = 0; index < fUsed; ++index) {
400        fPt[index] = cubic.ptAtT(fT[0][index]);
401    }
402    return fUsed;
403}
404