SkGeometry.cpp revision 562d0e1cd2286945cb73fca0233560071b052129 
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
11/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
12    involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
13    May also introduce overflow of fixed when we compute our setup.
14*/
15//    #define DIRECT_EVAL_OF_POLYNOMIALS
16
17////////////////////////////////////////////////////////////////////////
18
19static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
20    SkScalar ab = a - b;
21    SkScalar bc = b - c;
22    if (ab < 0) {
23        bc = -bc;
24    }
25    return ab == 0 || bc < 0;
26}
27
28////////////////////////////////////////////////////////////////////////
29
30static bool is_unit_interval(SkScalar x) {
31    return x > 0 && x < SK_Scalar1;
32}
33
34static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
35    SkASSERT(ratio);
36
37    if (numer < 0) {
38        numer = -numer;
39        denom = -denom;
40    }
41
42    if (denom == 0 || numer == 0 || numer >= denom) {
43        return 0;
44    }
45
46    SkScalar r = SkScalarDiv(numer, denom);
47    if (SkScalarIsNaN(r)) {
48        return 0;
49    }
50    SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
51    if (r == 0) { // catch underflow if numer <<<< denom
52        return 0;
53    }
54    *ratio = r;
55    return 1;
56}
57
58/** From Numerical Recipes in C.
59
60    Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
61    x1 = Q / A
62    x2 = C / Q
63*/
64int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
65    SkASSERT(roots);
66
67    if (A == 0) {
68        return valid_unit_divide(-C, B, roots);
69    }
70
71    SkScalar* r = roots;
72
73    SkScalar R = B*B - 4*A*C;
74    if (R < 0 || SkScalarIsNaN(R)) {  // complex roots
75        return 0;
76    }
77    R = SkScalarSqrt(R);
78
79    SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
80    r += valid_unit_divide(Q, A, r);
81    r += valid_unit_divide(C, Q, r);
82    if (r - roots == 2) {
83        if (roots[0] > roots[1])
84            SkTSwap<SkScalar>(roots[0], roots[1]);
85        else if (roots[0] == roots[1])  // nearly-equal?
86            r -= 1; // skip the double root
87    }
88    return (int)(r - roots);
89}
90
91///////////////////////////////////////////////////////////////////////////////
92///////////////////////////////////////////////////////////////////////////////
93
94static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
95    SkASSERT(src);
96    SkASSERT(t >= 0 && t <= SK_Scalar1);
97
98#ifdef DIRECT_EVAL_OF_POLYNOMIALS
99    SkScalar    C = src[0];
100    SkScalar    A = src[4] - 2 * src[2] + C;
101    SkScalar    B = 2 * (src[2] - C);
102    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
103#else
104    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
105    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
106    return SkScalarInterp(ab, bc, t);
107#endif
108}
109
110static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
111    SkScalar A = src[4] - 2 * src[2] + src[0];
112    SkScalar B = src[2] - src[0];
113
114    return 2 * SkScalarMulAdd(A, t, B);
115}
116
117static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) {
118    SkScalar A = src[4] - 2 * src[2] + src[0];
119    SkScalar B = src[2] - src[0];
120    return A + 2 * B;
121}
122
123void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt,
124                  SkVector* tangent) {
125    SkASSERT(src);
126    SkASSERT(t >= 0 && t <= SK_Scalar1);
127
128    if (pt) {
129        pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
130    }
131    if (tangent) {
132        tangent->set(eval_quad_derivative(&src[0].fX, t),
133                     eval_quad_derivative(&src[0].fY, t));
134    }
135}
136
137void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) {
138    SkASSERT(src);
139
140    if (pt) {
141        SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
142        SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
143        SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
144        SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
145        pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
146    }
147    if (tangent) {
148        tangent->set(eval_quad_derivative_at_half(&src[0].fX),
149                     eval_quad_derivative_at_half(&src[0].fY));
150    }
151}
152
153static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) {
154    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
155    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
156
157    dst[0] = src[0];
158    dst[2] = ab;
159    dst[4] = SkScalarInterp(ab, bc, t);
160    dst[6] = bc;
161    dst[8] = src[4];
162}
163
164void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
165    SkASSERT(t > 0 && t < SK_Scalar1);
166
167    interp_quad_coords(&src[0].fX, &dst[0].fX, t);
168    interp_quad_coords(&src[0].fY, &dst[0].fY, t);
169}
170
171void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
172    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
173    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
174    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
175    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
176
177    dst[0] = src[0];
178    dst[1].set(x01, y01);
179    dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
180    dst[3].set(x12, y12);
181    dst[4] = src[2];
182}
183
184/** Quad'(t) = At + B, where
185    A = 2(a - 2b + c)
186    B = 2(b - a)
187    Solve for t, only if it fits between 0 < t < 1
188*/
189int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
190    /*  At + B == 0
191        t = -B / A
192    */
193    return valid_unit_divide(a - b, a - b - b + c, tValue);
194}
195
196static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
197    coords[2] = coords[6] = coords[4];
198}
199
200/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
201 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
202 */
203int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
204    SkASSERT(src);
205    SkASSERT(dst);
206
207    SkScalar a = src[0].fY;
208    SkScalar b = src[1].fY;
209    SkScalar c = src[2].fY;
210
211    if (is_not_monotonic(a, b, c)) {
212        SkScalar    tValue;
213        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
214            SkChopQuadAt(src, dst, tValue);
215            flatten_double_quad_extrema(&dst[0].fY);
216            return 1;
217        }
218        // if we get here, we need to force dst to be monotonic, even though
219        // we couldn't compute a unit_divide value (probably underflow).
220        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
221    }
222    dst[0].set(src[0].fX, a);
223    dst[1].set(src[1].fX, b);
224    dst[2].set(src[2].fX, c);
225    return 0;
226}
227
228/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
229    stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
230 */
231int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
232    SkASSERT(src);
233    SkASSERT(dst);
234
235    SkScalar a = src[0].fX;
236    SkScalar b = src[1].fX;
237    SkScalar c = src[2].fX;
238
239    if (is_not_monotonic(a, b, c)) {
240        SkScalar tValue;
241        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
242            SkChopQuadAt(src, dst, tValue);
243            flatten_double_quad_extrema(&dst[0].fX);
244            return 1;
245        }
246        // if we get here, we need to force dst to be monotonic, even though
247        // we couldn't compute a unit_divide value (probably underflow).
248        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
249    }
250    dst[0].set(a, src[0].fY);
251    dst[1].set(b, src[1].fY);
252    dst[2].set(c, src[2].fY);
253    return 0;
254}
255
256//  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
257//  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
258//  F''(t)  = 2 (a - 2b + c)
259//
260//  A = 2 (b - a)
261//  B = 2 (a - 2b + c)
262//
263//  Maximum curvature for a quadratic means solving
264//  Fx' Fx'' + Fy' Fy'' = 0
265//
266//  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
267//
268SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
269    SkScalar    Ax = src[1].fX - src[0].fX;
270    SkScalar    Ay = src[1].fY - src[0].fY;
271    SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
272    SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
273    SkScalar    t = 0;  // 0 means don't chop
274
275    (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
276    return t;
277}
278
279int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
280    SkScalar t = SkFindQuadMaxCurvature(src);
281    if (t == 0) {
282        memcpy(dst, src, 3 * sizeof(SkPoint));
283        return 1;
284    } else {
285        SkChopQuadAt(src, dst, t);
286        return 2;
287    }
288}
289
290#define SK_ScalarTwoThirds  (0.666666666f)
291
292void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
293    const SkScalar scale = SK_ScalarTwoThirds;
294    dst[0] = src[0];
295    dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
296               src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
297    dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
298               src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
299    dst[3] = src[2];
300}
301
302//////////////////////////////////////////////////////////////////////////////
303///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
304//////////////////////////////////////////////////////////////////////////////
305
306static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
307    coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
308    coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
309    coeff[2] = 3*(pt[2] - pt[0]);
310    coeff[3] = pt[0];
311}
312
313void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
314    SkASSERT(pts);
315
316    if (cx) {
317        get_cubic_coeff(&pts[0].fX, cx);
318    }
319    if (cy) {
320        get_cubic_coeff(&pts[0].fY, cy);
321    }
322}
323
324static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
325    SkASSERT(src);
326    SkASSERT(t >= 0 && t <= SK_Scalar1);
327
328    if (t == 0) {
329        return src[0];
330    }
331
332#ifdef DIRECT_EVAL_OF_POLYNOMIALS
333    SkScalar D = src[0];
334    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
335    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
336    SkScalar C = 3*(src[2] - D);
337
338    return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
339#else
340    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
341    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
342    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
343    SkScalar    abc = SkScalarInterp(ab, bc, t);
344    SkScalar    bcd = SkScalarInterp(bc, cd, t);
345    return SkScalarInterp(abc, bcd, t);
346#endif
347}
348
349/** return At^2 + Bt + C
350*/
351static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
352    SkASSERT(t >= 0 && t <= SK_Scalar1);
353
354    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
355}
356
357static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
358    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
359    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
360    SkScalar C = src[2] - src[0];
361
362    return eval_quadratic(A, B, C, t);
363}
364
365static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
366    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
367    SkScalar B = src[4] - 2 * src[2] + src[0];
368
369    return SkScalarMulAdd(A, t, B);
370}
371
372void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
373                   SkVector* tangent, SkVector* curvature) {
374    SkASSERT(src);
375    SkASSERT(t >= 0 && t <= SK_Scalar1);
376
377    if (loc) {
378        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
379    }
380    if (tangent) {
381        tangent->set(eval_cubic_derivative(&src[0].fX, t),
382                     eval_cubic_derivative(&src[0].fY, t));
383    }
384    if (curvature) {
385        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
386                       eval_cubic_2ndDerivative(&src[0].fY, t));
387    }
388}
389
390/** Cubic'(t) = At^2 + Bt + C, where
391    A = 3(-a + 3(b - c) + d)
392    B = 6(a - 2b + c)
393    C = 3(b - a)
394    Solve for t, keeping only those that fit betwee 0 < t < 1
395*/
396int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
397                       SkScalar tValues[2]) {
398    // we divide A,B,C by 3 to simplify
399    SkScalar A = d - a + 3*(b - c);
400    SkScalar B = 2*(a - b - b + c);
401    SkScalar C = b - a;
402
403    return SkFindUnitQuadRoots(A, B, C, tValues);
404}
405
406static void interp_cubic_coords(const SkScalar* src, SkScalar* dst,
407                                SkScalar t) {
408    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
409    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
410    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
411    SkScalar    abc = SkScalarInterp(ab, bc, t);
412    SkScalar    bcd = SkScalarInterp(bc, cd, t);
413    SkScalar    abcd = SkScalarInterp(abc, bcd, t);
414
415    dst[0] = src[0];
416    dst[2] = ab;
417    dst[4] = abc;
418    dst[6] = abcd;
419    dst[8] = bcd;
420    dst[10] = cd;
421    dst[12] = src[6];
422}
423
424void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
425    SkASSERT(t > 0 && t < SK_Scalar1);
426
427    interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
428    interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
429}
430
431/*  http://code.google.com/p/skia/issues/detail?id=32
432
433    This test code would fail when we didn't check the return result of
434    valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
435    that after the first chop, the parameters to valid_unit_divide are equal
436    (thanks to finite float precision and rounding in the subtracts). Thus
437    even though the 2nd tValue looks < 1.0, after we renormalize it, we end
438    up with 1.0, hence the need to check and just return the last cubic as
439    a degenerate clump of 4 points in the sampe place.
440
441    static void test_cubic() {
442        SkPoint src[4] = {
443            { 556.25000, 523.03003 },
444            { 556.23999, 522.96002 },
445            { 556.21997, 522.89001 },
446            { 556.21997, 522.82001 }
447        };
448        SkPoint dst[10];
449        SkScalar tval[] = { 0.33333334f, 0.99999994f };
450        SkChopCubicAt(src, dst, tval, 2);
451    }
452 */
453
454void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
455                   const SkScalar tValues[], int roots) {
456#ifdef SK_DEBUG
457    {
458        for (int i = 0; i < roots - 1; i++)
459        {
460            SkASSERT(is_unit_interval(tValues[i]));
461            SkASSERT(is_unit_interval(tValues[i+1]));
462            SkASSERT(tValues[i] < tValues[i+1]);
463        }
464    }
465#endif
466
467    if (dst) {
468        if (roots == 0) { // nothing to chop
469            memcpy(dst, src, 4*sizeof(SkPoint));
470        } else {
471            SkScalar    t = tValues[0];
472            SkPoint     tmp[4];
473
474            for (int i = 0; i < roots; i++) {
475                SkChopCubicAt(src, dst, t);
476                if (i == roots - 1) {
477                    break;
478                }
479
480                dst += 3;
481                // have src point to the remaining cubic (after the chop)
482                memcpy(tmp, dst, 4 * sizeof(SkPoint));
483                src = tmp;
484
485                // watch out in case the renormalized t isn't in range
486                if (!valid_unit_divide(tValues[i+1] - tValues[i],
487                                       SK_Scalar1 - tValues[i], &t)) {
488                    // if we can't, just create a degenerate cubic
489                    dst[4] = dst[5] = dst[6] = src[3];
490                    break;
491                }
492            }
493        }
494    }
495}
496
497void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
498    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
499    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
500    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
501    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
502    SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
503    SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
504
505    SkScalar x012 = SkScalarAve(x01, x12);
506    SkScalar y012 = SkScalarAve(y01, y12);
507    SkScalar x123 = SkScalarAve(x12, x23);
508    SkScalar y123 = SkScalarAve(y12, y23);
509
510    dst[0] = src[0];
511    dst[1].set(x01, y01);
512    dst[2].set(x012, y012);
513    dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
514    dst[4].set(x123, y123);
515    dst[5].set(x23, y23);
516    dst[6] = src[3];
517}
518
519static void flatten_double_cubic_extrema(SkScalar coords[14]) {
520    coords[4] = coords[8] = coords[6];
521}
522
523/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
524    the resulting beziers are monotonic in Y. This is called by the scan
525    converter.  Depending on what is returned, dst[] is treated as follows:
526    0   dst[0..3] is the original cubic
527    1   dst[0..3] and dst[3..6] are the two new cubics
528    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
529    If dst == null, it is ignored and only the count is returned.
530*/
531int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
532    SkScalar    tValues[2];
533    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
534                                           src[3].fY, tValues);
535
536    SkChopCubicAt(src, dst, tValues, roots);
537    if (dst && roots > 0) {
538        // we do some cleanup to ensure our Y extrema are flat
539        flatten_double_cubic_extrema(&dst[0].fY);
540        if (roots == 2) {
541            flatten_double_cubic_extrema(&dst[3].fY);
542        }
543    }
544    return roots;
545}
546
547int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
548    SkScalar    tValues[2];
549    int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
550                                           src[3].fX, tValues);
551
552    SkChopCubicAt(src, dst, tValues, roots);
553    if (dst && roots > 0) {
554        // we do some cleanup to ensure our Y extrema are flat
555        flatten_double_cubic_extrema(&dst[0].fX);
556        if (roots == 2) {
557            flatten_double_cubic_extrema(&dst[3].fX);
558        }
559    }
560    return roots;
561}
562
563/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
564
565    Inflection means that curvature is zero.
566    Curvature is [F' x F''] / [F'^3]
567    So we solve F'x X F''y - F'y X F''y == 0
568    After some canceling of the cubic term, we get
569    A = b - a
570    B = c - 2b + a
571    C = d - 3c + 3b - a
572    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
573*/
574int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
575    SkScalar    Ax = src[1].fX - src[0].fX;
576    SkScalar    Ay = src[1].fY - src[0].fY;
577    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
578    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
579    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
580    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
581
582    return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
583                               Ax*Cy - Ay*Cx,
584                               Ax*By - Ay*Bx,
585                               tValues);
586}
587
588int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
589    SkScalar    tValues[2];
590    int         count = SkFindCubicInflections(src, tValues);
591
592    if (dst) {
593        if (count == 0) {
594            memcpy(dst, src, 4 * sizeof(SkPoint));
595        } else {
596            SkChopCubicAt(src, dst, tValues, count);
597        }
598    }
599    return count + 1;
600}
601
602// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
603// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
604// Classification:
605// discr(I) > 0        Serpentine
606// discr(I) = 0        Cusp
607// discr(I) < 0        Loop
608// d0 = d1 = 0         Quadratic
609// d0 = d1 = d2 = 0    Line
610// p0 = p1 = p2 = p3   Point
611static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
612    if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
613        return kPoint_SkCubicType;
614    }
615    const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
616    if (discr > SK_ScalarNearlyZero) {
617        return kSerpentine_SkCubicType;
618    } else if (discr < -SK_ScalarNearlyZero) {
619        return kLoop_SkCubicType;
620    } else {
621        if (0.f == d[0] && 0.f == d[1]) {
622            return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
623        } else {
624            return kCusp_SkCubicType;
625        }
626    }
627}
628
629// Assumes the third component of points is 1.
630// Calcs p0 . (p1 x p2)
631static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
632    const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
633    const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
634    const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
635    return (xComp + yComp + wComp);
636}
637
638// Calc coefficients of I(s,t) where roots of I are inflection points of curve
639// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
640// d0 = a1 - 2*a2+3*a3
641// d1 = -a2 + 3*a3
642// d2 = 3*a3
643// a1 = p0 . (p3 x p2)
644// a2 = p1 . (p0 x p3)
645// a3 = p2 . (p1 x p0)
646// Places the values of d1, d2, d3 in array d passed in
647static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
648    SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
649    SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
650    SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
651
652    // need to scale a's or values in later calculations will grow to high
653    SkScalar max = SkScalarAbs(a1);
654    max = SkMaxScalar(max, SkScalarAbs(a2));
655    max = SkMaxScalar(max, SkScalarAbs(a3));
656    max = 1.f/max;
657    a1 = a1 * max;
658    a2 = a2 * max;
659    a3 = a3 * max;
660
661    d[2] = 3.f * a3;
662    d[1] = d[2] - a2;
663    d[0] = d[1] - a2 + a1;
664}
665
666SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
667    calc_cubic_inflection_func(src, d);
668    return classify_cubic(src, d);
669}
670
671template <typename T> void bubble_sort(T array[], int count) {
672    for (int i = count - 1; i > 0; --i)
673        for (int j = i; j > 0; --j)
674            if (array[j] < array[j-1])
675            {
676                T   tmp(array[j]);
677                array[j] = array[j-1];
678                array[j-1] = tmp;
679            }
680}
681
682/**
683 *  Given an array and count, remove all pair-wise duplicates from the array,
684 *  keeping the existing sorting, and return the new count
685 */
686static int collaps_duplicates(SkScalar array[], int count) {
687    for (int n = count; n > 1; --n) {
688        if (array[0] == array[1]) {
689            for (int i = 1; i < n; ++i) {
690                array[i - 1] = array[i];
691            }
692            count -= 1;
693        } else {
694            array += 1;
695        }
696    }
697    return count;
698}
699
700#ifdef SK_DEBUG
701
702#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
703
704static void test_collaps_duplicates() {
705    static bool gOnce;
706    if (gOnce) { return; }
707    gOnce = true;
708    const SkScalar src0[] = { 0 };
709    const SkScalar src1[] = { 0, 0 };
710    const SkScalar src2[] = { 0, 1 };
711    const SkScalar src3[] = { 0, 0, 0 };
712    const SkScalar src4[] = { 0, 0, 1 };
713    const SkScalar src5[] = { 0, 1, 1 };
714    const SkScalar src6[] = { 0, 1, 2 };
715    const struct {
716        const SkScalar* fData;
717        int fCount;
718        int fCollapsedCount;
719    } data[] = {
720        { TEST_COLLAPS_ENTRY(src0), 1 },
721        { TEST_COLLAPS_ENTRY(src1), 1 },
722        { TEST_COLLAPS_ENTRY(src2), 2 },
723        { TEST_COLLAPS_ENTRY(src3), 1 },
724        { TEST_COLLAPS_ENTRY(src4), 2 },
725        { TEST_COLLAPS_ENTRY(src5), 2 },
726        { TEST_COLLAPS_ENTRY(src6), 3 },
727    };
728    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
729        SkScalar dst[3];
730        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
731        int count = collaps_duplicates(dst, data[i].fCount);
732        SkASSERT(data[i].fCollapsedCount == count);
733        for (int j = 1; j < count; ++j) {
734            SkASSERT(dst[j-1] < dst[j]);
735        }
736    }
737}
738#endif
739
740static SkScalar SkScalarCubeRoot(SkScalar x) {
741    return SkScalarPow(x, 0.3333333f);
742}
743
744/*  Solve coeff(t) == 0, returning the number of roots that
745    lie withing 0 < t < 1.
746    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
747
748    Eliminates repeated roots (so that all tValues are distinct, and are always
749    in increasing order.
750*/
751static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
752    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
753        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
754    }
755
756    SkScalar a, b, c, Q, R;
757
758    {
759        SkASSERT(coeff[0] != 0);
760
761        SkScalar inva = SkScalarInvert(coeff[0]);
762        a = coeff[1] * inva;
763        b = coeff[2] * inva;
764        c = coeff[3] * inva;
765    }
766    Q = (a*a - b*3) / 9;
767    R = (2*a*a*a - 9*a*b + 27*c) / 54;
768
769    SkScalar Q3 = Q * Q * Q;
770    SkScalar R2MinusQ3 = R * R - Q3;
771    SkScalar adiv3 = a / 3;
772
773    SkScalar*   roots = tValues;
774    SkScalar    r;
775
776    if (R2MinusQ3 < 0) { // we have 3 real roots
777        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
778        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
779
780        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
781        if (is_unit_interval(r)) {
782            *roots++ = r;
783        }
784        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
785        if (is_unit_interval(r)) {
786            *roots++ = r;
787        }
788        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
789        if (is_unit_interval(r)) {
790            *roots++ = r;
791        }
792        SkDEBUGCODE(test_collaps_duplicates();)
793
794        // now sort the roots
795        int count = (int)(roots - tValues);
796        SkASSERT((unsigned)count <= 3);
797        bubble_sort(tValues, count);
798        count = collaps_duplicates(tValues, count);
799        roots = tValues + count;    // so we compute the proper count below
800    } else {              // we have 1 real root
801        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
802        A = SkScalarCubeRoot(A);
803        if (R > 0) {
804            A = -A;
805        }
806        if (A != 0) {
807            A += Q / A;
808        }
809        r = A - adiv3;
810        if (is_unit_interval(r)) {
811            *roots++ = r;
812        }
813    }
814
815    return (int)(roots - tValues);
816}
817
818/*  Looking for F' dot F'' == 0
819
820    A = b - a
821    B = c - 2b + a
822    C = d - 3c + 3b - a
823
824    F' = 3Ct^2 + 6Bt + 3A
825    F'' = 6Ct + 6B
826
827    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
828*/
829static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
830    SkScalar    a = src[2] - src[0];
831    SkScalar    b = src[4] - 2 * src[2] + src[0];
832    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
833
834    coeff[0] = c * c;
835    coeff[1] = 3 * b * c;
836    coeff[2] = 2 * b * b + c * a;
837    coeff[3] = a * b;
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*/
851int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
852    SkScalar coeffX[4], coeffY[4];
853    int      i;
854
855    formulate_F1DotF2(&src[0].fX, coeffX);
856    formulate_F1DotF2(&src[0].fY, coeffY);
857
858    for (i = 0; i < 4; i++) {
859        coeffX[i] += coeffY[i];
860    }
861
862    SkScalar    t[3];
863    int         count = solve_cubic_poly(coeffX, t);
864    int         maxCount = 0;
865
866    // now remove extrema where the curvature is zero (mins)
867    // !!!! need a test for this !!!!
868    for (i = 0; i < count; i++) {
869        // if (not_min_curvature())
870        if (t[i] > 0 && t[i] < SK_Scalar1) {
871            tValues[maxCount++] = t[i];
872        }
873    }
874    return maxCount;
875}
876
877int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
878                              SkScalar tValues[3]) {
879    SkScalar    t_storage[3];
880
881    if (tValues == NULL) {
882        tValues = t_storage;
883    }
884
885    int count = SkFindCubicMaxCurvature(src, tValues);
886
887    if (dst) {
888        if (count == 0) {
889            memcpy(dst, src, 4 * sizeof(SkPoint));
890        } else {
891            SkChopCubicAt(src, dst, tValues, count);
892        }
893    }
894    return count + 1;
895}
896
897///////////////////////////////////////////////////////////////////////////////
898
899/*  Find t value for quadratic [a, b, c] = d.
900    Return 0 if there is no solution within [0, 1)
901*/
902static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
903    // At^2 + Bt + C = d
904    SkScalar A = a - 2 * b + c;
905    SkScalar B = 2 * (b - a);
906    SkScalar C = a - d;
907
908    SkScalar    roots[2];
909    int         count = SkFindUnitQuadRoots(A, B, C, roots);
910
911    SkASSERT(count <= 1);
912    return count == 1 ? roots[0] : 0;
913}
914
915/*  given a quad-curve and a point (x,y), chop the quad at that point and place
916    the new off-curve point and endpoint into 'dest'.
917    Should only return false if the computed pos is the start of the curve
918    (i.e. root == 0)
919*/
920static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
921                                SkPoint* dest) {
922    const SkScalar* base;
923    SkScalar        value;
924
925    if (SkScalarAbs(x) < SkScalarAbs(y)) {
926        base = &quad[0].fX;
927        value = x;
928    } else {
929        base = &quad[0].fY;
930        value = y;
931    }
932
933    // note: this returns 0 if it thinks value is out of range, meaning the
934    // root might return something outside of [0, 1)
935    SkScalar t = quad_solve(base[0], base[2], base[4], value);
936
937    if (t > 0) {
938        SkPoint tmp[5];
939        SkChopQuadAt(quad, tmp, t);
940        dest[0] = tmp[1];
941        dest[1].set(x, y);
942        return true;
943    } else {
944        /*  t == 0 means either the value triggered a root outside of [0, 1)
945            For our purposes, we can ignore the <= 0 roots, but we want to
946            catch the >= 1 roots (which given our caller, will basically mean
947            a root of 1, give-or-take numerical instability). If we are in the
948            >= 1 case, return the existing offCurve point.
949
950            The test below checks to see if we are close to the "end" of the
951            curve (near base[4]). Rather than specifying a tolerance, I just
952            check to see if value is on to the right/left of the middle point
953            (depending on the direction/sign of the end points).
954        */
955        if ((base[0] < base[4] && value > base[2]) ||
956            (base[0] > base[4] && value < base[2]))   // should root have been 1
957        {
958            dest[0] = quad[1];
959            dest[1].set(x, y);
960            return true;
961        }
962    }
963    return false;
964}
965
966static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
967// The mid point of the quadratic arc approximation is half way between the two
968// control points. The float epsilon adjustment moves the on curve point out by
969// two bits, distributing the convex test error between the round rect
970// approximation and the convex cross product sign equality test.
971#define SK_MID_RRECT_OFFSET \
972    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
973    { SK_Scalar1,            0                      },
974    { SK_Scalar1,            SK_ScalarTanPIOver8    },
975    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
976    { SK_ScalarTanPIOver8,   SK_Scalar1             },
977
978    { 0,                     SK_Scalar1             },
979    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
980    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
981    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
982
983    { -SK_Scalar1,           0                      },
984    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
985    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
986    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
987
988    { 0,                     -SK_Scalar1            },
989    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
990    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
991    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
992
993    { SK_Scalar1,            0                      }
994#undef SK_MID_RRECT_OFFSET
995};
996
997int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
998                   SkRotationDirection dir, const SkMatrix* userMatrix,
999                   SkPoint quadPoints[]) {
1000    // rotate by x,y so that uStart is (1.0)
1001    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1002    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1003
1004    SkScalar absX = SkScalarAbs(x);
1005    SkScalar absY = SkScalarAbs(y);
1006
1007    int pointCount;
1008
1009    // check for (effectively) coincident vectors
1010    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1011    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1012    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1013        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1014         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1015
1016        // just return the start-point
1017        quadPoints[0].set(SK_Scalar1, 0);
1018        pointCount = 1;
1019    } else {
1020        if (dir == kCCW_SkRotationDirection) {
1021            y = -y;
1022        }
1023        // what octant (quadratic curve) is [xy] in?
1024        int oct = 0;
1025        bool sameSign = true;
1026
1027        if (0 == y) {
1028            oct = 4;        // 180
1029            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1030        } else if (0 == x) {
1031            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1032            oct = y > 0 ? 2 : 6; // 90 : 270
1033        } else {
1034            if (y < 0) {
1035                oct += 4;
1036            }
1037            if ((x < 0) != (y < 0)) {
1038                oct += 2;
1039                sameSign = false;
1040            }
1041            if ((absX < absY) == sameSign) {
1042                oct += 1;
1043            }
1044        }
1045
1046        int wholeCount = oct << 1;
1047        memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1048
1049        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1050        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1051            wholeCount += 2;
1052        }
1053        pointCount = wholeCount + 1;
1054    }
1055
1056    // now handle counter-clockwise and the initial unitStart rotation
1057    SkMatrix    matrix;
1058    matrix.setSinCos(uStart.fY, uStart.fX);
1059    if (dir == kCCW_SkRotationDirection) {
1060        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1061    }
1062    if (userMatrix) {
1063        matrix.postConcat(*userMatrix);
1064    }
1065    matrix.mapPoints(quadPoints, pointCount);
1066    return pointCount;
1067}
1068
1069
1070///////////////////////////////////////////////////////////////////////////////
1071//
1072// NURB representation for conics.  Helpful explanations at:
1073//
1074// http://citeseerx.ist.psu.edu/viewdoc/
1075//   download?doi=10.1.1.44.5740&rep=rep1&type=ps
1076// and
1077// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1078//
1079// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1080//     ------------------------------------------
1081//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1082//
1083//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1084//     ------------------------------------------------
1085//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1086//
1087
1088static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1089    SkASSERT(src);
1090    SkASSERT(t >= 0 && t <= SK_Scalar1);
1091
1092    SkScalar    src2w = SkScalarMul(src[2], w);
1093    SkScalar    C = src[0];
1094    SkScalar    A = src[4] - 2 * src2w + C;
1095    SkScalar    B = 2 * (src2w - C);
1096    SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1097
1098    B = 2 * (w - SK_Scalar1);
1099    C = SK_Scalar1;
1100    A = -B;
1101    SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1102
1103    return SkScalarDiv(numer, denom);
1104}
1105
1106// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1107//
1108//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1109//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1110//  t^0 : -2 P0 w + 2 P1 w
1111//
1112//  We disregard magnitude, so we can freely ignore the denominator of F', and
1113//  divide the numerator by 2
1114//
1115//    coeff[0] for t^2
1116//    coeff[1] for t^1
1117//    coeff[2] for t^0
1118//
1119static void conic_deriv_coeff(const SkScalar src[],
1120                              SkScalar w,
1121                              SkScalar coeff[3]) {
1122    const SkScalar P20 = src[4] - src[0];
1123    const SkScalar P10 = src[2] - src[0];
1124    const SkScalar wP10 = w * P10;
1125    coeff[0] = w * P20 - P20;
1126    coeff[1] = P20 - 2 * wP10;
1127    coeff[2] = wP10;
1128}
1129
1130static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1131    SkScalar coeff[3];
1132    conic_deriv_coeff(coord, w, coeff);
1133    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1134}
1135
1136static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1137    SkScalar coeff[3];
1138    conic_deriv_coeff(src, w, coeff);
1139
1140    SkScalar tValues[2];
1141    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1142    SkASSERT(0 == roots || 1 == roots);
1143
1144    if (1 == roots) {
1145        *t = tValues[0];
1146        return true;
1147    }
1148    return false;
1149}
1150
1151struct SkP3D {
1152    SkScalar fX, fY, fZ;
1153
1154    void set(SkScalar x, SkScalar y, SkScalar z) {
1155        fX = x; fY = y; fZ = z;
1156    }
1157
1158    void projectDown(SkPoint* dst) const {
1159        dst->set(fX / fZ, fY / fZ);
1160    }
1161};
1162
1163// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1164static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1165    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1166    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1167    dst[0] = ab;
1168    dst[3] = SkScalarInterp(ab, bc, t);
1169    dst[6] = bc;
1170}
1171
1172static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1173    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1174    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1175    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1176}
1177
1178void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1179    SkASSERT(t >= 0 && t <= SK_Scalar1);
1180
1181    if (pt) {
1182        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1183                conic_eval_pos(&fPts[0].fY, fW, t));
1184    }
1185    if (tangent) {
1186        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1187                     conic_eval_tan(&fPts[0].fY, fW, t));
1188    }
1189}
1190
1191void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1192    SkP3D tmp[3], tmp2[3];
1193
1194    ratquad_mapTo3D(fPts, fW, tmp);
1195
1196    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1197    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1198    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1199
1200    dst[0].fPts[0] = fPts[0];
1201    tmp2[0].projectDown(&dst[0].fPts[1]);
1202    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1203    tmp2[2].projectDown(&dst[1].fPts[1]);
1204    dst[1].fPts[2] = fPts[2];
1205
1206    // to put in "standard form", where w0 and w2 are both 1, we compute the
1207    // new w1 as sqrt(w1*w1/w0*w2)
1208    // or
1209    // w1 /= sqrt(w0*w2)
1210    //
1211    // However, in our case, we know that for dst[0]:
1212    //     w0 == 1, and for dst[1], w2 == 1
1213    //
1214    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1215    dst[0].fW = tmp2[0].fZ / root;
1216    dst[1].fW = tmp2[2].fZ / root;
1217}
1218
1219static SkScalar subdivide_w_value(SkScalar w) {
1220    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1221}
1222
1223void SkConic::chop(SkConic dst[2]) const {
1224    SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
1225    SkScalar p1x = fW * fPts[1].fX;
1226    SkScalar p1y = fW * fPts[1].fY;
1227    SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
1228    SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
1229
1230    dst[0].fPts[0] = fPts[0];
1231    dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
1232                       (fPts[0].fY + p1y) * scale);
1233    dst[0].fPts[2].set(mx, my);
1234
1235    dst[1].fPts[0].set(mx, my);
1236    dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
1237                       (p1y + fPts[2].fY) * scale);
1238    dst[1].fPts[2] = fPts[2];
1239
1240    dst[0].fW = dst[1].fW = subdivide_w_value(fW);
1241}
1242
1243/*
1244 *  "High order approximation of conic sections by quadratic splines"
1245 *      by Michael Floater, 1993
1246 */
1247#define AS_QUAD_ERROR_SETUP                                         \
1248    SkScalar a = fW - 1;                                            \
1249    SkScalar k = a / (4 * (2 + a));                                 \
1250    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1251    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1252
1253void SkConic::computeAsQuadError(SkVector* err) const {
1254    AS_QUAD_ERROR_SETUP
1255    err->set(x, y);
1256}
1257
1258bool SkConic::asQuadTol(SkScalar tol) const {
1259    AS_QUAD_ERROR_SETUP
1260    return (x * x + y * y) <= tol * tol;
1261}
1262
1263// Limit the number of suggested quads to approximate a conic
1264#define kMaxConicToQuadPOW2     5
1265
1266int SkConic::computeQuadPOW2(SkScalar tol) const {
1267    if (tol < 0 || !SkScalarIsFinite(tol)) {
1268        return 0;
1269    }
1270
1271    AS_QUAD_ERROR_SETUP
1272
1273    SkScalar error = SkScalarSqrt(x * x + y * y);
1274    int pow2;
1275    for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
1276        if (error <= tol) {
1277            break;
1278        }
1279        error *= 0.25f;
1280    }
1281    // float version -- using ceil gives the same results as the above.
1282    if (false) {
1283        SkScalar err = SkScalarSqrt(x * x + y * y);
1284        if (err <= tol) {
1285            return 0;
1286        }
1287        SkScalar tol2 = tol * tol;
1288        if (tol2 == 0) {
1289            return kMaxConicToQuadPOW2;
1290        }
1291        SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
1292        int altPow2 = SkScalarCeilToInt(fpow2);
1293        if (altPow2 != pow2) {
1294            SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
1295        }
1296        pow2 = altPow2;
1297    }
1298    return pow2;
1299}
1300
1301static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1302    SkASSERT(level >= 0);
1303
1304    if (0 == level) {
1305        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1306        return pts + 2;
1307    } else {
1308        SkConic dst[2];
1309        src.chop(dst);
1310        --level;
1311        pts = subdivide(dst[0], pts, level);
1312        return subdivide(dst[1], pts, level);
1313    }
1314}
1315
1316int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1317    SkASSERT(pow2 >= 0);
1318    *pts = fPts[0];
1319    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1320    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1321    return 1 << pow2;
1322}
1323
1324bool SkConic::findXExtrema(SkScalar* t) const {
1325    return conic_find_extrema(&fPts[0].fX, fW, t);
1326}
1327
1328bool SkConic::findYExtrema(SkScalar* t) const {
1329    return conic_find_extrema(&fPts[0].fY, fW, t);
1330}
1331
1332bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1333    SkScalar t;
1334    if (this->findXExtrema(&t)) {
1335        this->chopAt(t, dst);
1336        // now clean-up the middle, since we know t was meant to be at
1337        // an X-extrema
1338        SkScalar value = dst[0].fPts[2].fX;
1339        dst[0].fPts[1].fX = value;
1340        dst[1].fPts[0].fX = value;
1341        dst[1].fPts[1].fX = value;
1342        return true;
1343    }
1344    return false;
1345}
1346
1347bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1348    SkScalar t;
1349    if (this->findYExtrema(&t)) {
1350        this->chopAt(t, dst);
1351        // now clean-up the middle, since we know t was meant to be at
1352        // an Y-extrema
1353        SkScalar value = dst[0].fPts[2].fY;
1354        dst[0].fPts[1].fY = value;
1355        dst[1].fPts[0].fY = value;
1356        dst[1].fPts[1].fY = value;
1357        return true;
1358    }
1359    return false;
1360}
1361
1362void SkConic::computeTightBounds(SkRect* bounds) const {
1363    SkPoint pts[4];
1364    pts[0] = fPts[0];
1365    pts[1] = fPts[2];
1366    int count = 2;
1367
1368    SkScalar t;
1369    if (this->findXExtrema(&t)) {
1370        this->evalAt(t, &pts[count++]);
1371    }
1372    if (this->findYExtrema(&t)) {
1373        this->evalAt(t, &pts[count++]);
1374    }
1375    bounds->set(pts, count);
1376}
1377
1378void SkConic::computeFastBounds(SkRect* bounds) const {
1379    bounds->set(fPts, 3);
1380}
1381
1382bool SkConic::findMaxCurvature(SkScalar* t) const {
1383    // TODO: Implement me
1384    return false;
1385}
1386
1387SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1388                             const SkMatrix& matrix) {
1389    if (!matrix.hasPerspective()) {
1390        return w;
1391    }
1392
1393    SkP3D src[3], dst[3];
1394
1395    ratquad_mapTo3D(pts, w, src);
1396
1397    matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1398
1399    // w' = sqrt(w1*w1/w0*w2)
1400    SkScalar w0 = dst[0].fZ;
1401    SkScalar w1 = dst[1].fZ;
1402    SkScalar w2 = dst[2].fZ;
1403    w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1404    return w;
1405}
1406
1407int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
1408                          const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
1409    // rotate by x,y so that uStart is (1.0)
1410    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1411    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1412
1413    SkScalar absY = SkScalarAbs(y);
1414
1415    // check for (effectively) coincident vectors
1416    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1417    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1418    if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) ||
1419                                                 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1420        return 0;
1421    }
1422
1423    if (dir == kCCW_SkRotationDirection) {
1424        y = -y;
1425    }
1426
1427    // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
1428    //      0 == [0  .. 90)
1429    //      1 == [90 ..180)
1430    //      2 == [180..270)
1431    //      3 == [270..360)
1432    //
1433    int quadrant = 0;
1434    if (0 == y) {
1435        quadrant = 2;        // 180
1436        SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1437    } else if (0 == x) {
1438        SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1439        quadrant = y > 0 ? 1 : 3; // 90 : 270
1440    } else {
1441        if (y < 0) {
1442            quadrant += 2;
1443        }
1444        if ((x < 0) != (y < 0)) {
1445            quadrant += 1;
1446        }
1447    }
1448
1449    const SkPoint quadrantPts[] = {
1450        { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
1451    };
1452    const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
1453
1454    int conicCount = quadrant;
1455    for (int i = 0; i < conicCount; ++i) {
1456        dst[i].set(&quadrantPts[i * 2], quadrantWeight);
1457    }
1458
1459    // Now compute any remaing (sub-90-degree) arc for the last conic
1460    const SkPoint finalP = { x, y };
1461    const SkPoint& lastQ = quadrantPts[quadrant * 2];  // will already be a unit-vector
1462    const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
1463    SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
1464
1465    if (dot < 1 - SK_ScalarNearlyZero) {
1466        SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
1467        // compute the bisector vector, and then rescale to be the off-curve point.
1468        // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
1469        // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
1470        // This is nice, since our computed weight is cos(theta/2) as well!
1471        //
1472        const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
1473        offCurve.setLength(SkScalarInvert(cosThetaOver2));
1474        dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
1475        conicCount += 1;
1476    }
1477
1478    // now handle counter-clockwise and the initial unitStart rotation
1479    SkMatrix    matrix;
1480    matrix.setSinCos(uStart.fY, uStart.fX);
1481    if (dir == kCCW_SkRotationDirection) {
1482        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1483    }
1484    if (userMatrix) {
1485        matrix.postConcat(*userMatrix);
1486    }
1487    for (int i = 0; i < conicCount; ++i) {
1488        matrix.mapPoints(dst[i].fPts, 3);
1489    }
1490    return conicCount;
1491}
1492