1/*
2 * Copyright 2011 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
8#include "GrPathUtils.h"
9
10#include "GrTypes.h"
11#include "SkGeometry.h"
12
13SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol,
14                                          const SkMatrix& viewM,
15                                          const SkRect& pathBounds) {
16    // In order to tesselate the path we get a bound on how much the matrix can
17    // scale when mapping to screen coordinates.
18    SkScalar stretch = viewM.getMaxScale();
19    SkScalar srcTol = devTol;
20
21    if (stretch < 0) {
22        // take worst case mapRadius amoung four corners.
23        // (less than perfect)
24        for (int i = 0; i < 4; ++i) {
25            SkMatrix mat;
26            mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight,
27                             (i < 2) ? pathBounds.fTop : pathBounds.fBottom);
28            mat.postConcat(viewM);
29            stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1));
30        }
31    }
32    return srcTol / stretch;
33}
34
35static const int MAX_POINTS_PER_CURVE = 1 << 10;
36static const SkScalar gMinCurveTol = 0.0001f;
37
38uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[],
39                                          SkScalar tol) {
40    if (tol < gMinCurveTol) {
41        tol = gMinCurveTol;
42    }
43    SkASSERT(tol > 0);
44
45    SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]);
46    if (d <= tol) {
47        return 1;
48    } else {
49        // Each time we subdivide, d should be cut in 4. So we need to
50        // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x)
51        // points.
52        // 2^(log4(x)) = sqrt(x);
53        SkScalar divSqrt = SkScalarSqrt(d / tol);
54        if (((SkScalar)SK_MaxS32) <= divSqrt) {
55            return MAX_POINTS_PER_CURVE;
56        } else {
57            int temp = SkScalarCeilToInt(divSqrt);
58            int pow2 = GrNextPow2(temp);
59            // Because of NaNs & INFs we can wind up with a degenerate temp
60            // such that pow2 comes out negative. Also, our point generator
61            // will always output at least one pt.
62            if (pow2 < 1) {
63                pow2 = 1;
64            }
65            return SkTMin(pow2, MAX_POINTS_PER_CURVE);
66        }
67    }
68}
69
70uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0,
71                                              const SkPoint& p1,
72                                              const SkPoint& p2,
73                                              SkScalar tolSqd,
74                                              SkPoint** points,
75                                              uint32_t pointsLeft) {
76    if (pointsLeft < 2 ||
77        (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) {
78        (*points)[0] = p2;
79        *points += 1;
80        return 1;
81    }
82
83    SkPoint q[] = {
84        { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
85        { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
86    };
87    SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) };
88
89    pointsLeft >>= 1;
90    uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft);
91    uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft);
92    return a + b;
93}
94
95uint32_t GrPathUtils::cubicPointCount(const SkPoint points[],
96                                           SkScalar tol) {
97    if (tol < gMinCurveTol) {
98        tol = gMinCurveTol;
99    }
100    SkASSERT(tol > 0);
101
102    SkScalar d = SkTMax(
103        points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]),
104        points[2].distanceToLineSegmentBetweenSqd(points[0], points[3]));
105    d = SkScalarSqrt(d);
106    if (d <= tol) {
107        return 1;
108    } else {
109        SkScalar divSqrt = SkScalarSqrt(d / tol);
110        if (((SkScalar)SK_MaxS32) <= divSqrt) {
111            return MAX_POINTS_PER_CURVE;
112        } else {
113            int temp = SkScalarCeilToInt(SkScalarSqrt(d / tol));
114            int pow2 = GrNextPow2(temp);
115            // Because of NaNs & INFs we can wind up with a degenerate temp
116            // such that pow2 comes out negative. Also, our point generator
117            // will always output at least one pt.
118            if (pow2 < 1) {
119                pow2 = 1;
120            }
121            return SkTMin(pow2, MAX_POINTS_PER_CURVE);
122        }
123    }
124}
125
126uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0,
127                                          const SkPoint& p1,
128                                          const SkPoint& p2,
129                                          const SkPoint& p3,
130                                          SkScalar tolSqd,
131                                          SkPoint** points,
132                                          uint32_t pointsLeft) {
133    if (pointsLeft < 2 ||
134        (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd &&
135         p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) {
136            (*points)[0] = p3;
137            *points += 1;
138            return 1;
139        }
140    SkPoint q[] = {
141        { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
142        { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
143        { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) }
144    };
145    SkPoint r[] = {
146        { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) },
147        { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) }
148    };
149    SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) };
150    pointsLeft >>= 1;
151    uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft);
152    uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft);
153    return a + b;
154}
155
156int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths,
157                                     SkScalar tol) {
158    if (tol < gMinCurveTol) {
159        tol = gMinCurveTol;
160    }
161    SkASSERT(tol > 0);
162
163    int pointCount = 0;
164    *subpaths = 1;
165
166    bool first = true;
167
168    SkPath::Iter iter(path, false);
169    SkPath::Verb verb;
170
171    SkPoint pts[4];
172    while ((verb = iter.next(pts)) != SkPath::kDone_Verb) {
173
174        switch (verb) {
175            case SkPath::kLine_Verb:
176                pointCount += 1;
177                break;
178            case SkPath::kConic_Verb: {
179                SkScalar weight = iter.conicWeight();
180                SkAutoConicToQuads converter;
181                const SkPoint* quadPts = converter.computeQuads(pts, weight, 0.25f);
182                for (int i = 0; i < converter.countQuads(); ++i) {
183                    pointCount += quadraticPointCount(quadPts + 2*i, tol);
184                }
185            }
186            case SkPath::kQuad_Verb:
187                pointCount += quadraticPointCount(pts, tol);
188                break;
189            case SkPath::kCubic_Verb:
190                pointCount += cubicPointCount(pts, tol);
191                break;
192            case SkPath::kMove_Verb:
193                pointCount += 1;
194                if (!first) {
195                    ++(*subpaths);
196                }
197                break;
198            default:
199                break;
200        }
201        first = false;
202    }
203    return pointCount;
204}
205
206void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) {
207    SkMatrix m;
208    // We want M such that M * xy_pt = uv_pt
209    // We know M * control_pts = [0  1/2 1]
210    //                           [0  0   1]
211    //                           [1  1   1]
212    // And control_pts = [x0 x1 x2]
213    //                   [y0 y1 y2]
214    //                   [1  1  1 ]
215    // We invert the control pt matrix and post concat to both sides to get M.
216    // Using the known form of the control point matrix and the result, we can
217    // optimize and improve precision.
218
219    double x0 = qPts[0].fX;
220    double y0 = qPts[0].fY;
221    double x1 = qPts[1].fX;
222    double y1 = qPts[1].fY;
223    double x2 = qPts[2].fX;
224    double y2 = qPts[2].fY;
225    double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2;
226
227    if (!sk_float_isfinite(det)
228        || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) {
229        // The quad is degenerate. Hopefully this is rare. Find the pts that are
230        // farthest apart to compute a line (unless it is really a pt).
231        SkScalar maxD = qPts[0].distanceToSqd(qPts[1]);
232        int maxEdge = 0;
233        SkScalar d = qPts[1].distanceToSqd(qPts[2]);
234        if (d > maxD) {
235            maxD = d;
236            maxEdge = 1;
237        }
238        d = qPts[2].distanceToSqd(qPts[0]);
239        if (d > maxD) {
240            maxD = d;
241            maxEdge = 2;
242        }
243        // We could have a tolerance here, not sure if it would improve anything
244        if (maxD > 0) {
245            // Set the matrix to give (u = 0, v = distance_to_line)
246            SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge];
247            // when looking from the point 0 down the line we want positive
248            // distances to be to the left. This matches the non-degenerate
249            // case.
250            lineVec.setOrthog(lineVec, SkPoint::kLeft_Side);
251            lineVec.dot(qPts[0]);
252            // first row
253            fM[0] = 0;
254            fM[1] = 0;
255            fM[2] = 0;
256            // second row
257            fM[3] = lineVec.fX;
258            fM[4] = lineVec.fY;
259            fM[5] = -lineVec.dot(qPts[maxEdge]);
260        } else {
261            // It's a point. It should cover zero area. Just set the matrix such
262            // that (u, v) will always be far away from the quad.
263            fM[0] = 0; fM[1] = 0; fM[2] = 100.f;
264            fM[3] = 0; fM[4] = 0; fM[5] = 100.f;
265        }
266    } else {
267        double scale = 1.0/det;
268
269        // compute adjugate matrix
270        double a0, a1, a2, a3, a4, a5, a6, a7, a8;
271        a0 = y1-y2;
272        a1 = x2-x1;
273        a2 = x1*y2-x2*y1;
274
275        a3 = y2-y0;
276        a4 = x0-x2;
277        a5 = x2*y0-x0*y2;
278
279        a6 = y0-y1;
280        a7 = x1-x0;
281        a8 = x0*y1-x1*y0;
282
283        // this performs the uv_pts*adjugate(control_pts) multiply,
284        // then does the scale by 1/det afterwards to improve precision
285        m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale);
286        m[SkMatrix::kMSkewX]  = (float)((0.5*a4 + a7)*scale);
287        m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale);
288
289        m[SkMatrix::kMSkewY]  = (float)(a6*scale);
290        m[SkMatrix::kMScaleY] = (float)(a7*scale);
291        m[SkMatrix::kMTransY] = (float)(a8*scale);
292
293        m[SkMatrix::kMPersp0] = (float)((a0 + a3 + a6)*scale);
294        m[SkMatrix::kMPersp1] = (float)((a1 + a4 + a7)*scale);
295        m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale);
296
297        // The matrix should not have perspective.
298        SkDEBUGCODE(static const SkScalar gTOL = 1.f / 100.f);
299        SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp0)) < gTOL);
300        SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp1)) < gTOL);
301
302        // It may not be normalized to have 1.0 in the bottom right
303        float m33 = m.get(SkMatrix::kMPersp2);
304        if (1.f != m33) {
305            m33 = 1.f / m33;
306            fM[0] = m33 * m.get(SkMatrix::kMScaleX);
307            fM[1] = m33 * m.get(SkMatrix::kMSkewX);
308            fM[2] = m33 * m.get(SkMatrix::kMTransX);
309            fM[3] = m33 * m.get(SkMatrix::kMSkewY);
310            fM[4] = m33 * m.get(SkMatrix::kMScaleY);
311            fM[5] = m33 * m.get(SkMatrix::kMTransY);
312        } else {
313            fM[0] = m.get(SkMatrix::kMScaleX);
314            fM[1] = m.get(SkMatrix::kMSkewX);
315            fM[2] = m.get(SkMatrix::kMTransX);
316            fM[3] = m.get(SkMatrix::kMSkewY);
317            fM[4] = m.get(SkMatrix::kMScaleY);
318            fM[5] = m.get(SkMatrix::kMTransY);
319        }
320    }
321}
322
323////////////////////////////////////////////////////////////////////////////////
324
325// k = (y2 - y0, x0 - x2, (x2 - x0)*y0 - (y2 - y0)*x0 )
326// l = (2*w * (y1 - y0), 2*w * (x0 - x1), 2*w * (x1*y0 - x0*y1))
327// m = (2*w * (y2 - y1), 2*w * (x1 - x2), 2*w * (x2*y1 - x1*y2))
328void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkScalar klm[9]) {
329    const SkScalar w2 = 2.f * weight;
330    klm[0] = p[2].fY - p[0].fY;
331    klm[1] = p[0].fX - p[2].fX;
332    klm[2] = (p[2].fX - p[0].fX) * p[0].fY - (p[2].fY - p[0].fY) * p[0].fX;
333
334    klm[3] = w2 * (p[1].fY - p[0].fY);
335    klm[4] = w2 * (p[0].fX - p[1].fX);
336    klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY);
337
338    klm[6] = w2 * (p[2].fY - p[1].fY);
339    klm[7] = w2 * (p[1].fX - p[2].fX);
340    klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY);
341
342    // scale the max absolute value of coeffs to 10
343    SkScalar scale = 0.f;
344    for (int i = 0; i < 9; ++i) {
345       scale = SkMaxScalar(scale, SkScalarAbs(klm[i]));
346    }
347    SkASSERT(scale > 0.f);
348    scale = 10.f / scale;
349    for (int i = 0; i < 9; ++i) {
350        klm[i] *= scale;
351    }
352}
353
354////////////////////////////////////////////////////////////////////////////////
355
356namespace {
357
358// a is the first control point of the cubic.
359// ab is the vector from a to the second control point.
360// dc is the vector from the fourth to the third control point.
361// d is the fourth control point.
362// p is the candidate quadratic control point.
363// this assumes that the cubic doesn't inflect and is simple
364bool is_point_within_cubic_tangents(const SkPoint& a,
365                                    const SkVector& ab,
366                                    const SkVector& dc,
367                                    const SkPoint& d,
368                                    SkPath::Direction dir,
369                                    const SkPoint p) {
370    SkVector ap = p - a;
371    SkScalar apXab = ap.cross(ab);
372    if (SkPath::kCW_Direction == dir) {
373        if (apXab > 0) {
374            return false;
375        }
376    } else {
377        SkASSERT(SkPath::kCCW_Direction == dir);
378        if (apXab < 0) {
379            return false;
380        }
381    }
382
383    SkVector dp = p - d;
384    SkScalar dpXdc = dp.cross(dc);
385    if (SkPath::kCW_Direction == dir) {
386        if (dpXdc < 0) {
387            return false;
388        }
389    } else {
390        SkASSERT(SkPath::kCCW_Direction == dir);
391        if (dpXdc > 0) {
392            return false;
393        }
394    }
395    return true;
396}
397
398void convert_noninflect_cubic_to_quads(const SkPoint p[4],
399                                       SkScalar toleranceSqd,
400                                       bool constrainWithinTangents,
401                                       SkPath::Direction dir,
402                                       SkTArray<SkPoint, true>* quads,
403                                       int sublevel = 0) {
404
405    // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
406    // p[2]. Point d is always p[3]. Point c is p[2] unless p[2] == p[3], in which case it is p[1].
407
408    SkVector ab = p[1] - p[0];
409    SkVector dc = p[2] - p[3];
410
411    if (ab.isZero()) {
412        if (dc.isZero()) {
413            SkPoint* degQuad = quads->push_back_n(3);
414            degQuad[0] = p[0];
415            degQuad[1] = p[0];
416            degQuad[2] = p[3];
417            return;
418        }
419        ab = p[2] - p[0];
420    }
421    if (dc.isZero()) {
422        dc = p[1] - p[3];
423    }
424
425    // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the
426    // constraint that the quad point falls between the tangents becomes hard to enforce and we are
427    // likely to hit the max subdivision count. However, in this case the cubic is approaching a
428    // line and the accuracy of the quad point isn't so important. We check if the two middle cubic
429    // control points are very close to the baseline vector. If so then we just pick quadratic
430    // points on the control polygon.
431
432    if (constrainWithinTangents) {
433        SkVector da = p[0] - p[3];
434        bool doQuads = dc.lengthSqd() < SK_ScalarNearlyZero ||
435                       ab.lengthSqd() < SK_ScalarNearlyZero;
436        if (!doQuads) {
437            SkScalar invDALengthSqd = da.lengthSqd();
438            if (invDALengthSqd > SK_ScalarNearlyZero) {
439                invDALengthSqd = SkScalarInvert(invDALengthSqd);
440                // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a.
441                // same goes for point c using vector cd.
442                SkScalar detABSqd = ab.cross(da);
443                detABSqd = SkScalarSquare(detABSqd);
444                SkScalar detDCSqd = dc.cross(da);
445                detDCSqd = SkScalarSquare(detDCSqd);
446                if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd &&
447                    SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) {
448                    doQuads = true;
449                }
450            }
451        }
452        if (doQuads) {
453            SkPoint b = p[0] + ab;
454            SkPoint c = p[3] + dc;
455            SkPoint mid = b + c;
456            mid.scale(SK_ScalarHalf);
457            // Insert two quadratics to cover the case when ab points away from d and/or dc
458            // points away from a.
459            if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) {
460                SkPoint* qpts = quads->push_back_n(6);
461                qpts[0] = p[0];
462                qpts[1] = b;
463                qpts[2] = mid;
464                qpts[3] = mid;
465                qpts[4] = c;
466                qpts[5] = p[3];
467            } else {
468                SkPoint* qpts = quads->push_back_n(3);
469                qpts[0] = p[0];
470                qpts[1] = mid;
471                qpts[2] = p[3];
472            }
473            return;
474        }
475    }
476
477    static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
478    static const int kMaxSubdivs = 10;
479
480    ab.scale(kLengthScale);
481    dc.scale(kLengthScale);
482
483    // e0 and e1 are extrapolations along vectors ab and dc.
484    SkVector c0 = p[0];
485    c0 += ab;
486    SkVector c1 = p[3];
487    c1 += dc;
488
489    SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1);
490    if (dSqd < toleranceSqd) {
491        SkPoint cAvg = c0;
492        cAvg += c1;
493        cAvg.scale(SK_ScalarHalf);
494
495        bool subdivide = false;
496
497        if (constrainWithinTangents &&
498            !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) {
499            // choose a new cAvg that is the intersection of the two tangent lines.
500            ab.setOrthog(ab);
501            SkScalar z0 = -ab.dot(p[0]);
502            dc.setOrthog(dc);
503            SkScalar z1 = -dc.dot(p[3]);
504            cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY);
505            cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1);
506            SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX);
507            z = SkScalarInvert(z);
508            cAvg.fX *= z;
509            cAvg.fY *= z;
510            if (sublevel <= kMaxSubdivs) {
511                SkScalar d0Sqd = c0.distanceToSqd(cAvg);
512                SkScalar d1Sqd = c1.distanceToSqd(cAvg);
513                // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know
514                // the distances and tolerance can't be negative.
515                // (d0 + d1)^2 > toleranceSqd
516                // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd
517                SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd));
518                subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd;
519            }
520        }
521        if (!subdivide) {
522            SkPoint* pts = quads->push_back_n(3);
523            pts[0] = p[0];
524            pts[1] = cAvg;
525            pts[2] = p[3];
526            return;
527        }
528    }
529    SkPoint choppedPts[7];
530    SkChopCubicAtHalf(p, choppedPts);
531    convert_noninflect_cubic_to_quads(choppedPts + 0,
532                                      toleranceSqd,
533                                      constrainWithinTangents,
534                                      dir,
535                                      quads,
536                                      sublevel + 1);
537    convert_noninflect_cubic_to_quads(choppedPts + 3,
538                                      toleranceSqd,
539                                      constrainWithinTangents,
540                                      dir,
541                                      quads,
542                                      sublevel + 1);
543}
544}
545
546void GrPathUtils::convertCubicToQuads(const SkPoint p[4],
547                                      SkScalar tolScale,
548                                      bool constrainWithinTangents,
549                                      SkPath::Direction dir,
550                                      SkTArray<SkPoint, true>* quads) {
551    SkPoint chopped[10];
552    int count = SkChopCubicAtInflections(p, chopped);
553
554    // base tolerance is 1 pixel.
555    static const SkScalar kTolerance = SK_Scalar1;
556    const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance));
557
558    for (int i = 0; i < count; ++i) {
559        SkPoint* cubic = chopped + 3*i;
560        convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads);
561    }
562
563}
564
565////////////////////////////////////////////////////////////////////////////////
566
567// Solves linear system to extract klm
568// P.K = k (similarly for l, m)
569// Where P is matrix of control points
570// K is coefficients for the line K
571// k is vector of values of K evaluated at the control points
572// Solving for K, thus K = P^(-1) . k
573static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4],
574                           const SkScalar controlL[4], const SkScalar controlM[4],
575                           SkScalar k[3], SkScalar l[3], SkScalar m[3]) {
576    SkMatrix matrix;
577    matrix.setAll(p[0].fX, p[0].fY, 1.f,
578                  p[1].fX, p[1].fY, 1.f,
579                  p[2].fX, p[2].fY, 1.f);
580    SkMatrix inverse;
581    if (matrix.invert(&inverse)) {
582       inverse.mapHomogeneousPoints(k, controlK, 1);
583       inverse.mapHomogeneousPoints(l, controlL, 1);
584       inverse.mapHomogeneousPoints(m, controlM, 1);
585    }
586
587}
588
589static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
590    SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]);
591    SkScalar ls = 3.f * d[1] - tempSqrt;
592    SkScalar lt = 6.f * d[0];
593    SkScalar ms = 3.f * d[1] + tempSqrt;
594    SkScalar mt = 6.f * d[0];
595
596    k[0] = ls * ms;
597    k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f;
598    k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f;
599    k[3] = (lt - ls) * (mt - ms);
600
601    l[0] = ls * ls * ls;
602    const SkScalar lt_ls = lt - ls;
603    l[1] = ls * ls * lt_ls * -1.f;
604    l[2] = lt_ls * lt_ls * ls;
605    l[3] = -1.f * lt_ls * lt_ls * lt_ls;
606
607    m[0] = ms * ms * ms;
608    const SkScalar mt_ms = mt - ms;
609    m[1] = ms * ms * mt_ms * -1.f;
610    m[2] = mt_ms * mt_ms * ms;
611    m[3] = -1.f * mt_ms * mt_ms * mt_ms;
612
613    // If d0 < 0 we need to flip the orientation of our curve
614    // This is done by negating the k and l values
615    // We want negative distance values to be on the inside
616    if ( d[0] > 0) {
617        for (int i = 0; i < 4; ++i) {
618            k[i] = -k[i];
619            l[i] = -l[i];
620        }
621    }
622}
623
624static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
625    SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]);
626    SkScalar ls = d[1] - tempSqrt;
627    SkScalar lt = 2.f * d[0];
628    SkScalar ms = d[1] + tempSqrt;
629    SkScalar mt = 2.f * d[0];
630
631    k[0] = ls * ms;
632    k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f;
633    k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f;
634    k[3] = (lt - ls) * (mt - ms);
635
636    l[0] = ls * ls * ms;
637    l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f;
638    l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f;
639    l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms);
640
641    m[0] = ls * ms * ms;
642    m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f;
643    m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f;
644    m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms);
645
646
647    // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0),
648    // we need to flip the orientation of our curve.
649    // This is done by negating the k and l values
650    if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) {
651        for (int i = 0; i < 4; ++i) {
652            k[i] = -k[i];
653            l[i] = -l[i];
654        }
655    }
656}
657
658static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
659    const SkScalar ls = d[2];
660    const SkScalar lt = 3.f * d[1];
661
662    k[0] = ls;
663    k[1] = ls - lt / 3.f;
664    k[2] = ls - 2.f * lt / 3.f;
665    k[3] = ls - lt;
666
667    l[0] = ls * ls * ls;
668    const SkScalar ls_lt = ls - lt;
669    l[1] = ls * ls * ls_lt;
670    l[2] = ls_lt * ls_lt * ls;
671    l[3] = ls_lt * ls_lt * ls_lt;
672
673    m[0] = 1.f;
674    m[1] = 1.f;
675    m[2] = 1.f;
676    m[3] = 1.f;
677}
678
679// For the case when a cubic is actually a quadratic
680// M =
681// 0     0     0
682// 1/3   0     1/3
683// 2/3   1/3   2/3
684// 1     1     1
685static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
686    k[0] = 0.f;
687    k[1] = 1.f/3.f;
688    k[2] = 2.f/3.f;
689    k[3] = 1.f;
690
691    l[0] = 0.f;
692    l[1] = 0.f;
693    l[2] = 1.f/3.f;
694    l[3] = 1.f;
695
696    m[0] = 0.f;
697    m[1] = 1.f/3.f;
698    m[2] = 2.f/3.f;
699    m[3] = 1.f;
700
701    // If d2 < 0 we need to flip the orientation of our curve
702    // This is done by negating the k and l values
703    if ( d[2] > 0) {
704        for (int i = 0; i < 4; ++i) {
705            k[i] = -k[i];
706            l[i] = -l[i];
707        }
708    }
709}
710
711int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9],
712                                             SkScalar klm_rev[3]) {
713    // Variable to store the two parametric values at the loop double point
714    SkScalar smallS = 0.f;
715    SkScalar largeS = 0.f;
716
717    SkScalar d[3];
718    SkCubicType cType = SkClassifyCubic(src, d);
719
720    int chop_count = 0;
721    if (kLoop_SkCubicType == cType) {
722        SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]);
723        SkScalar ls = d[1] - tempSqrt;
724        SkScalar lt = 2.f * d[0];
725        SkScalar ms = d[1] + tempSqrt;
726        SkScalar mt = 2.f * d[0];
727        ls = ls / lt;
728        ms = ms / mt;
729        // need to have t values sorted since this is what is expected by SkChopCubicAt
730        if (ls <= ms) {
731            smallS = ls;
732            largeS = ms;
733        } else {
734            smallS = ms;
735            largeS = ls;
736        }
737
738        SkScalar chop_ts[2];
739        if (smallS > 0.f && smallS < 1.f) {
740            chop_ts[chop_count++] = smallS;
741        }
742        if (largeS > 0.f && largeS < 1.f) {
743            chop_ts[chop_count++] = largeS;
744        }
745        if(dst) {
746            SkChopCubicAt(src, dst, chop_ts, chop_count);
747        }
748    } else {
749        if (dst) {
750            memcpy(dst, src, sizeof(SkPoint) * 4);
751        }
752    }
753
754    if (klm && klm_rev) {
755        // Set klm_rev to to match the sub_section of cubic that needs to have its orientation
756        // flipped. This will always be the section that is the "loop"
757        if (2 == chop_count) {
758            klm_rev[0] = 1.f;
759            klm_rev[1] = -1.f;
760            klm_rev[2] = 1.f;
761        } else if (1 == chop_count) {
762            if (smallS < 0.f) {
763                klm_rev[0] = -1.f;
764                klm_rev[1] = 1.f;
765            } else {
766                klm_rev[0] = 1.f;
767                klm_rev[1] = -1.f;
768            }
769        } else {
770            if (smallS < 0.f && largeS > 1.f) {
771                klm_rev[0] = -1.f;
772            } else {
773                klm_rev[0] = 1.f;
774            }
775        }
776        SkScalar controlK[4];
777        SkScalar controlL[4];
778        SkScalar controlM[4];
779
780        if (kSerpentine_SkCubicType == cType || (kCusp_SkCubicType == cType && 0.f != d[0])) {
781            set_serp_klm(d, controlK, controlL, controlM);
782        } else if (kLoop_SkCubicType == cType) {
783            set_loop_klm(d, controlK, controlL, controlM);
784        } else if (kCusp_SkCubicType == cType) {
785            SkASSERT(0.f == d[0]);
786            set_cusp_klm(d, controlK, controlL, controlM);
787        } else if (kQuadratic_SkCubicType == cType) {
788            set_quadratic_klm(d, controlK, controlL, controlM);
789        }
790
791        calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]);
792    }
793    return chop_count + 1;
794}
795
796void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) {
797    SkScalar d[3];
798    SkCubicType cType = SkClassifyCubic(p, d);
799
800    SkScalar controlK[4];
801    SkScalar controlL[4];
802    SkScalar controlM[4];
803
804    if (kSerpentine_SkCubicType == cType || (kCusp_SkCubicType == cType && 0.f != d[0])) {
805        set_serp_klm(d, controlK, controlL, controlM);
806    } else if (kLoop_SkCubicType == cType) {
807        set_loop_klm(d, controlK, controlL, controlM);
808    } else if (kCusp_SkCubicType == cType) {
809        SkASSERT(0.f == d[0]);
810        set_cusp_klm(d, controlK, controlL, controlM);
811    } else if (kQuadratic_SkCubicType == cType) {
812        set_quadratic_klm(d, controlK, controlL, controlM);
813    }
814
815    calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]);
816}
817