SkGeometry.cpp revision 220f926d9d4b38a9018c922c095847bbd261f583 
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
11bool SkXRayCrossesLine(const SkXRay& pt,
12                       const SkPoint pts[2],
13                       bool* ambiguous) {
14    if (ambiguous) {
15        *ambiguous = false;
16    }
17    // Determine quick discards.
18    // Consider query line going exactly through point 0 to not
19    // intersect, for symmetry with SkXRayCrossesMonotonicCubic.
20    if (pt.fY == pts[0].fY) {
21        if (ambiguous) {
22            *ambiguous = true;
23        }
24        return false;
25    }
26    if (pt.fY < pts[0].fY && pt.fY < pts[1].fY)
27        return false;
28    if (pt.fY > pts[0].fY && pt.fY > pts[1].fY)
29        return false;
30    if (pt.fX > pts[0].fX && pt.fX > pts[1].fX)
31        return false;
32    // Determine degenerate cases
33    if (SkScalarNearlyZero(pts[0].fY - pts[1].fY))
34        return false;
35    if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) {
36        // We've already determined the query point lies within the
37        // vertical range of the line segment.
38        if (pt.fX <= pts[0].fX) {
39            if (ambiguous) {
40                *ambiguous = (pt.fY == pts[1].fY);
41            }
42            return true;
43        }
44        return false;
45    }
46    // Ambiguity check
47    if (pt.fY == pts[1].fY) {
48        if (pt.fX <= pts[1].fX) {
49            if (ambiguous) {
50                *ambiguous = true;
51            }
52            return true;
53        }
54        return false;
55    }
56    // Full line segment evaluation
57    SkScalar delta_y = pts[1].fY - pts[0].fY;
58    SkScalar delta_x = pts[1].fX - pts[0].fX;
59    SkScalar slope = SkScalarDiv(delta_y, delta_x);
60    SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX);
61    // Solve for x coordinate at y = pt.fY
62    SkScalar x = SkScalarDiv(pt.fY - b, slope);
63    return pt.fX <= x;
64}
65
66/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
67    involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
68    May also introduce overflow of fixed when we compute our setup.
69*/
70//    #define DIRECT_EVAL_OF_POLYNOMIALS
71
72////////////////////////////////////////////////////////////////////////
73
74static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
75    SkScalar ab = a - b;
76    SkScalar bc = b - c;
77    if (ab < 0) {
78        bc = -bc;
79    }
80    return ab == 0 || bc < 0;
81}
82
83////////////////////////////////////////////////////////////////////////
84
85static bool is_unit_interval(SkScalar x) {
86    return x > 0 && x < SK_Scalar1;
87}
88
89static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
90    SkASSERT(ratio);
91
92    if (numer < 0) {
93        numer = -numer;
94        denom = -denom;
95    }
96
97    if (denom == 0 || numer == 0 || numer >= denom) {
98        return 0;
99    }
100
101    SkScalar r = SkScalarDiv(numer, denom);
102    if (SkScalarIsNaN(r)) {
103        return 0;
104    }
105    SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
106    if (r == 0) { // catch underflow if numer <<<< denom
107        return 0;
108    }
109    *ratio = r;
110    return 1;
111}
112
113/** From Numerical Recipes in C.
114
115    Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
116    x1 = Q / A
117    x2 = C / Q
118*/
119int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
120    SkASSERT(roots);
121
122    if (A == 0) {
123        return valid_unit_divide(-C, B, roots);
124    }
125
126    SkScalar* r = roots;
127
128    SkScalar R = B*B - 4*A*C;
129    if (R < 0 || SkScalarIsNaN(R)) {  // complex roots
130        return 0;
131    }
132    R = SkScalarSqrt(R);
133
134    SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
135    r += valid_unit_divide(Q, A, r);
136    r += valid_unit_divide(C, Q, r);
137    if (r - roots == 2) {
138        if (roots[0] > roots[1])
139            SkTSwap<SkScalar>(roots[0], roots[1]);
140        else if (roots[0] == roots[1])  // nearly-equal?
141            r -= 1; // skip the double root
142    }
143    return (int)(r - roots);
144}
145
146///////////////////////////////////////////////////////////////////////////////
147///////////////////////////////////////////////////////////////////////////////
148
149static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
150    SkASSERT(src);
151    SkASSERT(t >= 0 && t <= SK_Scalar1);
152
153#ifdef DIRECT_EVAL_OF_POLYNOMIALS
154    SkScalar    C = src[0];
155    SkScalar    A = src[4] - 2 * src[2] + C;
156    SkScalar    B = 2 * (src[2] - C);
157    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
158#else
159    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
160    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
161    return SkScalarInterp(ab, bc, t);
162#endif
163}
164
165static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
166    SkScalar A = src[4] - 2 * src[2] + src[0];
167    SkScalar B = src[2] - src[0];
168
169    return 2 * SkScalarMulAdd(A, t, B);
170}
171
172static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) {
173    SkScalar A = src[4] - 2 * src[2] + src[0];
174    SkScalar B = src[2] - src[0];
175    return A + 2 * B;
176}
177
178void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt,
179                  SkVector* tangent) {
180    SkASSERT(src);
181    SkASSERT(t >= 0 && t <= SK_Scalar1);
182
183    if (pt) {
184        pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
185    }
186    if (tangent) {
187        tangent->set(eval_quad_derivative(&src[0].fX, t),
188                     eval_quad_derivative(&src[0].fY, t));
189    }
190}
191
192void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) {
193    SkASSERT(src);
194
195    if (pt) {
196        SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
197        SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
198        SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
199        SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
200        pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
201    }
202    if (tangent) {
203        tangent->set(eval_quad_derivative_at_half(&src[0].fX),
204                     eval_quad_derivative_at_half(&src[0].fY));
205    }
206}
207
208static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) {
209    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
210    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
211
212    dst[0] = src[0];
213    dst[2] = ab;
214    dst[4] = SkScalarInterp(ab, bc, t);
215    dst[6] = bc;
216    dst[8] = src[4];
217}
218
219void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
220    SkASSERT(t > 0 && t < SK_Scalar1);
221
222    interp_quad_coords(&src[0].fX, &dst[0].fX, t);
223    interp_quad_coords(&src[0].fY, &dst[0].fY, t);
224}
225
226void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
227    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
228    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
229    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
230    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
231
232    dst[0] = src[0];
233    dst[1].set(x01, y01);
234    dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
235    dst[3].set(x12, y12);
236    dst[4] = src[2];
237}
238
239/** Quad'(t) = At + B, where
240    A = 2(a - 2b + c)
241    B = 2(b - a)
242    Solve for t, only if it fits between 0 < t < 1
243*/
244int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
245    /*  At + B == 0
246        t = -B / A
247    */
248    return valid_unit_divide(a - b, a - b - b + c, tValue);
249}
250
251static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
252    coords[2] = coords[6] = coords[4];
253}
254
255/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
256 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
257 */
258int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
259    SkASSERT(src);
260    SkASSERT(dst);
261
262    SkScalar a = src[0].fY;
263    SkScalar b = src[1].fY;
264    SkScalar c = src[2].fY;
265
266    if (is_not_monotonic(a, b, c)) {
267        SkScalar    tValue;
268        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
269            SkChopQuadAt(src, dst, tValue);
270            flatten_double_quad_extrema(&dst[0].fY);
271            return 1;
272        }
273        // if we get here, we need to force dst to be monotonic, even though
274        // we couldn't compute a unit_divide value (probably underflow).
275        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
276    }
277    dst[0].set(src[0].fX, a);
278    dst[1].set(src[1].fX, b);
279    dst[2].set(src[2].fX, c);
280    return 0;
281}
282
283/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
284    stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
285 */
286int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
287    SkASSERT(src);
288    SkASSERT(dst);
289
290    SkScalar a = src[0].fX;
291    SkScalar b = src[1].fX;
292    SkScalar c = src[2].fX;
293
294    if (is_not_monotonic(a, b, c)) {
295        SkScalar tValue;
296        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
297            SkChopQuadAt(src, dst, tValue);
298            flatten_double_quad_extrema(&dst[0].fX);
299            return 1;
300        }
301        // if we get here, we need to force dst to be monotonic, even though
302        // we couldn't compute a unit_divide value (probably underflow).
303        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
304    }
305    dst[0].set(a, src[0].fY);
306    dst[1].set(b, src[1].fY);
307    dst[2].set(c, src[2].fY);
308    return 0;
309}
310
311//  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
312//  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
313//  F''(t)  = 2 (a - 2b + c)
314//
315//  A = 2 (b - a)
316//  B = 2 (a - 2b + c)
317//
318//  Maximum curvature for a quadratic means solving
319//  Fx' Fx'' + Fy' Fy'' = 0
320//
321//  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
322//
323SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
324    SkScalar    Ax = src[1].fX - src[0].fX;
325    SkScalar    Ay = src[1].fY - src[0].fY;
326    SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
327    SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
328    SkScalar    t = 0;  // 0 means don't chop
329
330    (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
331    return t;
332}
333
334int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
335    SkScalar t = SkFindQuadMaxCurvature(src);
336    if (t == 0) {
337        memcpy(dst, src, 3 * sizeof(SkPoint));
338        return 1;
339    } else {
340        SkChopQuadAt(src, dst, t);
341        return 2;
342    }
343}
344
345#define SK_ScalarTwoThirds  (0.666666666f)
346
347void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
348    const SkScalar scale = SK_ScalarTwoThirds;
349    dst[0] = src[0];
350    dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
351               src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
352    dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
353               src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
354    dst[3] = src[2];
355}
356
357//////////////////////////////////////////////////////////////////////////////
358///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
359//////////////////////////////////////////////////////////////////////////////
360
361static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
362    coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
363    coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
364    coeff[2] = 3*(pt[2] - pt[0]);
365    coeff[3] = pt[0];
366}
367
368void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
369    SkASSERT(pts);
370
371    if (cx) {
372        get_cubic_coeff(&pts[0].fX, cx);
373    }
374    if (cy) {
375        get_cubic_coeff(&pts[0].fY, cy);
376    }
377}
378
379static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
380    SkASSERT(src);
381    SkASSERT(t >= 0 && t <= SK_Scalar1);
382
383    if (t == 0) {
384        return src[0];
385    }
386
387#ifdef DIRECT_EVAL_OF_POLYNOMIALS
388    SkScalar D = src[0];
389    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
390    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
391    SkScalar C = 3*(src[2] - D);
392
393    return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
394#else
395    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
396    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
397    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
398    SkScalar    abc = SkScalarInterp(ab, bc, t);
399    SkScalar    bcd = SkScalarInterp(bc, cd, t);
400    return SkScalarInterp(abc, bcd, t);
401#endif
402}
403
404/** return At^2 + Bt + C
405*/
406static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
407    SkASSERT(t >= 0 && t <= SK_Scalar1);
408
409    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
410}
411
412static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
413    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
414    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
415    SkScalar C = src[2] - src[0];
416
417    return eval_quadratic(A, B, C, t);
418}
419
420static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
421    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
422    SkScalar B = src[4] - 2 * src[2] + src[0];
423
424    return SkScalarMulAdd(A, t, B);
425}
426
427void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
428                   SkVector* tangent, SkVector* curvature) {
429    SkASSERT(src);
430    SkASSERT(t >= 0 && t <= SK_Scalar1);
431
432    if (loc) {
433        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
434    }
435    if (tangent) {
436        tangent->set(eval_cubic_derivative(&src[0].fX, t),
437                     eval_cubic_derivative(&src[0].fY, t));
438    }
439    if (curvature) {
440        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
441                       eval_cubic_2ndDerivative(&src[0].fY, t));
442    }
443}
444
445/** Cubic'(t) = At^2 + Bt + C, where
446    A = 3(-a + 3(b - c) + d)
447    B = 6(a - 2b + c)
448    C = 3(b - a)
449    Solve for t, keeping only those that fit betwee 0 < t < 1
450*/
451int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
452                       SkScalar tValues[2]) {
453    // we divide A,B,C by 3 to simplify
454    SkScalar A = d - a + 3*(b - c);
455    SkScalar B = 2*(a - b - b + c);
456    SkScalar C = b - a;
457
458    return SkFindUnitQuadRoots(A, B, C, tValues);
459}
460
461static void interp_cubic_coords(const SkScalar* src, SkScalar* dst,
462                                SkScalar t) {
463    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
464    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
465    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
466    SkScalar    abc = SkScalarInterp(ab, bc, t);
467    SkScalar    bcd = SkScalarInterp(bc, cd, t);
468    SkScalar    abcd = SkScalarInterp(abc, bcd, t);
469
470    dst[0] = src[0];
471    dst[2] = ab;
472    dst[4] = abc;
473    dst[6] = abcd;
474    dst[8] = bcd;
475    dst[10] = cd;
476    dst[12] = src[6];
477}
478
479void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
480    SkASSERT(t > 0 && t < SK_Scalar1);
481
482    interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
483    interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
484}
485
486/*  http://code.google.com/p/skia/issues/detail?id=32
487
488    This test code would fail when we didn't check the return result of
489    valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
490    that after the first chop, the parameters to valid_unit_divide are equal
491    (thanks to finite float precision and rounding in the subtracts). Thus
492    even though the 2nd tValue looks < 1.0, after we renormalize it, we end
493    up with 1.0, hence the need to check and just return the last cubic as
494    a degenerate clump of 4 points in the sampe place.
495
496    static void test_cubic() {
497        SkPoint src[4] = {
498            { 556.25000, 523.03003 },
499            { 556.23999, 522.96002 },
500            { 556.21997, 522.89001 },
501            { 556.21997, 522.82001 }
502        };
503        SkPoint dst[10];
504        SkScalar tval[] = { 0.33333334f, 0.99999994f };
505        SkChopCubicAt(src, dst, tval, 2);
506    }
507 */
508
509void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
510                   const SkScalar tValues[], int roots) {
511#ifdef SK_DEBUG
512    {
513        for (int i = 0; i < roots - 1; i++)
514        {
515            SkASSERT(is_unit_interval(tValues[i]));
516            SkASSERT(is_unit_interval(tValues[i+1]));
517            SkASSERT(tValues[i] < tValues[i+1]);
518        }
519    }
520#endif
521
522    if (dst) {
523        if (roots == 0) { // nothing to chop
524            memcpy(dst, src, 4*sizeof(SkPoint));
525        } else {
526            SkScalar    t = tValues[0];
527            SkPoint     tmp[4];
528
529            for (int i = 0; i < roots; i++) {
530                SkChopCubicAt(src, dst, t);
531                if (i == roots - 1) {
532                    break;
533                }
534
535                dst += 3;
536                // have src point to the remaining cubic (after the chop)
537                memcpy(tmp, dst, 4 * sizeof(SkPoint));
538                src = tmp;
539
540                // watch out in case the renormalized t isn't in range
541                if (!valid_unit_divide(tValues[i+1] - tValues[i],
542                                       SK_Scalar1 - tValues[i], &t)) {
543                    // if we can't, just create a degenerate cubic
544                    dst[4] = dst[5] = dst[6] = src[3];
545                    break;
546                }
547            }
548        }
549    }
550}
551
552void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
553    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
554    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
555    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
556    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
557    SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
558    SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
559
560    SkScalar x012 = SkScalarAve(x01, x12);
561    SkScalar y012 = SkScalarAve(y01, y12);
562    SkScalar x123 = SkScalarAve(x12, x23);
563    SkScalar y123 = SkScalarAve(y12, y23);
564
565    dst[0] = src[0];
566    dst[1].set(x01, y01);
567    dst[2].set(x012, y012);
568    dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
569    dst[4].set(x123, y123);
570    dst[5].set(x23, y23);
571    dst[6] = src[3];
572}
573
574static void flatten_double_cubic_extrema(SkScalar coords[14]) {
575    coords[4] = coords[8] = coords[6];
576}
577
578/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
579    the resulting beziers are monotonic in Y. This is called by the scan
580    converter.  Depending on what is returned, dst[] is treated as follows:
581    0   dst[0..3] is the original cubic
582    1   dst[0..3] and dst[3..6] are the two new cubics
583    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
584    If dst == null, it is ignored and only the count is returned.
585*/
586int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
587    SkScalar    tValues[2];
588    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
589                                           src[3].fY, tValues);
590
591    SkChopCubicAt(src, dst, tValues, roots);
592    if (dst && roots > 0) {
593        // we do some cleanup to ensure our Y extrema are flat
594        flatten_double_cubic_extrema(&dst[0].fY);
595        if (roots == 2) {
596            flatten_double_cubic_extrema(&dst[3].fY);
597        }
598    }
599    return roots;
600}
601
602int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
603    SkScalar    tValues[2];
604    int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
605                                           src[3].fX, tValues);
606
607    SkChopCubicAt(src, dst, tValues, roots);
608    if (dst && roots > 0) {
609        // we do some cleanup to ensure our Y extrema are flat
610        flatten_double_cubic_extrema(&dst[0].fX);
611        if (roots == 2) {
612            flatten_double_cubic_extrema(&dst[3].fX);
613        }
614    }
615    return roots;
616}
617
618/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
619
620    Inflection means that curvature is zero.
621    Curvature is [F' x F''] / [F'^3]
622    So we solve F'x X F''y - F'y X F''y == 0
623    After some canceling of the cubic term, we get
624    A = b - a
625    B = c - 2b + a
626    C = d - 3c + 3b - a
627    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
628*/
629int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
630    SkScalar    Ax = src[1].fX - src[0].fX;
631    SkScalar    Ay = src[1].fY - src[0].fY;
632    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
633    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
634    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
635    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
636
637    return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
638                               Ax*Cy - Ay*Cx,
639                               Ax*By - Ay*Bx,
640                               tValues);
641}
642
643int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
644    SkScalar    tValues[2];
645    int         count = SkFindCubicInflections(src, tValues);
646
647    if (dst) {
648        if (count == 0) {
649            memcpy(dst, src, 4 * sizeof(SkPoint));
650        } else {
651            SkChopCubicAt(src, dst, tValues, count);
652        }
653    }
654    return count + 1;
655}
656
657// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
658// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
659// Classification:
660// discr(I) > 0        Serpentine
661// discr(I) = 0        Cusp
662// discr(I) < 0        Loop
663// d0 = d1 = 0         Quadratic
664// d0 = d1 = d2 = 0    Line
665// p0 = p1 = p2 = p3   Point
666static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
667    if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
668        return kPoint_SkCubicType;
669    }
670    const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
671    if (discr > SK_ScalarNearlyZero) {
672        return kSerpentine_SkCubicType;
673    } else if (discr < -SK_ScalarNearlyZero) {
674        return kLoop_SkCubicType;
675    } else {
676        if (0.f == d[0] && 0.f == d[1]) {
677            return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
678        } else {
679            return kCusp_SkCubicType;
680        }
681    }
682}
683
684// Assumes the third component of points is 1.
685// Calcs p0 . (p1 x p2)
686static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
687    const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
688    const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
689    const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
690    return (xComp + yComp + wComp);
691}
692
693// Calc coefficients of I(s,t) where roots of I are inflection points of curve
694// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
695// d0 = a1 - 2*a2+3*a3
696// d1 = -a2 + 3*a3
697// d2 = 3*a3
698// a1 = p0 . (p3 x p2)
699// a2 = p1 . (p0 x p3)
700// a3 = p2 . (p1 x p0)
701// Places the values of d1, d2, d3 in array d passed in
702static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
703    SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
704    SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
705    SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
706
707    // need to scale a's or values in later calculations will grow to high
708    SkScalar max = SkScalarAbs(a1);
709    max = SkMaxScalar(max, SkScalarAbs(a2));
710    max = SkMaxScalar(max, SkScalarAbs(a3));
711    max = 1.f/max;
712    a1 = a1 * max;
713    a2 = a2 * max;
714    a3 = a3 * max;
715
716    d[2] = 3.f * a3;
717    d[1] = d[2] - a2;
718    d[0] = d[1] - a2 + a1;
719}
720
721SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
722    calc_cubic_inflection_func(src, d);
723    return classify_cubic(src, d);
724}
725
726template <typename T> void bubble_sort(T array[], int count) {
727    for (int i = count - 1; i > 0; --i)
728        for (int j = i; j > 0; --j)
729            if (array[j] < array[j-1])
730            {
731                T   tmp(array[j]);
732                array[j] = array[j-1];
733                array[j-1] = tmp;
734            }
735}
736
737/**
738 *  Given an array and count, remove all pair-wise duplicates from the array,
739 *  keeping the existing sorting, and return the new count
740 */
741static int collaps_duplicates(SkScalar array[], int count) {
742    for (int n = count; n > 1; --n) {
743        if (array[0] == array[1]) {
744            for (int i = 1; i < n; ++i) {
745                array[i - 1] = array[i];
746            }
747            count -= 1;
748        } else {
749            array += 1;
750        }
751    }
752    return count;
753}
754
755#ifdef SK_DEBUG
756
757#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
758
759static void test_collaps_duplicates() {
760    static bool gOnce;
761    if (gOnce) { return; }
762    gOnce = true;
763    const SkScalar src0[] = { 0 };
764    const SkScalar src1[] = { 0, 0 };
765    const SkScalar src2[] = { 0, 1 };
766    const SkScalar src3[] = { 0, 0, 0 };
767    const SkScalar src4[] = { 0, 0, 1 };
768    const SkScalar src5[] = { 0, 1, 1 };
769    const SkScalar src6[] = { 0, 1, 2 };
770    const struct {
771        const SkScalar* fData;
772        int fCount;
773        int fCollapsedCount;
774    } data[] = {
775        { TEST_COLLAPS_ENTRY(src0), 1 },
776        { TEST_COLLAPS_ENTRY(src1), 1 },
777        { TEST_COLLAPS_ENTRY(src2), 2 },
778        { TEST_COLLAPS_ENTRY(src3), 1 },
779        { TEST_COLLAPS_ENTRY(src4), 2 },
780        { TEST_COLLAPS_ENTRY(src5), 2 },
781        { TEST_COLLAPS_ENTRY(src6), 3 },
782    };
783    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
784        SkScalar dst[3];
785        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
786        int count = collaps_duplicates(dst, data[i].fCount);
787        SkASSERT(data[i].fCollapsedCount == count);
788        for (int j = 1; j < count; ++j) {
789            SkASSERT(dst[j-1] < dst[j]);
790        }
791    }
792}
793#endif
794
795static SkScalar SkScalarCubeRoot(SkScalar x) {
796    return SkScalarPow(x, 0.3333333f);
797}
798
799/*  Solve coeff(t) == 0, returning the number of roots that
800    lie withing 0 < t < 1.
801    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
802
803    Eliminates repeated roots (so that all tValues are distinct, and are always
804    in increasing order.
805*/
806static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
807    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
808        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
809    }
810
811    SkScalar a, b, c, Q, R;
812
813    {
814        SkASSERT(coeff[0] != 0);
815
816        SkScalar inva = SkScalarInvert(coeff[0]);
817        a = coeff[1] * inva;
818        b = coeff[2] * inva;
819        c = coeff[3] * inva;
820    }
821    Q = (a*a - b*3) / 9;
822    R = (2*a*a*a - 9*a*b + 27*c) / 54;
823
824    SkScalar Q3 = Q * Q * Q;
825    SkScalar R2MinusQ3 = R * R - Q3;
826    SkScalar adiv3 = a / 3;
827
828    SkScalar*   roots = tValues;
829    SkScalar    r;
830
831    if (R2MinusQ3 < 0) { // we have 3 real roots
832        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
833        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
834
835        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
836        if (is_unit_interval(r)) {
837            *roots++ = r;
838        }
839        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
840        if (is_unit_interval(r)) {
841            *roots++ = r;
842        }
843        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
844        if (is_unit_interval(r)) {
845            *roots++ = r;
846        }
847        SkDEBUGCODE(test_collaps_duplicates();)
848
849        // now sort the roots
850        int count = (int)(roots - tValues);
851        SkASSERT((unsigned)count <= 3);
852        bubble_sort(tValues, count);
853        count = collaps_duplicates(tValues, count);
854        roots = tValues + count;    // so we compute the proper count below
855    } else {              // we have 1 real root
856        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
857        A = SkScalarCubeRoot(A);
858        if (R > 0) {
859            A = -A;
860        }
861        if (A != 0) {
862            A += Q / A;
863        }
864        r = A - adiv3;
865        if (is_unit_interval(r)) {
866            *roots++ = r;
867        }
868    }
869
870    return (int)(roots - tValues);
871}
872
873/*  Looking for F' dot F'' == 0
874
875    A = b - a
876    B = c - 2b + a
877    C = d - 3c + 3b - a
878
879    F' = 3Ct^2 + 6Bt + 3A
880    F'' = 6Ct + 6B
881
882    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
883*/
884static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
885    SkScalar    a = src[2] - src[0];
886    SkScalar    b = src[4] - 2 * src[2] + src[0];
887    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
888
889    coeff[0] = c * c;
890    coeff[1] = 3 * b * c;
891    coeff[2] = 2 * b * b + c * a;
892    coeff[3] = a * b;
893}
894
895/*  Looking for F' dot F'' == 0
896
897    A = b - a
898    B = c - 2b + a
899    C = d - 3c + 3b - a
900
901    F' = 3Ct^2 + 6Bt + 3A
902    F'' = 6Ct + 6B
903
904    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
905*/
906int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
907    SkScalar coeffX[4], coeffY[4];
908    int      i;
909
910    formulate_F1DotF2(&src[0].fX, coeffX);
911    formulate_F1DotF2(&src[0].fY, coeffY);
912
913    for (i = 0; i < 4; i++) {
914        coeffX[i] += coeffY[i];
915    }
916
917    SkScalar    t[3];
918    int         count = solve_cubic_poly(coeffX, t);
919    int         maxCount = 0;
920
921    // now remove extrema where the curvature is zero (mins)
922    // !!!! need a test for this !!!!
923    for (i = 0; i < count; i++) {
924        // if (not_min_curvature())
925        if (t[i] > 0 && t[i] < SK_Scalar1) {
926            tValues[maxCount++] = t[i];
927        }
928    }
929    return maxCount;
930}
931
932int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
933                              SkScalar tValues[3]) {
934    SkScalar    t_storage[3];
935
936    if (tValues == NULL) {
937        tValues = t_storage;
938    }
939
940    int count = SkFindCubicMaxCurvature(src, tValues);
941
942    if (dst) {
943        if (count == 0) {
944            memcpy(dst, src, 4 * sizeof(SkPoint));
945        } else {
946            SkChopCubicAt(src, dst, tValues, count);
947        }
948    }
949    return count + 1;
950}
951
952bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4],
953                                 bool* ambiguous) {
954    if (ambiguous) {
955        *ambiguous = false;
956    }
957
958    // Find the minimum and maximum y of the extrema, which are the
959    // first and last points since this cubic is monotonic
960    SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
961    SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
962
963    if (pt.fY == cubic[0].fY
964        || pt.fY < min_y
965        || pt.fY > max_y) {
966        // The query line definitely does not cross the curve
967        if (ambiguous) {
968            *ambiguous = (pt.fY == cubic[0].fY);
969        }
970        return false;
971    }
972
973    bool pt_at_extremum = (pt.fY == cubic[3].fY);
974
975    SkScalar min_x =
976        SkMinScalar(
977            SkMinScalar(
978                SkMinScalar(cubic[0].fX, cubic[1].fX),
979                cubic[2].fX),
980            cubic[3].fX);
981    if (pt.fX < min_x) {
982        // The query line definitely crosses the curve
983        if (ambiguous) {
984            *ambiguous = pt_at_extremum;
985        }
986        return true;
987    }
988
989    SkScalar max_x =
990        SkMaxScalar(
991            SkMaxScalar(
992                SkMaxScalar(cubic[0].fX, cubic[1].fX),
993                cubic[2].fX),
994            cubic[3].fX);
995    if (pt.fX > max_x) {
996        // The query line definitely does not cross the curve
997        return false;
998    }
999
1000    // Do a binary search to find the parameter value which makes y as
1001    // close as possible to the query point. See whether the query
1002    // line's origin is to the left of the associated x coordinate.
1003
1004    // kMaxIter is chosen as the number of mantissa bits for a float,
1005    // since there's no way we are going to get more precision by
1006    // iterating more times than that.
1007    const int kMaxIter = 23;
1008    SkPoint eval;
1009    int iter = 0;
1010    SkScalar upper_t;
1011    SkScalar lower_t;
1012    // Need to invert direction of t parameter if cubic goes up
1013    // instead of down
1014    if (cubic[3].fY > cubic[0].fY) {
1015        upper_t = SK_Scalar1;
1016        lower_t = 0;
1017    } else {
1018        upper_t = 0;
1019        lower_t = SK_Scalar1;
1020    }
1021    do {
1022        SkScalar t = SkScalarAve(upper_t, lower_t);
1023        SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
1024        if (pt.fY > eval.fY) {
1025            lower_t = t;
1026        } else {
1027            upper_t = t;
1028        }
1029    } while (++iter < kMaxIter
1030             && !SkScalarNearlyZero(eval.fY - pt.fY));
1031    if (pt.fX <= eval.fX) {
1032        if (ambiguous) {
1033            *ambiguous = pt_at_extremum;
1034        }
1035        return true;
1036    }
1037    return false;
1038}
1039
1040int SkNumXRayCrossingsForCubic(const SkXRay& pt,
1041                               const SkPoint cubic[4],
1042                               bool* ambiguous) {
1043    int num_crossings = 0;
1044    SkPoint monotonic_cubics[10];
1045    int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
1046    if (ambiguous) {
1047        *ambiguous = false;
1048    }
1049    bool locally_ambiguous;
1050    if (SkXRayCrossesMonotonicCubic(pt,
1051                                    &monotonic_cubics[0],
1052                                    &locally_ambiguous))
1053        ++num_crossings;
1054    if (ambiguous) {
1055        *ambiguous |= locally_ambiguous;
1056    }
1057    if (num_monotonic_cubics > 0)
1058        if (SkXRayCrossesMonotonicCubic(pt,
1059                                        &monotonic_cubics[3],
1060                                        &locally_ambiguous))
1061            ++num_crossings;
1062    if (ambiguous) {
1063        *ambiguous |= locally_ambiguous;
1064    }
1065    if (num_monotonic_cubics > 1)
1066        if (SkXRayCrossesMonotonicCubic(pt,
1067                                        &monotonic_cubics[6],
1068                                        &locally_ambiguous))
1069            ++num_crossings;
1070    if (ambiguous) {
1071        *ambiguous |= locally_ambiguous;
1072    }
1073    return num_crossings;
1074}
1075
1076///////////////////////////////////////////////////////////////////////////////
1077
1078/*  Find t value for quadratic [a, b, c] = d.
1079    Return 0 if there is no solution within [0, 1)
1080*/
1081static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
1082    // At^2 + Bt + C = d
1083    SkScalar A = a - 2 * b + c;
1084    SkScalar B = 2 * (b - a);
1085    SkScalar C = a - d;
1086
1087    SkScalar    roots[2];
1088    int         count = SkFindUnitQuadRoots(A, B, C, roots);
1089
1090    SkASSERT(count <= 1);
1091    return count == 1 ? roots[0] : 0;
1092}
1093
1094/*  given a quad-curve and a point (x,y), chop the quad at that point and place
1095    the new off-curve point and endpoint into 'dest'.
1096    Should only return false if the computed pos is the start of the curve
1097    (i.e. root == 0)
1098*/
1099static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
1100                                SkPoint* dest) {
1101    const SkScalar* base;
1102    SkScalar        value;
1103
1104    if (SkScalarAbs(x) < SkScalarAbs(y)) {
1105        base = &quad[0].fX;
1106        value = x;
1107    } else {
1108        base = &quad[0].fY;
1109        value = y;
1110    }
1111
1112    // note: this returns 0 if it thinks value is out of range, meaning the
1113    // root might return something outside of [0, 1)
1114    SkScalar t = quad_solve(base[0], base[2], base[4], value);
1115
1116    if (t > 0) {
1117        SkPoint tmp[5];
1118        SkChopQuadAt(quad, tmp, t);
1119        dest[0] = tmp[1];
1120        dest[1].set(x, y);
1121        return true;
1122    } else {
1123        /*  t == 0 means either the value triggered a root outside of [0, 1)
1124            For our purposes, we can ignore the <= 0 roots, but we want to
1125            catch the >= 1 roots (which given our caller, will basically mean
1126            a root of 1, give-or-take numerical instability). If we are in the
1127            >= 1 case, return the existing offCurve point.
1128
1129            The test below checks to see if we are close to the "end" of the
1130            curve (near base[4]). Rather than specifying a tolerance, I just
1131            check to see if value is on to the right/left of the middle point
1132            (depending on the direction/sign of the end points).
1133        */
1134        if ((base[0] < base[4] && value > base[2]) ||
1135            (base[0] > base[4] && value < base[2]))   // should root have been 1
1136        {
1137            dest[0] = quad[1];
1138            dest[1].set(x, y);
1139            return true;
1140        }
1141    }
1142    return false;
1143}
1144
1145static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
1146// The mid point of the quadratic arc approximation is half way between the two
1147// control points. The float epsilon adjustment moves the on curve point out by
1148// two bits, distributing the convex test error between the round rect
1149// approximation and the convex cross product sign equality test.
1150#define SK_MID_RRECT_OFFSET \
1151    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1152    { SK_Scalar1,            0                      },
1153    { SK_Scalar1,            SK_ScalarTanPIOver8    },
1154    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
1155    { SK_ScalarTanPIOver8,   SK_Scalar1             },
1156
1157    { 0,                     SK_Scalar1             },
1158    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
1159    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
1160    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
1161
1162    { -SK_Scalar1,           0                      },
1163    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
1164    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
1165    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
1166
1167    { 0,                     -SK_Scalar1            },
1168    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
1169    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
1170    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
1171
1172    { SK_Scalar1,            0                      }
1173#undef SK_MID_RRECT_OFFSET
1174};
1175
1176int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1177                   SkRotationDirection dir, const SkMatrix* userMatrix,
1178                   SkPoint quadPoints[]) {
1179    // rotate by x,y so that uStart is (1.0)
1180    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1181    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1182
1183    SkScalar absX = SkScalarAbs(x);
1184    SkScalar absY = SkScalarAbs(y);
1185
1186    int pointCount;
1187
1188    // check for (effectively) coincident vectors
1189    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1190    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1191    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1192        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1193         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1194
1195        // just return the start-point
1196        quadPoints[0].set(SK_Scalar1, 0);
1197        pointCount = 1;
1198    } else {
1199        if (dir == kCCW_SkRotationDirection) {
1200            y = -y;
1201        }
1202        // what octant (quadratic curve) is [xy] in?
1203        int oct = 0;
1204        bool sameSign = true;
1205
1206        if (0 == y) {
1207            oct = 4;        // 180
1208            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1209        } else if (0 == x) {
1210            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1211            oct = y > 0 ? 2 : 6; // 90 : 270
1212        } else {
1213            if (y < 0) {
1214                oct += 4;
1215            }
1216            if ((x < 0) != (y < 0)) {
1217                oct += 2;
1218                sameSign = false;
1219            }
1220            if ((absX < absY) == sameSign) {
1221                oct += 1;
1222            }
1223        }
1224
1225        int wholeCount = oct << 1;
1226        memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1227
1228        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1229        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1230            wholeCount += 2;
1231        }
1232        pointCount = wholeCount + 1;
1233    }
1234
1235    // now handle counter-clockwise and the initial unitStart rotation
1236    SkMatrix    matrix;
1237    matrix.setSinCos(uStart.fY, uStart.fX);
1238    if (dir == kCCW_SkRotationDirection) {
1239        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1240    }
1241    if (userMatrix) {
1242        matrix.postConcat(*userMatrix);
1243    }
1244    matrix.mapPoints(quadPoints, pointCount);
1245    return pointCount;
1246}
1247
1248
1249///////////////////////////////////////////////////////////////////////////////
1250//
1251// NURB representation for conics.  Helpful explanations at:
1252//
1253// http://citeseerx.ist.psu.edu/viewdoc/
1254//   download?doi=10.1.1.44.5740&rep=rep1&type=ps
1255// and
1256// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1257//
1258// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1259//     ------------------------------------------
1260//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1261//
1262//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1263//     ------------------------------------------------
1264//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1265//
1266
1267static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1268    SkASSERT(src);
1269    SkASSERT(t >= 0 && t <= SK_Scalar1);
1270
1271    SkScalar    src2w = SkScalarMul(src[2], w);
1272    SkScalar    C = src[0];
1273    SkScalar    A = src[4] - 2 * src2w + C;
1274    SkScalar    B = 2 * (src2w - C);
1275    SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1276
1277    B = 2 * (w - SK_Scalar1);
1278    C = SK_Scalar1;
1279    A = -B;
1280    SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1281
1282    return SkScalarDiv(numer, denom);
1283}
1284
1285// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1286//
1287//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1288//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1289//  t^0 : -2 P0 w + 2 P1 w
1290//
1291//  We disregard magnitude, so we can freely ignore the denominator of F', and
1292//  divide the numerator by 2
1293//
1294//    coeff[0] for t^2
1295//    coeff[1] for t^1
1296//    coeff[2] for t^0
1297//
1298static void conic_deriv_coeff(const SkScalar src[],
1299                              SkScalar w,
1300                              SkScalar coeff[3]) {
1301    const SkScalar P20 = src[4] - src[0];
1302    const SkScalar P10 = src[2] - src[0];
1303    const SkScalar wP10 = w * P10;
1304    coeff[0] = w * P20 - P20;
1305    coeff[1] = P20 - 2 * wP10;
1306    coeff[2] = wP10;
1307}
1308
1309static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1310    SkScalar coeff[3];
1311    conic_deriv_coeff(coord, w, coeff);
1312    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1313}
1314
1315static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1316    SkScalar coeff[3];
1317    conic_deriv_coeff(src, w, coeff);
1318
1319    SkScalar tValues[2];
1320    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1321    SkASSERT(0 == roots || 1 == roots);
1322
1323    if (1 == roots) {
1324        *t = tValues[0];
1325        return true;
1326    }
1327    return false;
1328}
1329
1330struct SkP3D {
1331    SkScalar fX, fY, fZ;
1332
1333    void set(SkScalar x, SkScalar y, SkScalar z) {
1334        fX = x; fY = y; fZ = z;
1335    }
1336
1337    void projectDown(SkPoint* dst) const {
1338        dst->set(fX / fZ, fY / fZ);
1339    }
1340};
1341
1342// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1343static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1344    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1345    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1346    dst[0] = ab;
1347    dst[3] = SkScalarInterp(ab, bc, t);
1348    dst[6] = bc;
1349}
1350
1351static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1352    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1353    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1354    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1355}
1356
1357void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1358    SkASSERT(t >= 0 && t <= SK_Scalar1);
1359
1360    if (pt) {
1361        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1362                conic_eval_pos(&fPts[0].fY, fW, t));
1363    }
1364    if (tangent) {
1365        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1366                     conic_eval_tan(&fPts[0].fY, fW, t));
1367    }
1368}
1369
1370void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1371    SkP3D tmp[3], tmp2[3];
1372
1373    ratquad_mapTo3D(fPts, fW, tmp);
1374
1375    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1376    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1377    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1378
1379    dst[0].fPts[0] = fPts[0];
1380    tmp2[0].projectDown(&dst[0].fPts[1]);
1381    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1382    tmp2[2].projectDown(&dst[1].fPts[1]);
1383    dst[1].fPts[2] = fPts[2];
1384
1385    // to put in "standard form", where w0 and w2 are both 1, we compute the
1386    // new w1 as sqrt(w1*w1/w0*w2)
1387    // or
1388    // w1 /= sqrt(w0*w2)
1389    //
1390    // However, in our case, we know that for dst[0]:
1391    //     w0 == 1, and for dst[1], w2 == 1
1392    //
1393    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1394    dst[0].fW = tmp2[0].fZ / root;
1395    dst[1].fW = tmp2[2].fZ / root;
1396}
1397
1398static SkScalar subdivide_w_value(SkScalar w) {
1399    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1400}
1401
1402void SkConic::chop(SkConic dst[2]) const {
1403    SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
1404    SkScalar p1x = fW * fPts[1].fX;
1405    SkScalar p1y = fW * fPts[1].fY;
1406    SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
1407    SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
1408
1409    dst[0].fPts[0] = fPts[0];
1410    dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
1411                       (fPts[0].fY + p1y) * scale);
1412    dst[0].fPts[2].set(mx, my);
1413
1414    dst[1].fPts[0].set(mx, my);
1415    dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
1416                       (p1y + fPts[2].fY) * scale);
1417    dst[1].fPts[2] = fPts[2];
1418
1419    dst[0].fW = dst[1].fW = subdivide_w_value(fW);
1420}
1421
1422/*
1423 *  "High order approximation of conic sections by quadratic splines"
1424 *      by Michael Floater, 1993
1425 */
1426#define AS_QUAD_ERROR_SETUP                                         \
1427    SkScalar a = fW - 1;                                            \
1428    SkScalar k = a / (4 * (2 + a));                                 \
1429    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1430    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1431
1432void SkConic::computeAsQuadError(SkVector* err) const {
1433    AS_QUAD_ERROR_SETUP
1434    err->set(x, y);
1435}
1436
1437bool SkConic::asQuadTol(SkScalar tol) const {
1438    AS_QUAD_ERROR_SETUP
1439    return (x * x + y * y) <= tol * tol;
1440}
1441
1442int SkConic::computeQuadPOW2(SkScalar tol) const {
1443    AS_QUAD_ERROR_SETUP
1444    SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
1445
1446    if (error <= 0) {
1447        return 0;
1448    }
1449    uint32_t ierr = (uint32_t)error;
1450    return (34 - SkCLZ(ierr)) >> 1;
1451}
1452
1453static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1454    SkASSERT(level >= 0);
1455
1456    if (0 == level) {
1457        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1458        return pts + 2;
1459    } else {
1460        SkConic dst[2];
1461        src.chop(dst);
1462        --level;
1463        pts = subdivide(dst[0], pts, level);
1464        return subdivide(dst[1], pts, level);
1465    }
1466}
1467
1468int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1469    SkASSERT(pow2 >= 0);
1470    *pts = fPts[0];
1471    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1472    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1473    return 1 << pow2;
1474}
1475
1476bool SkConic::findXExtrema(SkScalar* t) const {
1477    return conic_find_extrema(&fPts[0].fX, fW, t);
1478}
1479
1480bool SkConic::findYExtrema(SkScalar* t) const {
1481    return conic_find_extrema(&fPts[0].fY, fW, t);
1482}
1483
1484bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1485    SkScalar t;
1486    if (this->findXExtrema(&t)) {
1487        this->chopAt(t, dst);
1488        // now clean-up the middle, since we know t was meant to be at
1489        // an X-extrema
1490        SkScalar value = dst[0].fPts[2].fX;
1491        dst[0].fPts[1].fX = value;
1492        dst[1].fPts[0].fX = value;
1493        dst[1].fPts[1].fX = value;
1494        return true;
1495    }
1496    return false;
1497}
1498
1499bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1500    SkScalar t;
1501    if (this->findYExtrema(&t)) {
1502        this->chopAt(t, dst);
1503        // now clean-up the middle, since we know t was meant to be at
1504        // an Y-extrema
1505        SkScalar value = dst[0].fPts[2].fY;
1506        dst[0].fPts[1].fY = value;
1507        dst[1].fPts[0].fY = value;
1508        dst[1].fPts[1].fY = value;
1509        return true;
1510    }
1511    return false;
1512}
1513
1514void SkConic::computeTightBounds(SkRect* bounds) const {
1515    SkPoint pts[4];
1516    pts[0] = fPts[0];
1517    pts[1] = fPts[2];
1518    int count = 2;
1519
1520    SkScalar t;
1521    if (this->findXExtrema(&t)) {
1522        this->evalAt(t, &pts[count++]);
1523    }
1524    if (this->findYExtrema(&t)) {
1525        this->evalAt(t, &pts[count++]);
1526    }
1527    bounds->set(pts, count);
1528}
1529
1530void SkConic::computeFastBounds(SkRect* bounds) const {
1531    bounds->set(fPts, 3);
1532}
1533
1534bool SkConic::findMaxCurvature(SkScalar* t) const {
1535    // TODO: Implement me
1536    return false;
1537}
1538
1539SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1540                             const SkMatrix& matrix) {
1541    if (!matrix.hasPerspective()) {
1542        return w;
1543    }
1544
1545    SkP3D src[3], dst[3];
1546
1547    ratquad_mapTo3D(pts, w, src);
1548
1549    matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1550
1551    // w' = sqrt(w1*w1/w0*w2)
1552    SkScalar w0 = dst[0].fZ;
1553    SkScalar w1 = dst[1].fZ;
1554    SkScalar w2 = dst[2].fZ;
1555    w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1556    return w;
1557}
1558