SkGeometry.cpp revision 6983f66d8b3a489133b751e2cef03e72a03bfeae 
1/*
2 * Copyright 2006 The Android Open Source Project
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 "SkGeometry.h"
9#include "SkMatrix.h"
10#include "SkNx.h"
11
12#if 0
13static Sk2s from_point(const SkPoint& point) {
14    return Sk2s::Load(&point.fX);
15}
16
17static SkPoint to_point(const Sk2s& x) {
18    SkPoint point;
19    x.store(&point.fX);
20    return point;
21}
22#endif
23
24static SkVector to_vector(const Sk2s& x) {
25    SkVector vector;
26    x.store(&vector.fX);
27    return vector;
28}
29
30/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
31    involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
32    May also introduce overflow of fixed when we compute our setup.
33*/
34//    #define DIRECT_EVAL_OF_POLYNOMIALS
35
36////////////////////////////////////////////////////////////////////////
37
38static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
39    SkScalar ab = a - b;
40    SkScalar bc = b - c;
41    if (ab < 0) {
42        bc = -bc;
43    }
44    return ab == 0 || bc < 0;
45}
46
47////////////////////////////////////////////////////////////////////////
48
49static bool is_unit_interval(SkScalar x) {
50    return x > 0 && x < SK_Scalar1;
51}
52
53static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
54    SkASSERT(ratio);
55
56    if (numer < 0) {
57        numer = -numer;
58        denom = -denom;
59    }
60
61    if (denom == 0 || numer == 0 || numer >= denom) {
62        return 0;
63    }
64
65    SkScalar r = SkScalarDiv(numer, denom);
66    if (SkScalarIsNaN(r)) {
67        return 0;
68    }
69    SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
70    if (r == 0) { // catch underflow if numer <<<< denom
71        return 0;
72    }
73    *ratio = r;
74    return 1;
75}
76
77/** From Numerical Recipes in C.
78
79    Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
80    x1 = Q / A
81    x2 = C / Q
82*/
83int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
84    SkASSERT(roots);
85
86    if (A == 0) {
87        return valid_unit_divide(-C, B, roots);
88    }
89
90    SkScalar* r = roots;
91
92    SkScalar R = B*B - 4*A*C;
93    if (R < 0 || SkScalarIsNaN(R)) {  // complex roots
94        return 0;
95    }
96    R = SkScalarSqrt(R);
97
98    SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
99    r += valid_unit_divide(Q, A, r);
100    r += valid_unit_divide(C, Q, r);
101    if (r - roots == 2) {
102        if (roots[0] > roots[1])
103            SkTSwap<SkScalar>(roots[0], roots[1]);
104        else if (roots[0] == roots[1])  // nearly-equal?
105            r -= 1; // skip the double root
106    }
107    return (int)(r - roots);
108}
109
110///////////////////////////////////////////////////////////////////////////////
111///////////////////////////////////////////////////////////////////////////////
112
113static Sk2s quad_poly_eval(const Sk2s& A, const Sk2s& B, const Sk2s& C, const Sk2s& t) {
114    return (A * t + B) * t + C;
115}
116
117static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
118    SkASSERT(src);
119    SkASSERT(t >= 0 && t <= SK_Scalar1);
120
121#ifdef DIRECT_EVAL_OF_POLYNOMIALS
122    SkScalar    C = src[0];
123    SkScalar    A = src[4] - 2 * src[2] + C;
124    SkScalar    B = 2 * (src[2] - C);
125    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
126#else
127    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
128    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
129    return SkScalarInterp(ab, bc, t);
130#endif
131}
132
133static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
134    SkScalar A = src[4] - 2 * src[2] + src[0];
135    SkScalar B = src[2] - src[0];
136
137    return 2 * SkScalarMulAdd(A, t, B);
138}
139
140void SkQuadToCoeff(const SkPoint pts[3], SkPoint coeff[3]) {
141    Sk2s p0 = from_point(pts[0]);
142    Sk2s p1 = from_point(pts[1]);
143    Sk2s p2 = from_point(pts[2]);
144
145    Sk2s p1minus2 = p1 - p0;
146
147    coeff[0] = to_point(p2 - p1 - p1 + p0);     // A * t^2
148    coeff[1] = to_point(p1minus2 + p1minus2);   // B * t
149    coeff[2] = pts[0];                          // C
150}
151
152void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) {
153    SkASSERT(src);
154    SkASSERT(t >= 0 && t <= SK_Scalar1);
155
156    if (pt) {
157        pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
158    }
159    if (tangent) {
160        tangent->set(eval_quad_derivative(&src[0].fX, t),
161                     eval_quad_derivative(&src[0].fY, t));
162    }
163}
164
165SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) {
166    SkASSERT(src);
167    SkASSERT(t >= 0 && t <= SK_Scalar1);
168
169    const Sk2s t2(t);
170
171    Sk2s P0 = from_point(src[0]);
172    Sk2s P1 = from_point(src[1]);
173    Sk2s P2 = from_point(src[2]);
174
175    Sk2s B = P1 - P0;
176    Sk2s A = P2 - P1 - B;
177
178    return to_point((A * t2 + B+B) * t2 + P0);
179}
180
181SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) {
182    SkASSERT(src);
183    SkASSERT(t >= 0 && t <= SK_Scalar1);
184
185    Sk2s P0 = from_point(src[0]);
186    Sk2s P1 = from_point(src[1]);
187    Sk2s P2 = from_point(src[2]);
188
189    Sk2s B = P1 - P0;
190    Sk2s A = P2 - P1 - B;
191    Sk2s T = A * Sk2s(t) + B;
192
193    return to_vector(T + T);
194}
195
196static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) {
197    return v0 + (v1 - v0) * t;
198}
199
200void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
201    SkASSERT(t > 0 && t < SK_Scalar1);
202
203    Sk2s p0 = from_point(src[0]);
204    Sk2s p1 = from_point(src[1]);
205    Sk2s p2 = from_point(src[2]);
206    Sk2s tt(t);
207
208    Sk2s p01 = interp(p0, p1, tt);
209    Sk2s p12 = interp(p1, p2, tt);
210
211    dst[0] = to_point(p0);
212    dst[1] = to_point(p01);
213    dst[2] = to_point(interp(p01, p12, tt));
214    dst[3] = to_point(p12);
215    dst[4] = to_point(p2);
216}
217
218void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
219    SkChopQuadAt(src, dst, 0.5f); return;
220}
221
222/** Quad'(t) = At + B, where
223    A = 2(a - 2b + c)
224    B = 2(b - a)
225    Solve for t, only if it fits between 0 < t < 1
226*/
227int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
228    /*  At + B == 0
229        t = -B / A
230    */
231    return valid_unit_divide(a - b, a - b - b + c, tValue);
232}
233
234static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
235    coords[2] = coords[6] = coords[4];
236}
237
238/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
239 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
240 */
241int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
242    SkASSERT(src);
243    SkASSERT(dst);
244
245    SkScalar a = src[0].fY;
246    SkScalar b = src[1].fY;
247    SkScalar c = src[2].fY;
248
249    if (is_not_monotonic(a, b, c)) {
250        SkScalar    tValue;
251        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
252            SkChopQuadAt(src, dst, tValue);
253            flatten_double_quad_extrema(&dst[0].fY);
254            return 1;
255        }
256        // if we get here, we need to force dst to be monotonic, even though
257        // we couldn't compute a unit_divide value (probably underflow).
258        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
259    }
260    dst[0].set(src[0].fX, a);
261    dst[1].set(src[1].fX, b);
262    dst[2].set(src[2].fX, c);
263    return 0;
264}
265
266/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
267    stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
268 */
269int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
270    SkASSERT(src);
271    SkASSERT(dst);
272
273    SkScalar a = src[0].fX;
274    SkScalar b = src[1].fX;
275    SkScalar c = src[2].fX;
276
277    if (is_not_monotonic(a, b, c)) {
278        SkScalar tValue;
279        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
280            SkChopQuadAt(src, dst, tValue);
281            flatten_double_quad_extrema(&dst[0].fX);
282            return 1;
283        }
284        // if we get here, we need to force dst to be monotonic, even though
285        // we couldn't compute a unit_divide value (probably underflow).
286        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
287    }
288    dst[0].set(a, src[0].fY);
289    dst[1].set(b, src[1].fY);
290    dst[2].set(c, src[2].fY);
291    return 0;
292}
293
294//  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
295//  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
296//  F''(t)  = 2 (a - 2b + c)
297//
298//  A = 2 (b - a)
299//  B = 2 (a - 2b + c)
300//
301//  Maximum curvature for a quadratic means solving
302//  Fx' Fx'' + Fy' Fy'' = 0
303//
304//  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
305//
306SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
307    SkScalar    Ax = src[1].fX - src[0].fX;
308    SkScalar    Ay = src[1].fY - src[0].fY;
309    SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
310    SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
311    SkScalar    t = 0;  // 0 means don't chop
312
313    (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
314    return t;
315}
316
317int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
318    SkScalar t = SkFindQuadMaxCurvature(src);
319    if (t == 0) {
320        memcpy(dst, src, 3 * sizeof(SkPoint));
321        return 1;
322    } else {
323        SkChopQuadAt(src, dst, t);
324        return 2;
325    }
326}
327
328void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
329    Sk2s scale(SkDoubleToScalar(2.0 / 3.0));
330    Sk2s s0 = from_point(src[0]);
331    Sk2s s1 = from_point(src[1]);
332    Sk2s s2 = from_point(src[2]);
333
334    dst[0] = src[0];
335    dst[1] = to_point(s0 + (s1 - s0) * scale);
336    dst[2] = to_point(s2 + (s1 - s2) * scale);
337    dst[3] = src[2];
338}
339
340//////////////////////////////////////////////////////////////////////////////
341///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
342//////////////////////////////////////////////////////////////////////////////
343
344static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
345    coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
346    coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
347    coeff[2] = 3*(pt[2] - pt[0]);
348    coeff[3] = pt[0];
349}
350
351void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
352    SkASSERT(pts);
353
354    if (cx) {
355        get_cubic_coeff(&pts[0].fX, cx);
356    }
357    if (cy) {
358        get_cubic_coeff(&pts[0].fY, cy);
359    }
360}
361
362static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
363    SkASSERT(src);
364    SkASSERT(t >= 0 && t <= SK_Scalar1);
365
366    if (t == 0) {
367        return src[0];
368    }
369
370#ifdef DIRECT_EVAL_OF_POLYNOMIALS
371    SkScalar D = src[0];
372    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
373    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
374    SkScalar C = 3*(src[2] - D);
375
376    return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
377#else
378    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
379    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
380    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
381    SkScalar    abc = SkScalarInterp(ab, bc, t);
382    SkScalar    bcd = SkScalarInterp(bc, cd, t);
383    return SkScalarInterp(abc, bcd, t);
384#endif
385}
386
387/** return At^2 + Bt + C
388*/
389static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
390    SkASSERT(t >= 0 && t <= SK_Scalar1);
391
392    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
393}
394
395static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
396    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
397    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
398    SkScalar C = src[2] - src[0];
399
400    return eval_quadratic(A, B, C, t);
401}
402
403static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
404    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
405    SkScalar B = src[4] - 2 * src[2] + src[0];
406
407    return SkScalarMulAdd(A, t, B);
408}
409
410void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
411                   SkVector* tangent, SkVector* curvature) {
412    SkASSERT(src);
413    SkASSERT(t >= 0 && t <= SK_Scalar1);
414
415    if (loc) {
416        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
417    }
418    if (tangent) {
419        tangent->set(eval_cubic_derivative(&src[0].fX, t),
420                     eval_cubic_derivative(&src[0].fY, t));
421    }
422    if (curvature) {
423        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
424                       eval_cubic_2ndDerivative(&src[0].fY, t));
425    }
426}
427
428/** Cubic'(t) = At^2 + Bt + C, where
429    A = 3(-a + 3(b - c) + d)
430    B = 6(a - 2b + c)
431    C = 3(b - a)
432    Solve for t, keeping only those that fit betwee 0 < t < 1
433*/
434int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
435                       SkScalar tValues[2]) {
436    // we divide A,B,C by 3 to simplify
437    SkScalar A = d - a + 3*(b - c);
438    SkScalar B = 2*(a - b - b + c);
439    SkScalar C = b - a;
440
441    return SkFindUnitQuadRoots(A, B, C, tValues);
442}
443
444void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
445    SkASSERT(t > 0 && t < SK_Scalar1);
446
447    Sk2s    p0 = from_point(src[0]);
448    Sk2s    p1 = from_point(src[1]);
449    Sk2s    p2 = from_point(src[2]);
450    Sk2s    p3 = from_point(src[3]);
451    Sk2s    tt(t);
452
453    Sk2s    ab = interp(p0, p1, tt);
454    Sk2s    bc = interp(p1, p2, tt);
455    Sk2s    cd = interp(p2, p3, tt);
456    Sk2s    abc = interp(ab, bc, tt);
457    Sk2s    bcd = interp(bc, cd, tt);
458    Sk2s    abcd = interp(abc, bcd, tt);
459
460    dst[0] = src[0];
461    dst[1] = to_point(ab);
462    dst[2] = to_point(abc);
463    dst[3] = to_point(abcd);
464    dst[4] = to_point(bcd);
465    dst[5] = to_point(cd);
466    dst[6] = src[3];
467}
468
469void SkCubicToCoeff(const SkPoint pts[4], SkPoint coeff[4]) {
470    Sk2s p0 = from_point(pts[0]);
471    Sk2s p1 = from_point(pts[1]);
472    Sk2s p2 = from_point(pts[2]);
473    Sk2s p3 = from_point(pts[3]);
474
475    const Sk2s three(3);
476    Sk2s p1minusp2 = p1 - p2;
477
478    Sk2s D = p0;
479    Sk2s A = p3 + three * p1minusp2 - D;
480    Sk2s B = three * (D - p1minusp2 - p1);
481    Sk2s C = three * (p1 - D);
482
483    coeff[0] = to_point(A);
484    coeff[1] = to_point(B);
485    coeff[2] = to_point(C);
486    coeff[3] = to_point(D);
487}
488
489/*  http://code.google.com/p/skia/issues/detail?id=32
490
491    This test code would fail when we didn't check the return result of
492    valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
493    that after the first chop, the parameters to valid_unit_divide are equal
494    (thanks to finite float precision and rounding in the subtracts). Thus
495    even though the 2nd tValue looks < 1.0, after we renormalize it, we end
496    up with 1.0, hence the need to check and just return the last cubic as
497    a degenerate clump of 4 points in the sampe place.
498
499    static void test_cubic() {
500        SkPoint src[4] = {
501            { 556.25000, 523.03003 },
502            { 556.23999, 522.96002 },
503            { 556.21997, 522.89001 },
504            { 556.21997, 522.82001 }
505        };
506        SkPoint dst[10];
507        SkScalar tval[] = { 0.33333334f, 0.99999994f };
508        SkChopCubicAt(src, dst, tval, 2);
509    }
510 */
511
512void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
513                   const SkScalar tValues[], int roots) {
514#ifdef SK_DEBUG
515    {
516        for (int i = 0; i < roots - 1; i++)
517        {
518            SkASSERT(is_unit_interval(tValues[i]));
519            SkASSERT(is_unit_interval(tValues[i+1]));
520            SkASSERT(tValues[i] < tValues[i+1]);
521        }
522    }
523#endif
524
525    if (dst) {
526        if (roots == 0) { // nothing to chop
527            memcpy(dst, src, 4*sizeof(SkPoint));
528        } else {
529            SkScalar    t = tValues[0];
530            SkPoint     tmp[4];
531
532            for (int i = 0; i < roots; i++) {
533                SkChopCubicAt(src, dst, t);
534                if (i == roots - 1) {
535                    break;
536                }
537
538                dst += 3;
539                // have src point to the remaining cubic (after the chop)
540                memcpy(tmp, dst, 4 * sizeof(SkPoint));
541                src = tmp;
542
543                // watch out in case the renormalized t isn't in range
544                if (!valid_unit_divide(tValues[i+1] - tValues[i],
545                                       SK_Scalar1 - tValues[i], &t)) {
546                    // if we can't, just create a degenerate cubic
547                    dst[4] = dst[5] = dst[6] = src[3];
548                    break;
549                }
550            }
551        }
552    }
553}
554
555void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
556    SkChopCubicAt(src, dst, 0.5f);
557}
558
559static void flatten_double_cubic_extrema(SkScalar coords[14]) {
560    coords[4] = coords[8] = coords[6];
561}
562
563/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
564    the resulting beziers are monotonic in Y. This is called by the scan
565    converter.  Depending on what is returned, dst[] is treated as follows:
566    0   dst[0..3] is the original cubic
567    1   dst[0..3] and dst[3..6] are the two new cubics
568    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
569    If dst == null, it is ignored and only the count is returned.
570*/
571int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
572    SkScalar    tValues[2];
573    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
574                                           src[3].fY, tValues);
575
576    SkChopCubicAt(src, dst, tValues, roots);
577    if (dst && roots > 0) {
578        // we do some cleanup to ensure our Y extrema are flat
579        flatten_double_cubic_extrema(&dst[0].fY);
580        if (roots == 2) {
581            flatten_double_cubic_extrema(&dst[3].fY);
582        }
583    }
584    return roots;
585}
586
587int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
588    SkScalar    tValues[2];
589    int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
590                                           src[3].fX, tValues);
591
592    SkChopCubicAt(src, dst, tValues, roots);
593    if (dst && roots > 0) {
594        // we do some cleanup to ensure our Y extrema are flat
595        flatten_double_cubic_extrema(&dst[0].fX);
596        if (roots == 2) {
597            flatten_double_cubic_extrema(&dst[3].fX);
598        }
599    }
600    return roots;
601}
602
603/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
604
605    Inflection means that curvature is zero.
606    Curvature is [F' x F''] / [F'^3]
607    So we solve F'x X F''y - F'y X F''y == 0
608    After some canceling of the cubic term, we get
609    A = b - a
610    B = c - 2b + a
611    C = d - 3c + 3b - a
612    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
613*/
614int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
615    SkScalar    Ax = src[1].fX - src[0].fX;
616    SkScalar    Ay = src[1].fY - src[0].fY;
617    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
618    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
619    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
620    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
621
622    return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
623                               Ax*Cy - Ay*Cx,
624                               Ax*By - Ay*Bx,
625                               tValues);
626}
627
628int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
629    SkScalar    tValues[2];
630    int         count = SkFindCubicInflections(src, tValues);
631
632    if (dst) {
633        if (count == 0) {
634            memcpy(dst, src, 4 * sizeof(SkPoint));
635        } else {
636            SkChopCubicAt(src, dst, tValues, count);
637        }
638    }
639    return count + 1;
640}
641
642// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
643// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
644// Classification:
645// discr(I) > 0        Serpentine
646// discr(I) = 0        Cusp
647// discr(I) < 0        Loop
648// d0 = d1 = 0         Quadratic
649// d0 = d1 = d2 = 0    Line
650// p0 = p1 = p2 = p3   Point
651static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
652    if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
653        return kPoint_SkCubicType;
654    }
655    const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
656    if (discr > SK_ScalarNearlyZero) {
657        return kSerpentine_SkCubicType;
658    } else if (discr < -SK_ScalarNearlyZero) {
659        return kLoop_SkCubicType;
660    } else {
661        if (0.f == d[0] && 0.f == d[1]) {
662            return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
663        } else {
664            return kCusp_SkCubicType;
665        }
666    }
667}
668
669// Assumes the third component of points is 1.
670// Calcs p0 . (p1 x p2)
671static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
672    const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
673    const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
674    const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
675    return (xComp + yComp + wComp);
676}
677
678// Calc coefficients of I(s,t) where roots of I are inflection points of curve
679// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
680// d0 = a1 - 2*a2+3*a3
681// d1 = -a2 + 3*a3
682// d2 = 3*a3
683// a1 = p0 . (p3 x p2)
684// a2 = p1 . (p0 x p3)
685// a3 = p2 . (p1 x p0)
686// Places the values of d1, d2, d3 in array d passed in
687static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
688    SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
689    SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
690    SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
691
692    // need to scale a's or values in later calculations will grow to high
693    SkScalar max = SkScalarAbs(a1);
694    max = SkMaxScalar(max, SkScalarAbs(a2));
695    max = SkMaxScalar(max, SkScalarAbs(a3));
696    max = 1.f/max;
697    a1 = a1 * max;
698    a2 = a2 * max;
699    a3 = a3 * max;
700
701    d[2] = 3.f * a3;
702    d[1] = d[2] - a2;
703    d[0] = d[1] - a2 + a1;
704}
705
706SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
707    calc_cubic_inflection_func(src, d);
708    return classify_cubic(src, d);
709}
710
711template <typename T> void bubble_sort(T array[], int count) {
712    for (int i = count - 1; i > 0; --i)
713        for (int j = i; j > 0; --j)
714            if (array[j] < array[j-1])
715            {
716                T   tmp(array[j]);
717                array[j] = array[j-1];
718                array[j-1] = tmp;
719            }
720}
721
722/**
723 *  Given an array and count, remove all pair-wise duplicates from the array,
724 *  keeping the existing sorting, and return the new count
725 */
726static int collaps_duplicates(SkScalar array[], int count) {
727    for (int n = count; n > 1; --n) {
728        if (array[0] == array[1]) {
729            for (int i = 1; i < n; ++i) {
730                array[i - 1] = array[i];
731            }
732            count -= 1;
733        } else {
734            array += 1;
735        }
736    }
737    return count;
738}
739
740#ifdef SK_DEBUG
741
742#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
743
744static void test_collaps_duplicates() {
745    static bool gOnce;
746    if (gOnce) { return; }
747    gOnce = true;
748    const SkScalar src0[] = { 0 };
749    const SkScalar src1[] = { 0, 0 };
750    const SkScalar src2[] = { 0, 1 };
751    const SkScalar src3[] = { 0, 0, 0 };
752    const SkScalar src4[] = { 0, 0, 1 };
753    const SkScalar src5[] = { 0, 1, 1 };
754    const SkScalar src6[] = { 0, 1, 2 };
755    const struct {
756        const SkScalar* fData;
757        int fCount;
758        int fCollapsedCount;
759    } data[] = {
760        { TEST_COLLAPS_ENTRY(src0), 1 },
761        { TEST_COLLAPS_ENTRY(src1), 1 },
762        { TEST_COLLAPS_ENTRY(src2), 2 },
763        { TEST_COLLAPS_ENTRY(src3), 1 },
764        { TEST_COLLAPS_ENTRY(src4), 2 },
765        { TEST_COLLAPS_ENTRY(src5), 2 },
766        { TEST_COLLAPS_ENTRY(src6), 3 },
767    };
768    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
769        SkScalar dst[3];
770        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
771        int count = collaps_duplicates(dst, data[i].fCount);
772        SkASSERT(data[i].fCollapsedCount == count);
773        for (int j = 1; j < count; ++j) {
774            SkASSERT(dst[j-1] < dst[j]);
775        }
776    }
777}
778#endif
779
780static SkScalar SkScalarCubeRoot(SkScalar x) {
781    return SkScalarPow(x, 0.3333333f);
782}
783
784/*  Solve coeff(t) == 0, returning the number of roots that
785    lie withing 0 < t < 1.
786    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
787
788    Eliminates repeated roots (so that all tValues are distinct, and are always
789    in increasing order.
790*/
791static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
792    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
793        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
794    }
795
796    SkScalar a, b, c, Q, R;
797
798    {
799        SkASSERT(coeff[0] != 0);
800
801        SkScalar inva = SkScalarInvert(coeff[0]);
802        a = coeff[1] * inva;
803        b = coeff[2] * inva;
804        c = coeff[3] * inva;
805    }
806    Q = (a*a - b*3) / 9;
807    R = (2*a*a*a - 9*a*b + 27*c) / 54;
808
809    SkScalar Q3 = Q * Q * Q;
810    SkScalar R2MinusQ3 = R * R - Q3;
811    SkScalar adiv3 = a / 3;
812
813    SkScalar*   roots = tValues;
814    SkScalar    r;
815
816    if (R2MinusQ3 < 0) { // we have 3 real roots
817        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
818        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
819
820        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
821        if (is_unit_interval(r)) {
822            *roots++ = r;
823        }
824        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
825        if (is_unit_interval(r)) {
826            *roots++ = r;
827        }
828        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
829        if (is_unit_interval(r)) {
830            *roots++ = r;
831        }
832        SkDEBUGCODE(test_collaps_duplicates();)
833
834        // now sort the roots
835        int count = (int)(roots - tValues);
836        SkASSERT((unsigned)count <= 3);
837        bubble_sort(tValues, count);
838        count = collaps_duplicates(tValues, count);
839        roots = tValues + count;    // so we compute the proper count below
840    } else {              // we have 1 real root
841        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
842        A = SkScalarCubeRoot(A);
843        if (R > 0) {
844            A = -A;
845        }
846        if (A != 0) {
847            A += Q / A;
848        }
849        r = A - adiv3;
850        if (is_unit_interval(r)) {
851            *roots++ = r;
852        }
853    }
854
855    return (int)(roots - tValues);
856}
857
858/*  Looking for F' dot F'' == 0
859
860    A = b - a
861    B = c - 2b + a
862    C = d - 3c + 3b - a
863
864    F' = 3Ct^2 + 6Bt + 3A
865    F'' = 6Ct + 6B
866
867    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
868*/
869static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
870    SkScalar    a = src[2] - src[0];
871    SkScalar    b = src[4] - 2 * src[2] + src[0];
872    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
873
874    coeff[0] = c * c;
875    coeff[1] = 3 * b * c;
876    coeff[2] = 2 * b * b + c * a;
877    coeff[3] = a * b;
878}
879
880/*  Looking for F' dot F'' == 0
881
882    A = b - a
883    B = c - 2b + a
884    C = d - 3c + 3b - a
885
886    F' = 3Ct^2 + 6Bt + 3A
887    F'' = 6Ct + 6B
888
889    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
890*/
891int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
892    SkScalar coeffX[4], coeffY[4];
893    int      i;
894
895    formulate_F1DotF2(&src[0].fX, coeffX);
896    formulate_F1DotF2(&src[0].fY, coeffY);
897
898    for (i = 0; i < 4; i++) {
899        coeffX[i] += coeffY[i];
900    }
901
902    SkScalar    t[3];
903    int         count = solve_cubic_poly(coeffX, t);
904    int         maxCount = 0;
905
906    // now remove extrema where the curvature is zero (mins)
907    // !!!! need a test for this !!!!
908    for (i = 0; i < count; i++) {
909        // if (not_min_curvature())
910        if (t[i] > 0 && t[i] < SK_Scalar1) {
911            tValues[maxCount++] = t[i];
912        }
913    }
914    return maxCount;
915}
916
917int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
918                              SkScalar tValues[3]) {
919    SkScalar    t_storage[3];
920
921    if (tValues == NULL) {
922        tValues = t_storage;
923    }
924
925    int count = SkFindCubicMaxCurvature(src, tValues);
926
927    if (dst) {
928        if (count == 0) {
929            memcpy(dst, src, 4 * sizeof(SkPoint));
930        } else {
931            SkChopCubicAt(src, dst, tValues, count);
932        }
933    }
934    return count + 1;
935}
936
937///////////////////////////////////////////////////////////////////////////////
938
939/*  Find t value for quadratic [a, b, c] = d.
940    Return 0 if there is no solution within [0, 1)
941*/
942static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
943    // At^2 + Bt + C = d
944    SkScalar A = a - 2 * b + c;
945    SkScalar B = 2 * (b - a);
946    SkScalar C = a - d;
947
948    SkScalar    roots[2];
949    int         count = SkFindUnitQuadRoots(A, B, C, roots);
950
951    SkASSERT(count <= 1);
952    return count == 1 ? roots[0] : 0;
953}
954
955/*  given a quad-curve and a point (x,y), chop the quad at that point and place
956    the new off-curve point and endpoint into 'dest'.
957    Should only return false if the computed pos is the start of the curve
958    (i.e. root == 0)
959*/
960static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
961                                SkPoint* dest) {
962    const SkScalar* base;
963    SkScalar        value;
964
965    if (SkScalarAbs(x) < SkScalarAbs(y)) {
966        base = &quad[0].fX;
967        value = x;
968    } else {
969        base = &quad[0].fY;
970        value = y;
971    }
972
973    // note: this returns 0 if it thinks value is out of range, meaning the
974    // root might return something outside of [0, 1)
975    SkScalar t = quad_solve(base[0], base[2], base[4], value);
976
977    if (t > 0) {
978        SkPoint tmp[5];
979        SkChopQuadAt(quad, tmp, t);
980        dest[0] = tmp[1];
981        dest[1].set(x, y);
982        return true;
983    } else {
984        /*  t == 0 means either the value triggered a root outside of [0, 1)
985            For our purposes, we can ignore the <= 0 roots, but we want to
986            catch the >= 1 roots (which given our caller, will basically mean
987            a root of 1, give-or-take numerical instability). If we are in the
988            >= 1 case, return the existing offCurve point.
989
990            The test below checks to see if we are close to the "end" of the
991            curve (near base[4]). Rather than specifying a tolerance, I just
992            check to see if value is on to the right/left of the middle point
993            (depending on the direction/sign of the end points).
994        */
995        if ((base[0] < base[4] && value > base[2]) ||
996            (base[0] > base[4] && value < base[2]))   // should root have been 1
997        {
998            dest[0] = quad[1];
999            dest[1].set(x, y);
1000            return true;
1001        }
1002    }
1003    return false;
1004}
1005
1006static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
1007// The mid point of the quadratic arc approximation is half way between the two
1008// control points. The float epsilon adjustment moves the on curve point out by
1009// two bits, distributing the convex test error between the round rect
1010// approximation and the convex cross product sign equality test.
1011#define SK_MID_RRECT_OFFSET \
1012    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1013    { SK_Scalar1,            0                      },
1014    { SK_Scalar1,            SK_ScalarTanPIOver8    },
1015    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
1016    { SK_ScalarTanPIOver8,   SK_Scalar1             },
1017
1018    { 0,                     SK_Scalar1             },
1019    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
1020    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
1021    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
1022
1023    { -SK_Scalar1,           0                      },
1024    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
1025    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
1026    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
1027
1028    { 0,                     -SK_Scalar1            },
1029    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
1030    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
1031    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
1032
1033    { SK_Scalar1,            0                      }
1034#undef SK_MID_RRECT_OFFSET
1035};
1036
1037int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1038                   SkRotationDirection dir, const SkMatrix* userMatrix,
1039                   SkPoint quadPoints[]) {
1040    // rotate by x,y so that uStart is (1.0)
1041    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1042    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1043
1044    SkScalar absX = SkScalarAbs(x);
1045    SkScalar absY = SkScalarAbs(y);
1046
1047    int pointCount;
1048
1049    // check for (effectively) coincident vectors
1050    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1051    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1052    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1053        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1054         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1055
1056        // just return the start-point
1057        quadPoints[0].set(SK_Scalar1, 0);
1058        pointCount = 1;
1059    } else {
1060        if (dir == kCCW_SkRotationDirection) {
1061            y = -y;
1062        }
1063        // what octant (quadratic curve) is [xy] in?
1064        int oct = 0;
1065        bool sameSign = true;
1066
1067        if (0 == y) {
1068            oct = 4;        // 180
1069            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1070        } else if (0 == x) {
1071            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1072            oct = y > 0 ? 2 : 6; // 90 : 270
1073        } else {
1074            if (y < 0) {
1075                oct += 4;
1076            }
1077            if ((x < 0) != (y < 0)) {
1078                oct += 2;
1079                sameSign = false;
1080            }
1081            if ((absX < absY) == sameSign) {
1082                oct += 1;
1083            }
1084        }
1085
1086        int wholeCount = oct << 1;
1087        memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1088
1089        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1090        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1091            wholeCount += 2;
1092        }
1093        pointCount = wholeCount + 1;
1094    }
1095
1096    // now handle counter-clockwise and the initial unitStart rotation
1097    SkMatrix    matrix;
1098    matrix.setSinCos(uStart.fY, uStart.fX);
1099    if (dir == kCCW_SkRotationDirection) {
1100        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1101    }
1102    if (userMatrix) {
1103        matrix.postConcat(*userMatrix);
1104    }
1105    matrix.mapPoints(quadPoints, pointCount);
1106    return pointCount;
1107}
1108
1109
1110///////////////////////////////////////////////////////////////////////////////
1111//
1112// NURB representation for conics.  Helpful explanations at:
1113//
1114// http://citeseerx.ist.psu.edu/viewdoc/
1115//   download?doi=10.1.1.44.5740&rep=rep1&type=ps
1116// and
1117// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1118//
1119// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1120//     ------------------------------------------
1121//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1122//
1123//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1124//     ------------------------------------------------
1125//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1126//
1127
1128static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1129    SkASSERT(src);
1130    SkASSERT(t >= 0 && t <= SK_Scalar1);
1131
1132    SkScalar    src2w = SkScalarMul(src[2], w);
1133    SkScalar    C = src[0];
1134    SkScalar    A = src[4] - 2 * src2w + C;
1135    SkScalar    B = 2 * (src2w - C);
1136    SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1137
1138    B = 2 * (w - SK_Scalar1);
1139    C = SK_Scalar1;
1140    A = -B;
1141    SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1142
1143    return SkScalarDiv(numer, denom);
1144}
1145
1146// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1147//
1148//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1149//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1150//  t^0 : -2 P0 w + 2 P1 w
1151//
1152//  We disregard magnitude, so we can freely ignore the denominator of F', and
1153//  divide the numerator by 2
1154//
1155//    coeff[0] for t^2
1156//    coeff[1] for t^1
1157//    coeff[2] for t^0
1158//
1159static void conic_deriv_coeff(const SkScalar src[],
1160                              SkScalar w,
1161                              SkScalar coeff[3]) {
1162    const SkScalar P20 = src[4] - src[0];
1163    const SkScalar P10 = src[2] - src[0];
1164    const SkScalar wP10 = w * P10;
1165    coeff[0] = w * P20 - P20;
1166    coeff[1] = P20 - 2 * wP10;
1167    coeff[2] = wP10;
1168}
1169
1170static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1171    SkScalar coeff[3];
1172    conic_deriv_coeff(coord, w, coeff);
1173    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1174}
1175
1176static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1177    SkScalar coeff[3];
1178    conic_deriv_coeff(src, w, coeff);
1179
1180    SkScalar tValues[2];
1181    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1182    SkASSERT(0 == roots || 1 == roots);
1183
1184    if (1 == roots) {
1185        *t = tValues[0];
1186        return true;
1187    }
1188    return false;
1189}
1190
1191struct SkP3D {
1192    SkScalar fX, fY, fZ;
1193
1194    void set(SkScalar x, SkScalar y, SkScalar z) {
1195        fX = x; fY = y; fZ = z;
1196    }
1197
1198    void projectDown(SkPoint* dst) const {
1199        dst->set(fX / fZ, fY / fZ);
1200    }
1201};
1202
1203// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1204static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1205    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1206    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1207    dst[0] = ab;
1208    dst[3] = SkScalarInterp(ab, bc, t);
1209    dst[6] = bc;
1210}
1211
1212static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1213    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1214    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1215    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1216}
1217
1218void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1219    SkASSERT(t >= 0 && t <= SK_Scalar1);
1220
1221    if (pt) {
1222        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1223                conic_eval_pos(&fPts[0].fY, fW, t));
1224    }
1225    if (tangent) {
1226        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1227                     conic_eval_tan(&fPts[0].fY, fW, t));
1228    }
1229}
1230
1231void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1232    SkP3D tmp[3], tmp2[3];
1233
1234    ratquad_mapTo3D(fPts, fW, tmp);
1235
1236    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1237    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1238    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1239
1240    dst[0].fPts[0] = fPts[0];
1241    tmp2[0].projectDown(&dst[0].fPts[1]);
1242    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1243    tmp2[2].projectDown(&dst[1].fPts[1]);
1244    dst[1].fPts[2] = fPts[2];
1245
1246    // to put in "standard form", where w0 and w2 are both 1, we compute the
1247    // new w1 as sqrt(w1*w1/w0*w2)
1248    // or
1249    // w1 /= sqrt(w0*w2)
1250    //
1251    // However, in our case, we know that for dst[0]:
1252    //     w0 == 1, and for dst[1], w2 == 1
1253    //
1254    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1255    dst[0].fW = tmp2[0].fZ / root;
1256    dst[1].fW = tmp2[2].fZ / root;
1257}
1258
1259static Sk2s times_2(const Sk2s& value) {
1260    return value + value;
1261}
1262
1263SkPoint SkConic::evalAt(SkScalar t) const {
1264    Sk2s p0 = from_point(fPts[0]);
1265    Sk2s p1 = from_point(fPts[1]);
1266    Sk2s p2 = from_point(fPts[2]);
1267    Sk2s tt(t);
1268    Sk2s ww(fW);
1269    Sk2s one(1);
1270
1271    Sk2s p1w = p1 * ww;
1272    Sk2s C = p0;
1273    Sk2s A = p2 - times_2(p1w) + p0;
1274    Sk2s B = times_2(p1w - C);
1275    Sk2s numer = quad_poly_eval(A, B, C, tt);
1276
1277    B = times_2(ww - one);
1278    A = -B;
1279    Sk2s denom = quad_poly_eval(A, B, one, tt);
1280
1281    return to_point(numer / denom);
1282}
1283
1284SkVector SkConic::evalTangentAt(SkScalar t) const {
1285    Sk2s p0 = from_point(fPts[0]);
1286    Sk2s p1 = from_point(fPts[1]);
1287    Sk2s p2 = from_point(fPts[2]);
1288    Sk2s ww(fW);
1289
1290    Sk2s p20 = p2 - p0;
1291    Sk2s p10 = p1 - p0;
1292
1293    Sk2s C = ww * p10;
1294    Sk2s A = ww * p20 - p20;
1295    Sk2s B = p20 - C - C;
1296
1297    return to_vector(quad_poly_eval(A, B, C, Sk2s(t)));
1298}
1299
1300static SkScalar subdivide_w_value(SkScalar w) {
1301    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1302}
1303
1304static Sk2s twice(const Sk2s& value) {
1305    return value + value;
1306}
1307
1308void SkConic::chop(SkConic * SK_RESTRICT dst) const {
1309    Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW));
1310    SkScalar newW = subdivide_w_value(fW);
1311
1312    Sk2s p0 = from_point(fPts[0]);
1313    Sk2s p1 = from_point(fPts[1]);
1314    Sk2s p2 = from_point(fPts[2]);
1315    Sk2s ww(fW);
1316
1317    Sk2s wp1 = ww * p1;
1318    Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f);
1319
1320    dst[0].fPts[0] = fPts[0];
1321    dst[0].fPts[1] = to_point((p0 + wp1) * scale);
1322    dst[0].fPts[2] = dst[1].fPts[0] = to_point(m);
1323    dst[1].fPts[1] = to_point((wp1 + p2) * scale);
1324    dst[1].fPts[2] = fPts[2];
1325
1326    dst[0].fW = dst[1].fW = newW;
1327}
1328
1329/*
1330 *  "High order approximation of conic sections by quadratic splines"
1331 *      by Michael Floater, 1993
1332 */
1333#define AS_QUAD_ERROR_SETUP                                         \
1334    SkScalar a = fW - 1;                                            \
1335    SkScalar k = a / (4 * (2 + a));                                 \
1336    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1337    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1338
1339void SkConic::computeAsQuadError(SkVector* err) const {
1340    AS_QUAD_ERROR_SETUP
1341    err->set(x, y);
1342}
1343
1344bool SkConic::asQuadTol(SkScalar tol) const {
1345    AS_QUAD_ERROR_SETUP
1346    return (x * x + y * y) <= tol * tol;
1347}
1348
1349// Limit the number of suggested quads to approximate a conic
1350#define kMaxConicToQuadPOW2     5
1351
1352int SkConic::computeQuadPOW2(SkScalar tol) const {
1353    if (tol < 0 || !SkScalarIsFinite(tol)) {
1354        return 0;
1355    }
1356
1357    AS_QUAD_ERROR_SETUP
1358
1359    SkScalar error = SkScalarSqrt(x * x + y * y);
1360    int pow2;
1361    for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
1362        if (error <= tol) {
1363            break;
1364        }
1365        error *= 0.25f;
1366    }
1367    // float version -- using ceil gives the same results as the above.
1368    if (false) {
1369        SkScalar err = SkScalarSqrt(x * x + y * y);
1370        if (err <= tol) {
1371            return 0;
1372        }
1373        SkScalar tol2 = tol * tol;
1374        if (tol2 == 0) {
1375            return kMaxConicToQuadPOW2;
1376        }
1377        SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
1378        int altPow2 = SkScalarCeilToInt(fpow2);
1379        if (altPow2 != pow2) {
1380            SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
1381        }
1382        pow2 = altPow2;
1383    }
1384    return pow2;
1385}
1386
1387static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1388    SkASSERT(level >= 0);
1389
1390    if (0 == level) {
1391        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1392        return pts + 2;
1393    } else {
1394        SkConic dst[2];
1395        src.chop(dst);
1396        --level;
1397        pts = subdivide(dst[0], pts, level);
1398        return subdivide(dst[1], pts, level);
1399    }
1400}
1401
1402int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1403    SkASSERT(pow2 >= 0);
1404    *pts = fPts[0];
1405    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1406    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1407    return 1 << pow2;
1408}
1409
1410bool SkConic::findXExtrema(SkScalar* t) const {
1411    return conic_find_extrema(&fPts[0].fX, fW, t);
1412}
1413
1414bool SkConic::findYExtrema(SkScalar* t) const {
1415    return conic_find_extrema(&fPts[0].fY, fW, t);
1416}
1417
1418bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1419    SkScalar t;
1420    if (this->findXExtrema(&t)) {
1421        this->chopAt(t, dst);
1422        // now clean-up the middle, since we know t was meant to be at
1423        // an X-extrema
1424        SkScalar value = dst[0].fPts[2].fX;
1425        dst[0].fPts[1].fX = value;
1426        dst[1].fPts[0].fX = value;
1427        dst[1].fPts[1].fX = value;
1428        return true;
1429    }
1430    return false;
1431}
1432
1433bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1434    SkScalar t;
1435    if (this->findYExtrema(&t)) {
1436        this->chopAt(t, dst);
1437        // now clean-up the middle, since we know t was meant to be at
1438        // an Y-extrema
1439        SkScalar value = dst[0].fPts[2].fY;
1440        dst[0].fPts[1].fY = value;
1441        dst[1].fPts[0].fY = value;
1442        dst[1].fPts[1].fY = value;
1443        return true;
1444    }
1445    return false;
1446}
1447
1448void SkConic::computeTightBounds(SkRect* bounds) const {
1449    SkPoint pts[4];
1450    pts[0] = fPts[0];
1451    pts[1] = fPts[2];
1452    int count = 2;
1453
1454    SkScalar t;
1455    if (this->findXExtrema(&t)) {
1456        this->evalAt(t, &pts[count++]);
1457    }
1458    if (this->findYExtrema(&t)) {
1459        this->evalAt(t, &pts[count++]);
1460    }
1461    bounds->set(pts, count);
1462}
1463
1464void SkConic::computeFastBounds(SkRect* bounds) const {
1465    bounds->set(fPts, 3);
1466}
1467
1468bool SkConic::findMaxCurvature(SkScalar* t) const {
1469    // TODO: Implement me
1470    return false;
1471}
1472
1473SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1474                             const SkMatrix& matrix) {
1475    if (!matrix.hasPerspective()) {
1476        return w;
1477    }
1478
1479    SkP3D src[3], dst[3];
1480
1481    ratquad_mapTo3D(pts, w, src);
1482
1483    matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1484
1485    // w' = sqrt(w1*w1/w0*w2)
1486    SkScalar w0 = dst[0].fZ;
1487    SkScalar w1 = dst[1].fZ;
1488    SkScalar w2 = dst[2].fZ;
1489    w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1490    return w;
1491}
1492
1493int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
1494                          const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
1495    // rotate by x,y so that uStart is (1.0)
1496    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1497    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1498
1499    SkScalar absY = SkScalarAbs(y);
1500
1501    // check for (effectively) coincident vectors
1502    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1503    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1504    if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) ||
1505                                                 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1506        return 0;
1507    }
1508
1509    if (dir == kCCW_SkRotationDirection) {
1510        y = -y;
1511    }
1512
1513    // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
1514    //      0 == [0  .. 90)
1515    //      1 == [90 ..180)
1516    //      2 == [180..270)
1517    //      3 == [270..360)
1518    //
1519    int quadrant = 0;
1520    if (0 == y) {
1521        quadrant = 2;        // 180
1522        SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1523    } else if (0 == x) {
1524        SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1525        quadrant = y > 0 ? 1 : 3; // 90 : 270
1526    } else {
1527        if (y < 0) {
1528            quadrant += 2;
1529        }
1530        if ((x < 0) != (y < 0)) {
1531            quadrant += 1;
1532        }
1533    }
1534
1535    const SkPoint quadrantPts[] = {
1536        { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
1537    };
1538    const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
1539
1540    int conicCount = quadrant;
1541    for (int i = 0; i < conicCount; ++i) {
1542        dst[i].set(&quadrantPts[i * 2], quadrantWeight);
1543    }
1544
1545    // Now compute any remaing (sub-90-degree) arc for the last conic
1546    const SkPoint finalP = { x, y };
1547    const SkPoint& lastQ = quadrantPts[quadrant * 2];  // will already be a unit-vector
1548    const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
1549    SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
1550
1551    if (dot < 1 - SK_ScalarNearlyZero) {
1552        SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
1553        // compute the bisector vector, and then rescale to be the off-curve point.
1554        // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
1555        // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
1556        // This is nice, since our computed weight is cos(theta/2) as well!
1557        //
1558        const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
1559        offCurve.setLength(SkScalarInvert(cosThetaOver2));
1560        dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
1561        conicCount += 1;
1562    }
1563
1564    // now handle counter-clockwise and the initial unitStart rotation
1565    SkMatrix    matrix;
1566    matrix.setSinCos(uStart.fY, uStart.fX);
1567    if (dir == kCCW_SkRotationDirection) {
1568        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1569    }
1570    if (userMatrix) {
1571        matrix.postConcat(*userMatrix);
1572    }
1573    for (int i = 0; i < conicCount; ++i) {
1574        matrix.mapPoints(dst[i].fPts, 3);
1575    }
1576    return conicCount;
1577}
1578