SkGeometry.cpp revision c6325cde797baeedebc515bc8610f96eacbd36c9 
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 SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
345    SkASSERT(src);
346    SkASSERT(t >= 0 && t <= SK_Scalar1);
347
348    if (t == 0) {
349        return src[0];
350    }
351
352#ifdef DIRECT_EVAL_OF_POLYNOMIALS
353    SkScalar D = src[0];
354    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
355    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
356    SkScalar C = 3*(src[2] - D);
357
358    return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
359#else
360    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
361    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
362    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
363    SkScalar    abc = SkScalarInterp(ab, bc, t);
364    SkScalar    bcd = SkScalarInterp(bc, cd, t);
365    return SkScalarInterp(abc, bcd, t);
366#endif
367}
368
369/** return At^2 + Bt + C
370*/
371static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
372    SkASSERT(t >= 0 && t <= SK_Scalar1);
373
374    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
375}
376
377static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
378    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
379    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
380    SkScalar C = src[2] - src[0];
381
382    return eval_quadratic(A, B, C, t);
383}
384
385static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
386    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
387    SkScalar B = src[4] - 2 * src[2] + src[0];
388
389    return SkScalarMulAdd(A, t, B);
390}
391
392void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
393                   SkVector* tangent, SkVector* curvature) {
394    SkASSERT(src);
395    SkASSERT(t >= 0 && t <= SK_Scalar1);
396
397    if (loc) {
398        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
399    }
400    if (tangent) {
401        tangent->set(eval_cubic_derivative(&src[0].fX, t),
402                     eval_cubic_derivative(&src[0].fY, t));
403    }
404    if (curvature) {
405        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
406                       eval_cubic_2ndDerivative(&src[0].fY, t));
407    }
408}
409
410/** Cubic'(t) = At^2 + Bt + C, where
411    A = 3(-a + 3(b - c) + d)
412    B = 6(a - 2b + c)
413    C = 3(b - a)
414    Solve for t, keeping only those that fit betwee 0 < t < 1
415*/
416int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
417                       SkScalar tValues[2]) {
418    // we divide A,B,C by 3 to simplify
419    SkScalar A = d - a + 3*(b - c);
420    SkScalar B = 2*(a - b - b + c);
421    SkScalar C = b - a;
422
423    return SkFindUnitQuadRoots(A, B, C, tValues);
424}
425
426void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
427    SkASSERT(t > 0 && t < SK_Scalar1);
428
429    Sk2s    p0 = from_point(src[0]);
430    Sk2s    p1 = from_point(src[1]);
431    Sk2s    p2 = from_point(src[2]);
432    Sk2s    p3 = from_point(src[3]);
433    Sk2s    tt(t);
434
435    Sk2s    ab = interp(p0, p1, tt);
436    Sk2s    bc = interp(p1, p2, tt);
437    Sk2s    cd = interp(p2, p3, tt);
438    Sk2s    abc = interp(ab, bc, tt);
439    Sk2s    bcd = interp(bc, cd, tt);
440    Sk2s    abcd = interp(abc, bcd, tt);
441
442    dst[0] = src[0];
443    dst[1] = to_point(ab);
444    dst[2] = to_point(abc);
445    dst[3] = to_point(abcd);
446    dst[4] = to_point(bcd);
447    dst[5] = to_point(cd);
448    dst[6] = src[3];
449}
450
451void SkCubicToCoeff(const SkPoint pts[4], SkPoint coeff[4]) {
452    Sk2s p0 = from_point(pts[0]);
453    Sk2s p1 = from_point(pts[1]);
454    Sk2s p2 = from_point(pts[2]);
455    Sk2s p3 = from_point(pts[3]);
456
457    const Sk2s three(3);
458    Sk2s p1minusp2 = p1 - p2;
459
460    Sk2s D = p0;
461    Sk2s A = p3 + three * p1minusp2 - D;
462    Sk2s B = three * (D - p1minusp2 - p1);
463    Sk2s C = three * (p1 - D);
464
465    coeff[0] = to_point(A);
466    coeff[1] = to_point(B);
467    coeff[2] = to_point(C);
468    coeff[3] = to_point(D);
469}
470
471/*  http://code.google.com/p/skia/issues/detail?id=32
472
473    This test code would fail when we didn't check the return result of
474    valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
475    that after the first chop, the parameters to valid_unit_divide are equal
476    (thanks to finite float precision and rounding in the subtracts). Thus
477    even though the 2nd tValue looks < 1.0, after we renormalize it, we end
478    up with 1.0, hence the need to check and just return the last cubic as
479    a degenerate clump of 4 points in the sampe place.
480
481    static void test_cubic() {
482        SkPoint src[4] = {
483            { 556.25000, 523.03003 },
484            { 556.23999, 522.96002 },
485            { 556.21997, 522.89001 },
486            { 556.21997, 522.82001 }
487        };
488        SkPoint dst[10];
489        SkScalar tval[] = { 0.33333334f, 0.99999994f };
490        SkChopCubicAt(src, dst, tval, 2);
491    }
492 */
493
494void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
495                   const SkScalar tValues[], int roots) {
496#ifdef SK_DEBUG
497    {
498        for (int i = 0; i < roots - 1; i++)
499        {
500            SkASSERT(is_unit_interval(tValues[i]));
501            SkASSERT(is_unit_interval(tValues[i+1]));
502            SkASSERT(tValues[i] < tValues[i+1]);
503        }
504    }
505#endif
506
507    if (dst) {
508        if (roots == 0) { // nothing to chop
509            memcpy(dst, src, 4*sizeof(SkPoint));
510        } else {
511            SkScalar    t = tValues[0];
512            SkPoint     tmp[4];
513
514            for (int i = 0; i < roots; i++) {
515                SkChopCubicAt(src, dst, t);
516                if (i == roots - 1) {
517                    break;
518                }
519
520                dst += 3;
521                // have src point to the remaining cubic (after the chop)
522                memcpy(tmp, dst, 4 * sizeof(SkPoint));
523                src = tmp;
524
525                // watch out in case the renormalized t isn't in range
526                if (!valid_unit_divide(tValues[i+1] - tValues[i],
527                                       SK_Scalar1 - tValues[i], &t)) {
528                    // if we can't, just create a degenerate cubic
529                    dst[4] = dst[5] = dst[6] = src[3];
530                    break;
531                }
532            }
533        }
534    }
535}
536
537void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
538    SkChopCubicAt(src, dst, 0.5f);
539}
540
541static void flatten_double_cubic_extrema(SkScalar coords[14]) {
542    coords[4] = coords[8] = coords[6];
543}
544
545/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
546    the resulting beziers are monotonic in Y. This is called by the scan
547    converter.  Depending on what is returned, dst[] is treated as follows:
548    0   dst[0..3] is the original cubic
549    1   dst[0..3] and dst[3..6] are the two new cubics
550    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
551    If dst == null, it is ignored and only the count is returned.
552*/
553int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
554    SkScalar    tValues[2];
555    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
556                                           src[3].fY, tValues);
557
558    SkChopCubicAt(src, dst, tValues, roots);
559    if (dst && roots > 0) {
560        // we do some cleanup to ensure our Y extrema are flat
561        flatten_double_cubic_extrema(&dst[0].fY);
562        if (roots == 2) {
563            flatten_double_cubic_extrema(&dst[3].fY);
564        }
565    }
566    return roots;
567}
568
569int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
570    SkScalar    tValues[2];
571    int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
572                                           src[3].fX, tValues);
573
574    SkChopCubicAt(src, dst, tValues, roots);
575    if (dst && roots > 0) {
576        // we do some cleanup to ensure our Y extrema are flat
577        flatten_double_cubic_extrema(&dst[0].fX);
578        if (roots == 2) {
579            flatten_double_cubic_extrema(&dst[3].fX);
580        }
581    }
582    return roots;
583}
584
585/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
586
587    Inflection means that curvature is zero.
588    Curvature is [F' x F''] / [F'^3]
589    So we solve F'x X F''y - F'y X F''y == 0
590    After some canceling of the cubic term, we get
591    A = b - a
592    B = c - 2b + a
593    C = d - 3c + 3b - a
594    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
595*/
596int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
597    SkScalar    Ax = src[1].fX - src[0].fX;
598    SkScalar    Ay = src[1].fY - src[0].fY;
599    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
600    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
601    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
602    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
603
604    return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
605                               Ax*Cy - Ay*Cx,
606                               Ax*By - Ay*Bx,
607                               tValues);
608}
609
610int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
611    SkScalar    tValues[2];
612    int         count = SkFindCubicInflections(src, tValues);
613
614    if (dst) {
615        if (count == 0) {
616            memcpy(dst, src, 4 * sizeof(SkPoint));
617        } else {
618            SkChopCubicAt(src, dst, tValues, count);
619        }
620    }
621    return count + 1;
622}
623
624// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
625// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
626// Classification:
627// discr(I) > 0        Serpentine
628// discr(I) = 0        Cusp
629// discr(I) < 0        Loop
630// d0 = d1 = 0         Quadratic
631// d0 = d1 = d2 = 0    Line
632// p0 = p1 = p2 = p3   Point
633static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
634    if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
635        return kPoint_SkCubicType;
636    }
637    const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
638    if (discr > SK_ScalarNearlyZero) {
639        return kSerpentine_SkCubicType;
640    } else if (discr < -SK_ScalarNearlyZero) {
641        return kLoop_SkCubicType;
642    } else {
643        if (0.f == d[0] && 0.f == d[1]) {
644            return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
645        } else {
646            return kCusp_SkCubicType;
647        }
648    }
649}
650
651// Assumes the third component of points is 1.
652// Calcs p0 . (p1 x p2)
653static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
654    const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
655    const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
656    const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
657    return (xComp + yComp + wComp);
658}
659
660// Calc coefficients of I(s,t) where roots of I are inflection points of curve
661// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
662// d0 = a1 - 2*a2+3*a3
663// d1 = -a2 + 3*a3
664// d2 = 3*a3
665// a1 = p0 . (p3 x p2)
666// a2 = p1 . (p0 x p3)
667// a3 = p2 . (p1 x p0)
668// Places the values of d1, d2, d3 in array d passed in
669static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
670    SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
671    SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
672    SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
673
674    // need to scale a's or values in later calculations will grow to high
675    SkScalar max = SkScalarAbs(a1);
676    max = SkMaxScalar(max, SkScalarAbs(a2));
677    max = SkMaxScalar(max, SkScalarAbs(a3));
678    max = 1.f/max;
679    a1 = a1 * max;
680    a2 = a2 * max;
681    a3 = a3 * max;
682
683    d[2] = 3.f * a3;
684    d[1] = d[2] - a2;
685    d[0] = d[1] - a2 + a1;
686}
687
688SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
689    calc_cubic_inflection_func(src, d);
690    return classify_cubic(src, d);
691}
692
693template <typename T> void bubble_sort(T array[], int count) {
694    for (int i = count - 1; i > 0; --i)
695        for (int j = i; j > 0; --j)
696            if (array[j] < array[j-1])
697            {
698                T   tmp(array[j]);
699                array[j] = array[j-1];
700                array[j-1] = tmp;
701            }
702}
703
704/**
705 *  Given an array and count, remove all pair-wise duplicates from the array,
706 *  keeping the existing sorting, and return the new count
707 */
708static int collaps_duplicates(SkScalar array[], int count) {
709    for (int n = count; n > 1; --n) {
710        if (array[0] == array[1]) {
711            for (int i = 1; i < n; ++i) {
712                array[i - 1] = array[i];
713            }
714            count -= 1;
715        } else {
716            array += 1;
717        }
718    }
719    return count;
720}
721
722#ifdef SK_DEBUG
723
724#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
725
726static void test_collaps_duplicates() {
727    static bool gOnce;
728    if (gOnce) { return; }
729    gOnce = true;
730    const SkScalar src0[] = { 0 };
731    const SkScalar src1[] = { 0, 0 };
732    const SkScalar src2[] = { 0, 1 };
733    const SkScalar src3[] = { 0, 0, 0 };
734    const SkScalar src4[] = { 0, 0, 1 };
735    const SkScalar src5[] = { 0, 1, 1 };
736    const SkScalar src6[] = { 0, 1, 2 };
737    const struct {
738        const SkScalar* fData;
739        int fCount;
740        int fCollapsedCount;
741    } data[] = {
742        { TEST_COLLAPS_ENTRY(src0), 1 },
743        { TEST_COLLAPS_ENTRY(src1), 1 },
744        { TEST_COLLAPS_ENTRY(src2), 2 },
745        { TEST_COLLAPS_ENTRY(src3), 1 },
746        { TEST_COLLAPS_ENTRY(src4), 2 },
747        { TEST_COLLAPS_ENTRY(src5), 2 },
748        { TEST_COLLAPS_ENTRY(src6), 3 },
749    };
750    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
751        SkScalar dst[3];
752        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
753        int count = collaps_duplicates(dst, data[i].fCount);
754        SkASSERT(data[i].fCollapsedCount == count);
755        for (int j = 1; j < count; ++j) {
756            SkASSERT(dst[j-1] < dst[j]);
757        }
758    }
759}
760#endif
761
762static SkScalar SkScalarCubeRoot(SkScalar x) {
763    return SkScalarPow(x, 0.3333333f);
764}
765
766/*  Solve coeff(t) == 0, returning the number of roots that
767    lie withing 0 < t < 1.
768    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
769
770    Eliminates repeated roots (so that all tValues are distinct, and are always
771    in increasing order.
772*/
773static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
774    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
775        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
776    }
777
778    SkScalar a, b, c, Q, R;
779
780    {
781        SkASSERT(coeff[0] != 0);
782
783        SkScalar inva = SkScalarInvert(coeff[0]);
784        a = coeff[1] * inva;
785        b = coeff[2] * inva;
786        c = coeff[3] * inva;
787    }
788    Q = (a*a - b*3) / 9;
789    R = (2*a*a*a - 9*a*b + 27*c) / 54;
790
791    SkScalar Q3 = Q * Q * Q;
792    SkScalar R2MinusQ3 = R * R - Q3;
793    SkScalar adiv3 = a / 3;
794
795    SkScalar*   roots = tValues;
796    SkScalar    r;
797
798    if (R2MinusQ3 < 0) { // we have 3 real roots
799        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
800        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
801
802        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
803        if (is_unit_interval(r)) {
804            *roots++ = r;
805        }
806        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
807        if (is_unit_interval(r)) {
808            *roots++ = r;
809        }
810        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
811        if (is_unit_interval(r)) {
812            *roots++ = r;
813        }
814        SkDEBUGCODE(test_collaps_duplicates();)
815
816        // now sort the roots
817        int count = (int)(roots - tValues);
818        SkASSERT((unsigned)count <= 3);
819        bubble_sort(tValues, count);
820        count = collaps_duplicates(tValues, count);
821        roots = tValues + count;    // so we compute the proper count below
822    } else {              // we have 1 real root
823        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
824        A = SkScalarCubeRoot(A);
825        if (R > 0) {
826            A = -A;
827        }
828        if (A != 0) {
829            A += Q / A;
830        }
831        r = A - adiv3;
832        if (is_unit_interval(r)) {
833            *roots++ = r;
834        }
835    }
836
837    return (int)(roots - tValues);
838}
839
840/*  Looking for F' dot F'' == 0
841
842    A = b - a
843    B = c - 2b + a
844    C = d - 3c + 3b - a
845
846    F' = 3Ct^2 + 6Bt + 3A
847    F'' = 6Ct + 6B
848
849    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
850*/
851static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
852    SkScalar    a = src[2] - src[0];
853    SkScalar    b = src[4] - 2 * src[2] + src[0];
854    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
855
856    coeff[0] = c * c;
857    coeff[1] = 3 * b * c;
858    coeff[2] = 2 * b * b + c * a;
859    coeff[3] = a * b;
860}
861
862/*  Looking for F' dot F'' == 0
863
864    A = b - a
865    B = c - 2b + a
866    C = d - 3c + 3b - a
867
868    F' = 3Ct^2 + 6Bt + 3A
869    F'' = 6Ct + 6B
870
871    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
872*/
873int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
874    SkScalar coeffX[4], coeffY[4];
875    int      i;
876
877    formulate_F1DotF2(&src[0].fX, coeffX);
878    formulate_F1DotF2(&src[0].fY, coeffY);
879
880    for (i = 0; i < 4; i++) {
881        coeffX[i] += coeffY[i];
882    }
883
884    SkScalar    t[3];
885    int         count = solve_cubic_poly(coeffX, t);
886    int         maxCount = 0;
887
888    // now remove extrema where the curvature is zero (mins)
889    // !!!! need a test for this !!!!
890    for (i = 0; i < count; i++) {
891        // if (not_min_curvature())
892        if (t[i] > 0 && t[i] < SK_Scalar1) {
893            tValues[maxCount++] = t[i];
894        }
895    }
896    return maxCount;
897}
898
899int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
900                              SkScalar tValues[3]) {
901    SkScalar    t_storage[3];
902
903    if (tValues == NULL) {
904        tValues = t_storage;
905    }
906
907    int count = SkFindCubicMaxCurvature(src, tValues);
908
909    if (dst) {
910        if (count == 0) {
911            memcpy(dst, src, 4 * sizeof(SkPoint));
912        } else {
913            SkChopCubicAt(src, dst, tValues, count);
914        }
915    }
916    return count + 1;
917}
918
919#include "../pathops/SkPathOpsCubic.h"
920
921typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const;
922
923static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7],
924                                     InterceptProc method) {
925    SkDCubic cubic;
926    double roots[3];
927    int count = (cubic.set(src).*method)(intercept, roots);
928    if (count > 0) {
929        SkDCubicPair pair = cubic.chopAt(roots[0]);
930        for (int i = 0; i < 7; ++i) {
931            dst[i] = pair.pts[i].asSkPoint();
932        }
933        return true;
934    }
935    return false;
936}
937
938bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) {
939    return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect);
940}
941
942bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) {
943    return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect);
944}
945
946///////////////////////////////////////////////////////////////////////////////
947
948/*  Find t value for quadratic [a, b, c] = d.
949    Return 0 if there is no solution within [0, 1)
950*/
951static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
952    // At^2 + Bt + C = d
953    SkScalar A = a - 2 * b + c;
954    SkScalar B = 2 * (b - a);
955    SkScalar C = a - d;
956
957    SkScalar    roots[2];
958    int         count = SkFindUnitQuadRoots(A, B, C, roots);
959
960    SkASSERT(count <= 1);
961    return count == 1 ? roots[0] : 0;
962}
963
964/*  given a quad-curve and a point (x,y), chop the quad at that point and place
965    the new off-curve point and endpoint into 'dest'.
966    Should only return false if the computed pos is the start of the curve
967    (i.e. root == 0)
968*/
969static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
970                                SkPoint* dest) {
971    const SkScalar* base;
972    SkScalar        value;
973
974    if (SkScalarAbs(x) < SkScalarAbs(y)) {
975        base = &quad[0].fX;
976        value = x;
977    } else {
978        base = &quad[0].fY;
979        value = y;
980    }
981
982    // note: this returns 0 if it thinks value is out of range, meaning the
983    // root might return something outside of [0, 1)
984    SkScalar t = quad_solve(base[0], base[2], base[4], value);
985
986    if (t > 0) {
987        SkPoint tmp[5];
988        SkChopQuadAt(quad, tmp, t);
989        dest[0] = tmp[1];
990        dest[1].set(x, y);
991        return true;
992    } else {
993        /*  t == 0 means either the value triggered a root outside of [0, 1)
994            For our purposes, we can ignore the <= 0 roots, but we want to
995            catch the >= 1 roots (which given our caller, will basically mean
996            a root of 1, give-or-take numerical instability). If we are in the
997            >= 1 case, return the existing offCurve point.
998
999            The test below checks to see if we are close to the "end" of the
1000            curve (near base[4]). Rather than specifying a tolerance, I just
1001            check to see if value is on to the right/left of the middle point
1002            (depending on the direction/sign of the end points).
1003        */
1004        if ((base[0] < base[4] && value > base[2]) ||
1005            (base[0] > base[4] && value < base[2]))   // should root have been 1
1006        {
1007            dest[0] = quad[1];
1008            dest[1].set(x, y);
1009            return true;
1010        }
1011    }
1012    return false;
1013}
1014
1015static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
1016// The mid point of the quadratic arc approximation is half way between the two
1017// control points. The float epsilon adjustment moves the on curve point out by
1018// two bits, distributing the convex test error between the round rect
1019// approximation and the convex cross product sign equality test.
1020#define SK_MID_RRECT_OFFSET \
1021    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1022    { SK_Scalar1,            0                      },
1023    { SK_Scalar1,            SK_ScalarTanPIOver8    },
1024    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
1025    { SK_ScalarTanPIOver8,   SK_Scalar1             },
1026
1027    { 0,                     SK_Scalar1             },
1028    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
1029    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
1030    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
1031
1032    { -SK_Scalar1,           0                      },
1033    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
1034    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
1035    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
1036
1037    { 0,                     -SK_Scalar1            },
1038    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
1039    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
1040    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
1041
1042    { SK_Scalar1,            0                      }
1043#undef SK_MID_RRECT_OFFSET
1044};
1045
1046int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1047                   SkRotationDirection dir, const SkMatrix* userMatrix,
1048                   SkPoint quadPoints[]) {
1049    // rotate by x,y so that uStart is (1.0)
1050    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1051    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1052
1053    SkScalar absX = SkScalarAbs(x);
1054    SkScalar absY = SkScalarAbs(y);
1055
1056    int pointCount;
1057
1058    // check for (effectively) coincident vectors
1059    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1060    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1061    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1062        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1063         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1064
1065        // just return the start-point
1066        quadPoints[0].set(SK_Scalar1, 0);
1067        pointCount = 1;
1068    } else {
1069        if (dir == kCCW_SkRotationDirection) {
1070            y = -y;
1071        }
1072        // what octant (quadratic curve) is [xy] in?
1073        int oct = 0;
1074        bool sameSign = true;
1075
1076        if (0 == y) {
1077            oct = 4;        // 180
1078            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1079        } else if (0 == x) {
1080            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1081            oct = y > 0 ? 2 : 6; // 90 : 270
1082        } else {
1083            if (y < 0) {
1084                oct += 4;
1085            }
1086            if ((x < 0) != (y < 0)) {
1087                oct += 2;
1088                sameSign = false;
1089            }
1090            if ((absX < absY) == sameSign) {
1091                oct += 1;
1092            }
1093        }
1094
1095        int wholeCount = oct << 1;
1096        memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1097
1098        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1099        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1100            wholeCount += 2;
1101        }
1102        pointCount = wholeCount + 1;
1103    }
1104
1105    // now handle counter-clockwise and the initial unitStart rotation
1106    SkMatrix    matrix;
1107    matrix.setSinCos(uStart.fY, uStart.fX);
1108    if (dir == kCCW_SkRotationDirection) {
1109        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1110    }
1111    if (userMatrix) {
1112        matrix.postConcat(*userMatrix);
1113    }
1114    matrix.mapPoints(quadPoints, pointCount);
1115    return pointCount;
1116}
1117
1118
1119///////////////////////////////////////////////////////////////////////////////
1120//
1121// NURB representation for conics.  Helpful explanations at:
1122//
1123// http://citeseerx.ist.psu.edu/viewdoc/
1124//   download?doi=10.1.1.44.5740&rep=rep1&type=ps
1125// and
1126// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1127//
1128// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1129//     ------------------------------------------
1130//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1131//
1132//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1133//     ------------------------------------------------
1134//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1135//
1136
1137static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1138    SkASSERT(src);
1139    SkASSERT(t >= 0 && t <= SK_Scalar1);
1140
1141    SkScalar    src2w = SkScalarMul(src[2], w);
1142    SkScalar    C = src[0];
1143    SkScalar    A = src[4] - 2 * src2w + C;
1144    SkScalar    B = 2 * (src2w - C);
1145    SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1146
1147    B = 2 * (w - SK_Scalar1);
1148    C = SK_Scalar1;
1149    A = -B;
1150    SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1151
1152    return SkScalarDiv(numer, denom);
1153}
1154
1155// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1156//
1157//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1158//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1159//  t^0 : -2 P0 w + 2 P1 w
1160//
1161//  We disregard magnitude, so we can freely ignore the denominator of F', and
1162//  divide the numerator by 2
1163//
1164//    coeff[0] for t^2
1165//    coeff[1] for t^1
1166//    coeff[2] for t^0
1167//
1168static void conic_deriv_coeff(const SkScalar src[],
1169                              SkScalar w,
1170                              SkScalar coeff[3]) {
1171    const SkScalar P20 = src[4] - src[0];
1172    const SkScalar P10 = src[2] - src[0];
1173    const SkScalar wP10 = w * P10;
1174    coeff[0] = w * P20 - P20;
1175    coeff[1] = P20 - 2 * wP10;
1176    coeff[2] = wP10;
1177}
1178
1179static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1180    SkScalar coeff[3];
1181    conic_deriv_coeff(coord, w, coeff);
1182    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1183}
1184
1185static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1186    SkScalar coeff[3];
1187    conic_deriv_coeff(src, w, coeff);
1188
1189    SkScalar tValues[2];
1190    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1191    SkASSERT(0 == roots || 1 == roots);
1192
1193    if (1 == roots) {
1194        *t = tValues[0];
1195        return true;
1196    }
1197    return false;
1198}
1199
1200struct SkP3D {
1201    SkScalar fX, fY, fZ;
1202
1203    void set(SkScalar x, SkScalar y, SkScalar z) {
1204        fX = x; fY = y; fZ = z;
1205    }
1206
1207    void projectDown(SkPoint* dst) const {
1208        dst->set(fX / fZ, fY / fZ);
1209    }
1210};
1211
1212// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1213static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1214    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1215    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1216    dst[0] = ab;
1217    dst[3] = SkScalarInterp(ab, bc, t);
1218    dst[6] = bc;
1219}
1220
1221static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1222    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1223    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1224    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1225}
1226
1227void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1228    SkASSERT(t >= 0 && t <= SK_Scalar1);
1229
1230    if (pt) {
1231        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1232                conic_eval_pos(&fPts[0].fY, fW, t));
1233    }
1234    if (tangent) {
1235        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1236                     conic_eval_tan(&fPts[0].fY, fW, t));
1237    }
1238}
1239
1240void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1241    SkP3D tmp[3], tmp2[3];
1242
1243    ratquad_mapTo3D(fPts, fW, tmp);
1244
1245    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1246    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1247    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1248
1249    dst[0].fPts[0] = fPts[0];
1250    tmp2[0].projectDown(&dst[0].fPts[1]);
1251    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1252    tmp2[2].projectDown(&dst[1].fPts[1]);
1253    dst[1].fPts[2] = fPts[2];
1254
1255    // to put in "standard form", where w0 and w2 are both 1, we compute the
1256    // new w1 as sqrt(w1*w1/w0*w2)
1257    // or
1258    // w1 /= sqrt(w0*w2)
1259    //
1260    // However, in our case, we know that for dst[0]:
1261    //     w0 == 1, and for dst[1], w2 == 1
1262    //
1263    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1264    dst[0].fW = tmp2[0].fZ / root;
1265    dst[1].fW = tmp2[2].fZ / root;
1266}
1267
1268static Sk2s times_2(const Sk2s& value) {
1269    return value + value;
1270}
1271
1272SkPoint SkConic::evalAt(SkScalar t) const {
1273    Sk2s p0 = from_point(fPts[0]);
1274    Sk2s p1 = from_point(fPts[1]);
1275    Sk2s p2 = from_point(fPts[2]);
1276    Sk2s tt(t);
1277    Sk2s ww(fW);
1278    Sk2s one(1);
1279
1280    Sk2s p1w = p1 * ww;
1281    Sk2s C = p0;
1282    Sk2s A = p2 - times_2(p1w) + p0;
1283    Sk2s B = times_2(p1w - C);
1284    Sk2s numer = quad_poly_eval(A, B, C, tt);
1285
1286    B = times_2(ww - one);
1287    A = -B;
1288    Sk2s denom = quad_poly_eval(A, B, one, tt);
1289
1290    return to_point(numer / denom);
1291}
1292
1293SkVector SkConic::evalTangentAt(SkScalar t) const {
1294    Sk2s p0 = from_point(fPts[0]);
1295    Sk2s p1 = from_point(fPts[1]);
1296    Sk2s p2 = from_point(fPts[2]);
1297    Sk2s ww(fW);
1298
1299    Sk2s p20 = p2 - p0;
1300    Sk2s p10 = p1 - p0;
1301
1302    Sk2s C = ww * p10;
1303    Sk2s A = ww * p20 - p20;
1304    Sk2s B = p20 - C - C;
1305
1306    return to_vector(quad_poly_eval(A, B, C, Sk2s(t)));
1307}
1308
1309static SkScalar subdivide_w_value(SkScalar w) {
1310    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1311}
1312
1313static Sk2s twice(const Sk2s& value) {
1314    return value + value;
1315}
1316
1317void SkConic::chop(SkConic * SK_RESTRICT dst) const {
1318    Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW));
1319    SkScalar newW = subdivide_w_value(fW);
1320
1321    Sk2s p0 = from_point(fPts[0]);
1322    Sk2s p1 = from_point(fPts[1]);
1323    Sk2s p2 = from_point(fPts[2]);
1324    Sk2s ww(fW);
1325
1326    Sk2s wp1 = ww * p1;
1327    Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f);
1328
1329    dst[0].fPts[0] = fPts[0];
1330    dst[0].fPts[1] = to_point((p0 + wp1) * scale);
1331    dst[0].fPts[2] = dst[1].fPts[0] = to_point(m);
1332    dst[1].fPts[1] = to_point((wp1 + p2) * scale);
1333    dst[1].fPts[2] = fPts[2];
1334
1335    dst[0].fW = dst[1].fW = newW;
1336}
1337
1338/*
1339 *  "High order approximation of conic sections by quadratic splines"
1340 *      by Michael Floater, 1993
1341 */
1342#define AS_QUAD_ERROR_SETUP                                         \
1343    SkScalar a = fW - 1;                                            \
1344    SkScalar k = a / (4 * (2 + a));                                 \
1345    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1346    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1347
1348void SkConic::computeAsQuadError(SkVector* err) const {
1349    AS_QUAD_ERROR_SETUP
1350    err->set(x, y);
1351}
1352
1353bool SkConic::asQuadTol(SkScalar tol) const {
1354    AS_QUAD_ERROR_SETUP
1355    return (x * x + y * y) <= tol * tol;
1356}
1357
1358// Limit the number of suggested quads to approximate a conic
1359#define kMaxConicToQuadPOW2     5
1360
1361int SkConic::computeQuadPOW2(SkScalar tol) const {
1362    if (tol < 0 || !SkScalarIsFinite(tol)) {
1363        return 0;
1364    }
1365
1366    AS_QUAD_ERROR_SETUP
1367
1368    SkScalar error = SkScalarSqrt(x * x + y * y);
1369    int pow2;
1370    for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
1371        if (error <= tol) {
1372            break;
1373        }
1374        error *= 0.25f;
1375    }
1376    // float version -- using ceil gives the same results as the above.
1377    if (false) {
1378        SkScalar err = SkScalarSqrt(x * x + y * y);
1379        if (err <= tol) {
1380            return 0;
1381        }
1382        SkScalar tol2 = tol * tol;
1383        if (tol2 == 0) {
1384            return kMaxConicToQuadPOW2;
1385        }
1386        SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
1387        int altPow2 = SkScalarCeilToInt(fpow2);
1388        if (altPow2 != pow2) {
1389            SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
1390        }
1391        pow2 = altPow2;
1392    }
1393    return pow2;
1394}
1395
1396static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1397    SkASSERT(level >= 0);
1398
1399    if (0 == level) {
1400        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1401        return pts + 2;
1402    } else {
1403        SkConic dst[2];
1404        src.chop(dst);
1405        --level;
1406        pts = subdivide(dst[0], pts, level);
1407        return subdivide(dst[1], pts, level);
1408    }
1409}
1410
1411int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1412    SkASSERT(pow2 >= 0);
1413    *pts = fPts[0];
1414    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1415    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1416    return 1 << pow2;
1417}
1418
1419bool SkConic::findXExtrema(SkScalar* t) const {
1420    return conic_find_extrema(&fPts[0].fX, fW, t);
1421}
1422
1423bool SkConic::findYExtrema(SkScalar* t) const {
1424    return conic_find_extrema(&fPts[0].fY, fW, t);
1425}
1426
1427bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1428    SkScalar t;
1429    if (this->findXExtrema(&t)) {
1430        this->chopAt(t, dst);
1431        // now clean-up the middle, since we know t was meant to be at
1432        // an X-extrema
1433        SkScalar value = dst[0].fPts[2].fX;
1434        dst[0].fPts[1].fX = value;
1435        dst[1].fPts[0].fX = value;
1436        dst[1].fPts[1].fX = value;
1437        return true;
1438    }
1439    return false;
1440}
1441
1442bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1443    SkScalar t;
1444    if (this->findYExtrema(&t)) {
1445        this->chopAt(t, dst);
1446        // now clean-up the middle, since we know t was meant to be at
1447        // an Y-extrema
1448        SkScalar value = dst[0].fPts[2].fY;
1449        dst[0].fPts[1].fY = value;
1450        dst[1].fPts[0].fY = value;
1451        dst[1].fPts[1].fY = value;
1452        return true;
1453    }
1454    return false;
1455}
1456
1457void SkConic::computeTightBounds(SkRect* bounds) const {
1458    SkPoint pts[4];
1459    pts[0] = fPts[0];
1460    pts[1] = fPts[2];
1461    int count = 2;
1462
1463    SkScalar t;
1464    if (this->findXExtrema(&t)) {
1465        this->evalAt(t, &pts[count++]);
1466    }
1467    if (this->findYExtrema(&t)) {
1468        this->evalAt(t, &pts[count++]);
1469    }
1470    bounds->set(pts, count);
1471}
1472
1473void SkConic::computeFastBounds(SkRect* bounds) const {
1474    bounds->set(fPts, 3);
1475}
1476
1477bool SkConic::findMaxCurvature(SkScalar* t) const {
1478    // TODO: Implement me
1479    return false;
1480}
1481
1482SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1483                             const SkMatrix& matrix) {
1484    if (!matrix.hasPerspective()) {
1485        return w;
1486    }
1487
1488    SkP3D src[3], dst[3];
1489
1490    ratquad_mapTo3D(pts, w, src);
1491
1492    matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1493
1494    // w' = sqrt(w1*w1/w0*w2)
1495    SkScalar w0 = dst[0].fZ;
1496    SkScalar w1 = dst[1].fZ;
1497    SkScalar w2 = dst[2].fZ;
1498    w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1499    return w;
1500}
1501
1502int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
1503                          const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
1504    // rotate by x,y so that uStart is (1.0)
1505    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1506    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1507
1508    SkScalar absY = SkScalarAbs(y);
1509
1510    // check for (effectively) coincident vectors
1511    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1512    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1513    if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) ||
1514                                                 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1515        return 0;
1516    }
1517
1518    if (dir == kCCW_SkRotationDirection) {
1519        y = -y;
1520    }
1521
1522    // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
1523    //      0 == [0  .. 90)
1524    //      1 == [90 ..180)
1525    //      2 == [180..270)
1526    //      3 == [270..360)
1527    //
1528    int quadrant = 0;
1529    if (0 == y) {
1530        quadrant = 2;        // 180
1531        SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1532    } else if (0 == x) {
1533        SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1534        quadrant = y > 0 ? 1 : 3; // 90 : 270
1535    } else {
1536        if (y < 0) {
1537            quadrant += 2;
1538        }
1539        if ((x < 0) != (y < 0)) {
1540            quadrant += 1;
1541        }
1542    }
1543
1544    const SkPoint quadrantPts[] = {
1545        { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
1546    };
1547    const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
1548
1549    int conicCount = quadrant;
1550    for (int i = 0; i < conicCount; ++i) {
1551        dst[i].set(&quadrantPts[i * 2], quadrantWeight);
1552    }
1553
1554    // Now compute any remaing (sub-90-degree) arc for the last conic
1555    const SkPoint finalP = { x, y };
1556    const SkPoint& lastQ = quadrantPts[quadrant * 2];  // will already be a unit-vector
1557    const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
1558    SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
1559
1560    if (dot < 1) {
1561        SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
1562        // compute the bisector vector, and then rescale to be the off-curve point.
1563        // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
1564        // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
1565        // This is nice, since our computed weight is cos(theta/2) as well!
1566        //
1567        const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
1568        offCurve.setLength(SkScalarInvert(cosThetaOver2));
1569        dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
1570        conicCount += 1;
1571    }
1572
1573    // now handle counter-clockwise and the initial unitStart rotation
1574    SkMatrix    matrix;
1575    matrix.setSinCos(uStart.fY, uStart.fX);
1576    if (dir == kCCW_SkRotationDirection) {
1577        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1578    }
1579    if (userMatrix) {
1580        matrix.postConcat(*userMatrix);
1581    }
1582    for (int i = 0; i < conicCount; ++i) {
1583        matrix.mapPoints(dst[i].fPts, 3);
1584    }
1585    return conicCount;
1586}
1587