SkGeometry.cpp revision c9adb05b64fa0bfadf9d1a782afcda470da68c9e 
1/*
2 * Copyright 2006 The Android Open Source Project
3 *
4 * Use of this source code is governed by a BSD-style license that can be
5 * found in the LICENSE file.
6 */
7
8#include "SkGeometry.h"
9#include "SkMatrix.h"
10#include "SkNx.h"
11
12static Sk2s from_point(const SkPoint& point) {
13    return Sk2s::Load(&point.fX);
14}
15
16static SkPoint to_point(const Sk2s& x) {
17    SkPoint point;
18    x.store(&point.fX);
19    return point;
20}
21
22static SkVector to_vector(const Sk2s& x) {
23    SkVector vector;
24    x.store(&vector.fX);
25    return vector;
26}
27
28/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
29    involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
30    May also introduce overflow of fixed when we compute our setup.
31*/
32//    #define DIRECT_EVAL_OF_POLYNOMIALS
33
34////////////////////////////////////////////////////////////////////////
35
36static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
37    SkScalar ab = a - b;
38    SkScalar bc = b - c;
39    if (ab < 0) {
40        bc = -bc;
41    }
42    return ab == 0 || bc < 0;
43}
44
45////////////////////////////////////////////////////////////////////////
46
47static bool is_unit_interval(SkScalar x) {
48    return x > 0 && x < SK_Scalar1;
49}
50
51static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
52    SkASSERT(ratio);
53
54    if (numer < 0) {
55        numer = -numer;
56        denom = -denom;
57    }
58
59    if (denom == 0 || numer == 0 || numer >= denom) {
60        return 0;
61    }
62
63    SkScalar r = SkScalarDiv(numer, denom);
64    if (SkScalarIsNaN(r)) {
65        return 0;
66    }
67    SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
68    if (r == 0) { // catch underflow if numer <<<< denom
69        return 0;
70    }
71    *ratio = r;
72    return 1;
73}
74
75/** From Numerical Recipes in C.
76
77    Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
78    x1 = Q / A
79    x2 = C / Q
80*/
81int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
82    SkASSERT(roots);
83
84    if (A == 0) {
85        return valid_unit_divide(-C, B, roots);
86    }
87
88    SkScalar* r = roots;
89
90    SkScalar R = B*B - 4*A*C;
91    if (R < 0 || SkScalarIsNaN(R)) {  // complex roots
92        return 0;
93    }
94    R = SkScalarSqrt(R);
95
96    SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
97    r += valid_unit_divide(Q, A, r);
98    r += valid_unit_divide(C, Q, r);
99    if (r - roots == 2) {
100        if (roots[0] > roots[1])
101            SkTSwap<SkScalar>(roots[0], roots[1]);
102        else if (roots[0] == roots[1])  // nearly-equal?
103            r -= 1; // skip the double root
104    }
105    return (int)(r - roots);
106}
107
108///////////////////////////////////////////////////////////////////////////////
109///////////////////////////////////////////////////////////////////////////////
110
111static Sk2s quad_poly_eval(const Sk2s& A, const Sk2s& B, const Sk2s& C, const Sk2s& t) {
112    return (A * t + B) * t + C;
113}
114
115static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
116    SkASSERT(src);
117    SkASSERT(t >= 0 && t <= SK_Scalar1);
118
119#ifdef DIRECT_EVAL_OF_POLYNOMIALS
120    SkScalar    C = src[0];
121    SkScalar    A = src[4] - 2 * src[2] + C;
122    SkScalar    B = 2 * (src[2] - C);
123    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
124#else
125    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
126    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
127    return SkScalarInterp(ab, bc, t);
128#endif
129}
130
131static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
132    SkScalar A = src[4] - 2 * src[2] + src[0];
133    SkScalar B = src[2] - src[0];
134
135    return 2 * SkScalarMulAdd(A, t, B);
136}
137
138void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) {
139    SkASSERT(src);
140    SkASSERT(t >= 0 && t <= SK_Scalar1);
141
142    if (pt) {
143        pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
144    }
145    if (tangent) {
146        tangent->set(eval_quad_derivative(&src[0].fX, t),
147                     eval_quad_derivative(&src[0].fY, t));
148    }
149}
150
151SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) {
152    SkASSERT(src);
153    SkASSERT(t >= 0 && t <= SK_Scalar1);
154
155    const Sk2s t2(t);
156
157    Sk2s P0 = from_point(src[0]);
158    Sk2s P1 = from_point(src[1]);
159    Sk2s P2 = from_point(src[2]);
160
161    Sk2s B = P1 - P0;
162    Sk2s A = P2 - P1 - B;
163
164    return to_point((A * t2 + B+B) * t2 + P0);
165}
166
167SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) {
168    SkASSERT(src);
169    SkASSERT(t >= 0 && t <= SK_Scalar1);
170
171    Sk2s P0 = from_point(src[0]);
172    Sk2s P1 = from_point(src[1]);
173    Sk2s P2 = from_point(src[2]);
174
175    Sk2s B = P1 - P0;
176    Sk2s A = P2 - P1 - B;
177    Sk2s T = A * Sk2s(t) + B;
178
179    return to_vector(T + T);
180}
181
182static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) {
183    return v0 + (v1 - v0) * t;
184}
185
186void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
187    SkASSERT(t > 0 && t < SK_Scalar1);
188
189    Sk2s p0 = from_point(src[0]);
190    Sk2s p1 = from_point(src[1]);
191    Sk2s p2 = from_point(src[2]);
192    Sk2s tt(t);
193
194    Sk2s p01 = interp(p0, p1, tt);
195    Sk2s p12 = interp(p1, p2, tt);
196
197    dst[0] = to_point(p0);
198    dst[1] = to_point(p01);
199    dst[2] = to_point(interp(p01, p12, tt));
200    dst[3] = to_point(p12);
201    dst[4] = to_point(p2);
202}
203
204void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
205    SkChopQuadAt(src, dst, 0.5f); return;
206}
207
208/** Quad'(t) = At + B, where
209    A = 2(a - 2b + c)
210    B = 2(b - a)
211    Solve for t, only if it fits between 0 < t < 1
212*/
213int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
214    /*  At + B == 0
215        t = -B / A
216    */
217    return valid_unit_divide(a - b, a - b - b + c, tValue);
218}
219
220static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
221    coords[2] = coords[6] = coords[4];
222}
223
224/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
225 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
226 */
227int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
228    SkASSERT(src);
229    SkASSERT(dst);
230
231    SkScalar a = src[0].fY;
232    SkScalar b = src[1].fY;
233    SkScalar c = src[2].fY;
234
235    if (is_not_monotonic(a, b, c)) {
236        SkScalar    tValue;
237        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
238            SkChopQuadAt(src, dst, tValue);
239            flatten_double_quad_extrema(&dst[0].fY);
240            return 1;
241        }
242        // if we get here, we need to force dst to be monotonic, even though
243        // we couldn't compute a unit_divide value (probably underflow).
244        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
245    }
246    dst[0].set(src[0].fX, a);
247    dst[1].set(src[1].fX, b);
248    dst[2].set(src[2].fX, c);
249    return 0;
250}
251
252/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
253    stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
254 */
255int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
256    SkASSERT(src);
257    SkASSERT(dst);
258
259    SkScalar a = src[0].fX;
260    SkScalar b = src[1].fX;
261    SkScalar c = src[2].fX;
262
263    if (is_not_monotonic(a, b, c)) {
264        SkScalar tValue;
265        if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
266            SkChopQuadAt(src, dst, tValue);
267            flatten_double_quad_extrema(&dst[0].fX);
268            return 1;
269        }
270        // if we get here, we need to force dst to be monotonic, even though
271        // we couldn't compute a unit_divide value (probably underflow).
272        b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
273    }
274    dst[0].set(a, src[0].fY);
275    dst[1].set(b, src[1].fY);
276    dst[2].set(c, src[2].fY);
277    return 0;
278}
279
280//  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
281//  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
282//  F''(t)  = 2 (a - 2b + c)
283//
284//  A = 2 (b - a)
285//  B = 2 (a - 2b + c)
286//
287//  Maximum curvature for a quadratic means solving
288//  Fx' Fx'' + Fy' Fy'' = 0
289//
290//  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
291//
292SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
293    SkScalar    Ax = src[1].fX - src[0].fX;
294    SkScalar    Ay = src[1].fY - src[0].fY;
295    SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
296    SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
297    SkScalar    t = 0;  // 0 means don't chop
298
299    (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
300    return t;
301}
302
303int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
304    SkScalar t = SkFindQuadMaxCurvature(src);
305    if (t == 0) {
306        memcpy(dst, src, 3 * sizeof(SkPoint));
307        return 1;
308    } else {
309        SkChopQuadAt(src, dst, t);
310        return 2;
311    }
312}
313
314void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
315    Sk2s scale(SkDoubleToScalar(2.0 / 3.0));
316    Sk2s s0 = from_point(src[0]);
317    Sk2s s1 = from_point(src[1]);
318    Sk2s s2 = from_point(src[2]);
319
320    dst[0] = src[0];
321    dst[1] = to_point(s0 + (s1 - s0) * scale);
322    dst[2] = to_point(s2 + (s1 - s2) * scale);
323    dst[3] = src[2];
324}
325
326//////////////////////////////////////////////////////////////////////////////
327///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
328//////////////////////////////////////////////////////////////////////////////
329
330static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
331    coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
332    coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
333    coeff[2] = 3*(pt[2] - pt[0]);
334    coeff[3] = pt[0];
335}
336
337void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
338    SkASSERT(pts);
339
340    if (cx) {
341        get_cubic_coeff(&pts[0].fX, cx);
342    }
343    if (cy) {
344        get_cubic_coeff(&pts[0].fY, cy);
345    }
346}
347
348static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
349    SkASSERT(src);
350    SkASSERT(t >= 0 && t <= SK_Scalar1);
351
352    if (t == 0) {
353        return src[0];
354    }
355
356#ifdef DIRECT_EVAL_OF_POLYNOMIALS
357    SkScalar D = src[0];
358    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
359    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
360    SkScalar C = 3*(src[2] - D);
361
362    return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
363#else
364    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
365    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
366    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
367    SkScalar    abc = SkScalarInterp(ab, bc, t);
368    SkScalar    bcd = SkScalarInterp(bc, cd, t);
369    return SkScalarInterp(abc, bcd, t);
370#endif
371}
372
373/** return At^2 + Bt + C
374*/
375static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
376    SkASSERT(t >= 0 && t <= SK_Scalar1);
377
378    return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
379}
380
381static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
382    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
383    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
384    SkScalar C = src[2] - src[0];
385
386    return eval_quadratic(A, B, C, t);
387}
388
389static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
390    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
391    SkScalar B = src[4] - 2 * src[2] + src[0];
392
393    return SkScalarMulAdd(A, t, B);
394}
395
396void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
397                   SkVector* tangent, SkVector* curvature) {
398    SkASSERT(src);
399    SkASSERT(t >= 0 && t <= SK_Scalar1);
400
401    if (loc) {
402        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
403    }
404    if (tangent) {
405        tangent->set(eval_cubic_derivative(&src[0].fX, t),
406                     eval_cubic_derivative(&src[0].fY, t));
407    }
408    if (curvature) {
409        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
410                       eval_cubic_2ndDerivative(&src[0].fY, t));
411    }
412}
413
414/** Cubic'(t) = At^2 + Bt + C, where
415    A = 3(-a + 3(b - c) + d)
416    B = 6(a - 2b + c)
417    C = 3(b - a)
418    Solve for t, keeping only those that fit betwee 0 < t < 1
419*/
420int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
421                       SkScalar tValues[2]) {
422    // we divide A,B,C by 3 to simplify
423    SkScalar A = d - a + 3*(b - c);
424    SkScalar B = 2*(a - b - b + c);
425    SkScalar C = b - a;
426
427    return SkFindUnitQuadRoots(A, B, C, tValues);
428}
429
430void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
431    SkASSERT(t > 0 && t < SK_Scalar1);
432
433    Sk2s    p0 = from_point(src[0]);
434    Sk2s    p1 = from_point(src[1]);
435    Sk2s    p2 = from_point(src[2]);
436    Sk2s    p3 = from_point(src[3]);
437    Sk2s    tt(t);
438
439    Sk2s    ab = interp(p0, p1, tt);
440    Sk2s    bc = interp(p1, p2, tt);
441    Sk2s    cd = interp(p2, p3, tt);
442    Sk2s    abc = interp(ab, bc, tt);
443    Sk2s    bcd = interp(bc, cd, tt);
444    Sk2s    abcd = interp(abc, bcd, tt);
445
446    dst[0] = src[0];
447    dst[1] = to_point(ab);
448    dst[2] = to_point(abc);
449    dst[3] = to_point(abcd);
450    dst[4] = to_point(bcd);
451    dst[5] = to_point(cd);
452    dst[6] = src[3];
453}
454
455/*  http://code.google.com/p/skia/issues/detail?id=32
456
457    This test code would fail when we didn't check the return result of
458    valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
459    that after the first chop, the parameters to valid_unit_divide are equal
460    (thanks to finite float precision and rounding in the subtracts). Thus
461    even though the 2nd tValue looks < 1.0, after we renormalize it, we end
462    up with 1.0, hence the need to check and just return the last cubic as
463    a degenerate clump of 4 points in the sampe place.
464
465    static void test_cubic() {
466        SkPoint src[4] = {
467            { 556.25000, 523.03003 },
468            { 556.23999, 522.96002 },
469            { 556.21997, 522.89001 },
470            { 556.21997, 522.82001 }
471        };
472        SkPoint dst[10];
473        SkScalar tval[] = { 0.33333334f, 0.99999994f };
474        SkChopCubicAt(src, dst, tval, 2);
475    }
476 */
477
478void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
479                   const SkScalar tValues[], int roots) {
480#ifdef SK_DEBUG
481    {
482        for (int i = 0; i < roots - 1; i++)
483        {
484            SkASSERT(is_unit_interval(tValues[i]));
485            SkASSERT(is_unit_interval(tValues[i+1]));
486            SkASSERT(tValues[i] < tValues[i+1]);
487        }
488    }
489#endif
490
491    if (dst) {
492        if (roots == 0) { // nothing to chop
493            memcpy(dst, src, 4*sizeof(SkPoint));
494        } else {
495            SkScalar    t = tValues[0];
496            SkPoint     tmp[4];
497
498            for (int i = 0; i < roots; i++) {
499                SkChopCubicAt(src, dst, t);
500                if (i == roots - 1) {
501                    break;
502                }
503
504                dst += 3;
505                // have src point to the remaining cubic (after the chop)
506                memcpy(tmp, dst, 4 * sizeof(SkPoint));
507                src = tmp;
508
509                // watch out in case the renormalized t isn't in range
510                if (!valid_unit_divide(tValues[i+1] - tValues[i],
511                                       SK_Scalar1 - tValues[i], &t)) {
512                    // if we can't, just create a degenerate cubic
513                    dst[4] = dst[5] = dst[6] = src[3];
514                    break;
515                }
516            }
517        }
518    }
519}
520
521void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
522    SkChopCubicAt(src, dst, 0.5f);
523}
524
525static void flatten_double_cubic_extrema(SkScalar coords[14]) {
526    coords[4] = coords[8] = coords[6];
527}
528
529/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
530    the resulting beziers are monotonic in Y. This is called by the scan
531    converter.  Depending on what is returned, dst[] is treated as follows:
532    0   dst[0..3] is the original cubic
533    1   dst[0..3] and dst[3..6] are the two new cubics
534    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
535    If dst == null, it is ignored and only the count is returned.
536*/
537int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
538    SkScalar    tValues[2];
539    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
540                                           src[3].fY, tValues);
541
542    SkChopCubicAt(src, dst, tValues, roots);
543    if (dst && roots > 0) {
544        // we do some cleanup to ensure our Y extrema are flat
545        flatten_double_cubic_extrema(&dst[0].fY);
546        if (roots == 2) {
547            flatten_double_cubic_extrema(&dst[3].fY);
548        }
549    }
550    return roots;
551}
552
553int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
554    SkScalar    tValues[2];
555    int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
556                                           src[3].fX, tValues);
557
558    SkChopCubicAt(src, dst, tValues, roots);
559    if (dst && roots > 0) {
560        // we do some cleanup to ensure our Y extrema are flat
561        flatten_double_cubic_extrema(&dst[0].fX);
562        if (roots == 2) {
563            flatten_double_cubic_extrema(&dst[3].fX);
564        }
565    }
566    return roots;
567}
568
569/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
570
571    Inflection means that curvature is zero.
572    Curvature is [F' x F''] / [F'^3]
573    So we solve F'x X F''y - F'y X F''y == 0
574    After some canceling of the cubic term, we get
575    A = b - a
576    B = c - 2b + a
577    C = d - 3c + 3b - a
578    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
579*/
580int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
581    SkScalar    Ax = src[1].fX - src[0].fX;
582    SkScalar    Ay = src[1].fY - src[0].fY;
583    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
584    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
585    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
586    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
587
588    return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
589                               Ax*Cy - Ay*Cx,
590                               Ax*By - Ay*Bx,
591                               tValues);
592}
593
594int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
595    SkScalar    tValues[2];
596    int         count = SkFindCubicInflections(src, tValues);
597
598    if (dst) {
599        if (count == 0) {
600            memcpy(dst, src, 4 * sizeof(SkPoint));
601        } else {
602            SkChopCubicAt(src, dst, tValues, count);
603        }
604    }
605    return count + 1;
606}
607
608// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
609// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
610// Classification:
611// discr(I) > 0        Serpentine
612// discr(I) = 0        Cusp
613// discr(I) < 0        Loop
614// d0 = d1 = 0         Quadratic
615// d0 = d1 = d2 = 0    Line
616// p0 = p1 = p2 = p3   Point
617static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
618    if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
619        return kPoint_SkCubicType;
620    }
621    const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
622    if (discr > SK_ScalarNearlyZero) {
623        return kSerpentine_SkCubicType;
624    } else if (discr < -SK_ScalarNearlyZero) {
625        return kLoop_SkCubicType;
626    } else {
627        if (0.f == d[0] && 0.f == d[1]) {
628            return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
629        } else {
630            return kCusp_SkCubicType;
631        }
632    }
633}
634
635// Assumes the third component of points is 1.
636// Calcs p0 . (p1 x p2)
637static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
638    const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
639    const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
640    const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
641    return (xComp + yComp + wComp);
642}
643
644// Calc coefficients of I(s,t) where roots of I are inflection points of curve
645// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
646// d0 = a1 - 2*a2+3*a3
647// d1 = -a2 + 3*a3
648// d2 = 3*a3
649// a1 = p0 . (p3 x p2)
650// a2 = p1 . (p0 x p3)
651// a3 = p2 . (p1 x p0)
652// Places the values of d1, d2, d3 in array d passed in
653static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
654    SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
655    SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
656    SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
657
658    // need to scale a's or values in later calculations will grow to high
659    SkScalar max = SkScalarAbs(a1);
660    max = SkMaxScalar(max, SkScalarAbs(a2));
661    max = SkMaxScalar(max, SkScalarAbs(a3));
662    max = 1.f/max;
663    a1 = a1 * max;
664    a2 = a2 * max;
665    a3 = a3 * max;
666
667    d[2] = 3.f * a3;
668    d[1] = d[2] - a2;
669    d[0] = d[1] - a2 + a1;
670}
671
672SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
673    calc_cubic_inflection_func(src, d);
674    return classify_cubic(src, d);
675}
676
677template <typename T> void bubble_sort(T array[], int count) {
678    for (int i = count - 1; i > 0; --i)
679        for (int j = i; j > 0; --j)
680            if (array[j] < array[j-1])
681            {
682                T   tmp(array[j]);
683                array[j] = array[j-1];
684                array[j-1] = tmp;
685            }
686}
687
688/**
689 *  Given an array and count, remove all pair-wise duplicates from the array,
690 *  keeping the existing sorting, and return the new count
691 */
692static int collaps_duplicates(SkScalar array[], int count) {
693    for (int n = count; n > 1; --n) {
694        if (array[0] == array[1]) {
695            for (int i = 1; i < n; ++i) {
696                array[i - 1] = array[i];
697            }
698            count -= 1;
699        } else {
700            array += 1;
701        }
702    }
703    return count;
704}
705
706#ifdef SK_DEBUG
707
708#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
709
710static void test_collaps_duplicates() {
711    static bool gOnce;
712    if (gOnce) { return; }
713    gOnce = true;
714    const SkScalar src0[] = { 0 };
715    const SkScalar src1[] = { 0, 0 };
716    const SkScalar src2[] = { 0, 1 };
717    const SkScalar src3[] = { 0, 0, 0 };
718    const SkScalar src4[] = { 0, 0, 1 };
719    const SkScalar src5[] = { 0, 1, 1 };
720    const SkScalar src6[] = { 0, 1, 2 };
721    const struct {
722        const SkScalar* fData;
723        int fCount;
724        int fCollapsedCount;
725    } data[] = {
726        { TEST_COLLAPS_ENTRY(src0), 1 },
727        { TEST_COLLAPS_ENTRY(src1), 1 },
728        { TEST_COLLAPS_ENTRY(src2), 2 },
729        { TEST_COLLAPS_ENTRY(src3), 1 },
730        { TEST_COLLAPS_ENTRY(src4), 2 },
731        { TEST_COLLAPS_ENTRY(src5), 2 },
732        { TEST_COLLAPS_ENTRY(src6), 3 },
733    };
734    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
735        SkScalar dst[3];
736        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
737        int count = collaps_duplicates(dst, data[i].fCount);
738        SkASSERT(data[i].fCollapsedCount == count);
739        for (int j = 1; j < count; ++j) {
740            SkASSERT(dst[j-1] < dst[j]);
741        }
742    }
743}
744#endif
745
746static SkScalar SkScalarCubeRoot(SkScalar x) {
747    return SkScalarPow(x, 0.3333333f);
748}
749
750/*  Solve coeff(t) == 0, returning the number of roots that
751    lie withing 0 < t < 1.
752    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
753
754    Eliminates repeated roots (so that all tValues are distinct, and are always
755    in increasing order.
756*/
757static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
758    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
759        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
760    }
761
762    SkScalar a, b, c, Q, R;
763
764    {
765        SkASSERT(coeff[0] != 0);
766
767        SkScalar inva = SkScalarInvert(coeff[0]);
768        a = coeff[1] * inva;
769        b = coeff[2] * inva;
770        c = coeff[3] * inva;
771    }
772    Q = (a*a - b*3) / 9;
773    R = (2*a*a*a - 9*a*b + 27*c) / 54;
774
775    SkScalar Q3 = Q * Q * Q;
776    SkScalar R2MinusQ3 = R * R - Q3;
777    SkScalar adiv3 = a / 3;
778
779    SkScalar*   roots = tValues;
780    SkScalar    r;
781
782    if (R2MinusQ3 < 0) { // we have 3 real roots
783        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
784        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
785
786        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
787        if (is_unit_interval(r)) {
788            *roots++ = r;
789        }
790        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
791        if (is_unit_interval(r)) {
792            *roots++ = r;
793        }
794        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
795        if (is_unit_interval(r)) {
796            *roots++ = r;
797        }
798        SkDEBUGCODE(test_collaps_duplicates();)
799
800        // now sort the roots
801        int count = (int)(roots - tValues);
802        SkASSERT((unsigned)count <= 3);
803        bubble_sort(tValues, count);
804        count = collaps_duplicates(tValues, count);
805        roots = tValues + count;    // so we compute the proper count below
806    } else {              // we have 1 real root
807        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
808        A = SkScalarCubeRoot(A);
809        if (R > 0) {
810            A = -A;
811        }
812        if (A != 0) {
813            A += Q / A;
814        }
815        r = A - adiv3;
816        if (is_unit_interval(r)) {
817            *roots++ = r;
818        }
819    }
820
821    return (int)(roots - tValues);
822}
823
824/*  Looking for F' dot F'' == 0
825
826    A = b - a
827    B = c - 2b + a
828    C = d - 3c + 3b - a
829
830    F' = 3Ct^2 + 6Bt + 3A
831    F'' = 6Ct + 6B
832
833    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
834*/
835static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
836    SkScalar    a = src[2] - src[0];
837    SkScalar    b = src[4] - 2 * src[2] + src[0];
838    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
839
840    coeff[0] = c * c;
841    coeff[1] = 3 * b * c;
842    coeff[2] = 2 * b * b + c * a;
843    coeff[3] = a * b;
844}
845
846/*  Looking for F' dot F'' == 0
847
848    A = b - a
849    B = c - 2b + a
850    C = d - 3c + 3b - a
851
852    F' = 3Ct^2 + 6Bt + 3A
853    F'' = 6Ct + 6B
854
855    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
856*/
857int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
858    SkScalar coeffX[4], coeffY[4];
859    int      i;
860
861    formulate_F1DotF2(&src[0].fX, coeffX);
862    formulate_F1DotF2(&src[0].fY, coeffY);
863
864    for (i = 0; i < 4; i++) {
865        coeffX[i] += coeffY[i];
866    }
867
868    SkScalar    t[3];
869    int         count = solve_cubic_poly(coeffX, t);
870    int         maxCount = 0;
871
872    // now remove extrema where the curvature is zero (mins)
873    // !!!! need a test for this !!!!
874    for (i = 0; i < count; i++) {
875        // if (not_min_curvature())
876        if (t[i] > 0 && t[i] < SK_Scalar1) {
877            tValues[maxCount++] = t[i];
878        }
879    }
880    return maxCount;
881}
882
883int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
884                              SkScalar tValues[3]) {
885    SkScalar    t_storage[3];
886
887    if (tValues == NULL) {
888        tValues = t_storage;
889    }
890
891    int count = SkFindCubicMaxCurvature(src, tValues);
892
893    if (dst) {
894        if (count == 0) {
895            memcpy(dst, src, 4 * sizeof(SkPoint));
896        } else {
897            SkChopCubicAt(src, dst, tValues, count);
898        }
899    }
900    return count + 1;
901}
902
903///////////////////////////////////////////////////////////////////////////////
904
905/*  Find t value for quadratic [a, b, c] = d.
906    Return 0 if there is no solution within [0, 1)
907*/
908static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
909    // At^2 + Bt + C = d
910    SkScalar A = a - 2 * b + c;
911    SkScalar B = 2 * (b - a);
912    SkScalar C = a - d;
913
914    SkScalar    roots[2];
915    int         count = SkFindUnitQuadRoots(A, B, C, roots);
916
917    SkASSERT(count <= 1);
918    return count == 1 ? roots[0] : 0;
919}
920
921/*  given a quad-curve and a point (x,y), chop the quad at that point and place
922    the new off-curve point and endpoint into 'dest'.
923    Should only return false if the computed pos is the start of the curve
924    (i.e. root == 0)
925*/
926static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
927                                SkPoint* dest) {
928    const SkScalar* base;
929    SkScalar        value;
930
931    if (SkScalarAbs(x) < SkScalarAbs(y)) {
932        base = &quad[0].fX;
933        value = x;
934    } else {
935        base = &quad[0].fY;
936        value = y;
937    }
938
939    // note: this returns 0 if it thinks value is out of range, meaning the
940    // root might return something outside of [0, 1)
941    SkScalar t = quad_solve(base[0], base[2], base[4], value);
942
943    if (t > 0) {
944        SkPoint tmp[5];
945        SkChopQuadAt(quad, tmp, t);
946        dest[0] = tmp[1];
947        dest[1].set(x, y);
948        return true;
949    } else {
950        /*  t == 0 means either the value triggered a root outside of [0, 1)
951            For our purposes, we can ignore the <= 0 roots, but we want to
952            catch the >= 1 roots (which given our caller, will basically mean
953            a root of 1, give-or-take numerical instability). If we are in the
954            >= 1 case, return the existing offCurve point.
955
956            The test below checks to see if we are close to the "end" of the
957            curve (near base[4]). Rather than specifying a tolerance, I just
958            check to see if value is on to the right/left of the middle point
959            (depending on the direction/sign of the end points).
960        */
961        if ((base[0] < base[4] && value > base[2]) ||
962            (base[0] > base[4] && value < base[2]))   // should root have been 1
963        {
964            dest[0] = quad[1];
965            dest[1].set(x, y);
966            return true;
967        }
968    }
969    return false;
970}
971
972static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
973// The mid point of the quadratic arc approximation is half way between the two
974// control points. The float epsilon adjustment moves the on curve point out by
975// two bits, distributing the convex test error between the round rect
976// approximation and the convex cross product sign equality test.
977#define SK_MID_RRECT_OFFSET \
978    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
979    { SK_Scalar1,            0                      },
980    { SK_Scalar1,            SK_ScalarTanPIOver8    },
981    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
982    { SK_ScalarTanPIOver8,   SK_Scalar1             },
983
984    { 0,                     SK_Scalar1             },
985    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
986    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
987    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
988
989    { -SK_Scalar1,           0                      },
990    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
991    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
992    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
993
994    { 0,                     -SK_Scalar1            },
995    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
996    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
997    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
998
999    { SK_Scalar1,            0                      }
1000#undef SK_MID_RRECT_OFFSET
1001};
1002
1003int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1004                   SkRotationDirection dir, const SkMatrix* userMatrix,
1005                   SkPoint quadPoints[]) {
1006    // rotate by x,y so that uStart is (1.0)
1007    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1008    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1009
1010    SkScalar absX = SkScalarAbs(x);
1011    SkScalar absY = SkScalarAbs(y);
1012
1013    int pointCount;
1014
1015    // check for (effectively) coincident vectors
1016    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1017    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1018    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1019        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1020         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1021
1022        // just return the start-point
1023        quadPoints[0].set(SK_Scalar1, 0);
1024        pointCount = 1;
1025    } else {
1026        if (dir == kCCW_SkRotationDirection) {
1027            y = -y;
1028        }
1029        // what octant (quadratic curve) is [xy] in?
1030        int oct = 0;
1031        bool sameSign = true;
1032
1033        if (0 == y) {
1034            oct = 4;        // 180
1035            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1036        } else if (0 == x) {
1037            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1038            oct = y > 0 ? 2 : 6; // 90 : 270
1039        } else {
1040            if (y < 0) {
1041                oct += 4;
1042            }
1043            if ((x < 0) != (y < 0)) {
1044                oct += 2;
1045                sameSign = false;
1046            }
1047            if ((absX < absY) == sameSign) {
1048                oct += 1;
1049            }
1050        }
1051
1052        int wholeCount = oct << 1;
1053        memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1054
1055        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1056        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1057            wholeCount += 2;
1058        }
1059        pointCount = wholeCount + 1;
1060    }
1061
1062    // now handle counter-clockwise and the initial unitStart rotation
1063    SkMatrix    matrix;
1064    matrix.setSinCos(uStart.fY, uStart.fX);
1065    if (dir == kCCW_SkRotationDirection) {
1066        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1067    }
1068    if (userMatrix) {
1069        matrix.postConcat(*userMatrix);
1070    }
1071    matrix.mapPoints(quadPoints, pointCount);
1072    return pointCount;
1073}
1074
1075
1076///////////////////////////////////////////////////////////////////////////////
1077//
1078// NURB representation for conics.  Helpful explanations at:
1079//
1080// http://citeseerx.ist.psu.edu/viewdoc/
1081//   download?doi=10.1.1.44.5740&rep=rep1&type=ps
1082// and
1083// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1084//
1085// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1086//     ------------------------------------------
1087//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1088//
1089//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1090//     ------------------------------------------------
1091//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1092//
1093
1094static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1095    SkASSERT(src);
1096    SkASSERT(t >= 0 && t <= SK_Scalar1);
1097
1098    SkScalar    src2w = SkScalarMul(src[2], w);
1099    SkScalar    C = src[0];
1100    SkScalar    A = src[4] - 2 * src2w + C;
1101    SkScalar    B = 2 * (src2w - C);
1102    SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1103
1104    B = 2 * (w - SK_Scalar1);
1105    C = SK_Scalar1;
1106    A = -B;
1107    SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1108
1109    return SkScalarDiv(numer, denom);
1110}
1111
1112// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1113//
1114//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1115//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1116//  t^0 : -2 P0 w + 2 P1 w
1117//
1118//  We disregard magnitude, so we can freely ignore the denominator of F', and
1119//  divide the numerator by 2
1120//
1121//    coeff[0] for t^2
1122//    coeff[1] for t^1
1123//    coeff[2] for t^0
1124//
1125static void conic_deriv_coeff(const SkScalar src[],
1126                              SkScalar w,
1127                              SkScalar coeff[3]) {
1128    const SkScalar P20 = src[4] - src[0];
1129    const SkScalar P10 = src[2] - src[0];
1130    const SkScalar wP10 = w * P10;
1131    coeff[0] = w * P20 - P20;
1132    coeff[1] = P20 - 2 * wP10;
1133    coeff[2] = wP10;
1134}
1135
1136static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1137    SkScalar coeff[3];
1138    conic_deriv_coeff(coord, w, coeff);
1139    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1140}
1141
1142static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1143    SkScalar coeff[3];
1144    conic_deriv_coeff(src, w, coeff);
1145
1146    SkScalar tValues[2];
1147    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1148    SkASSERT(0 == roots || 1 == roots);
1149
1150    if (1 == roots) {
1151        *t = tValues[0];
1152        return true;
1153    }
1154    return false;
1155}
1156
1157struct SkP3D {
1158    SkScalar fX, fY, fZ;
1159
1160    void set(SkScalar x, SkScalar y, SkScalar z) {
1161        fX = x; fY = y; fZ = z;
1162    }
1163
1164    void projectDown(SkPoint* dst) const {
1165        dst->set(fX / fZ, fY / fZ);
1166    }
1167};
1168
1169// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1170static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1171    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1172    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1173    dst[0] = ab;
1174    dst[3] = SkScalarInterp(ab, bc, t);
1175    dst[6] = bc;
1176}
1177
1178static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1179    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1180    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1181    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1182}
1183
1184void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1185    SkASSERT(t >= 0 && t <= SK_Scalar1);
1186
1187    if (pt) {
1188        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1189                conic_eval_pos(&fPts[0].fY, fW, t));
1190    }
1191    if (tangent) {
1192        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1193                     conic_eval_tan(&fPts[0].fY, fW, t));
1194    }
1195}
1196
1197void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1198    SkP3D tmp[3], tmp2[3];
1199
1200    ratquad_mapTo3D(fPts, fW, tmp);
1201
1202    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1203    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1204    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1205
1206    dst[0].fPts[0] = fPts[0];
1207    tmp2[0].projectDown(&dst[0].fPts[1]);
1208    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1209    tmp2[2].projectDown(&dst[1].fPts[1]);
1210    dst[1].fPts[2] = fPts[2];
1211
1212    // to put in "standard form", where w0 and w2 are both 1, we compute the
1213    // new w1 as sqrt(w1*w1/w0*w2)
1214    // or
1215    // w1 /= sqrt(w0*w2)
1216    //
1217    // However, in our case, we know that for dst[0]:
1218    //     w0 == 1, and for dst[1], w2 == 1
1219    //
1220    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1221    dst[0].fW = tmp2[0].fZ / root;
1222    dst[1].fW = tmp2[2].fZ / root;
1223}
1224
1225static Sk2s times_2(const Sk2s& value) {
1226    return value + value;
1227}
1228
1229SkPoint SkConic::evalAt(SkScalar t) const {
1230    Sk2s p0 = from_point(fPts[0]);
1231    Sk2s p1 = from_point(fPts[1]);
1232    Sk2s p2 = from_point(fPts[2]);
1233    Sk2s tt(t);
1234    Sk2s ww(fW);
1235    Sk2s one(1);
1236
1237    Sk2s p1w = p1 * ww;
1238    Sk2s C = p0;
1239    Sk2s A = p2 - times_2(p1w) + p0;
1240    Sk2s B = times_2(p1w - C);
1241    Sk2s numer = quad_poly_eval(A, B, C, tt);
1242
1243    B = times_2(ww - one);
1244    A = -B;
1245    Sk2s denom = quad_poly_eval(A, B, one, tt);
1246
1247    return to_point(numer / denom);
1248}
1249
1250SkVector SkConic::evalTangentAt(SkScalar t) const {
1251    Sk2s p0 = from_point(fPts[0]);
1252    Sk2s p1 = from_point(fPts[1]);
1253    Sk2s p2 = from_point(fPts[2]);
1254    Sk2s ww(fW);
1255
1256    Sk2s p20 = p2 - p0;
1257    Sk2s p10 = p1 - p0;
1258
1259    Sk2s C = ww * p10;
1260    Sk2s A = ww * p20 - p20;
1261    Sk2s B = p20 - C - C;
1262
1263    return to_vector(quad_poly_eval(A, B, C, Sk2s(t)));
1264}
1265
1266static SkScalar subdivide_w_value(SkScalar w) {
1267    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1268}
1269
1270static Sk2s twice(const Sk2s& value) {
1271    return value + value;
1272}
1273
1274void SkConic::chop(SkConic * SK_RESTRICT dst) const {
1275    Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW));
1276    SkScalar newW = subdivide_w_value(fW);
1277
1278    Sk2s p0 = from_point(fPts[0]);
1279    Sk2s p1 = from_point(fPts[1]);
1280    Sk2s p2 = from_point(fPts[2]);
1281    Sk2s ww(fW);
1282
1283    Sk2s wp1 = ww * p1;
1284    Sk2s m = (p0 + twice(wp1) + p2) * scale * Sk2s(0.5f);
1285
1286    dst[0].fPts[0] = fPts[0];
1287    dst[0].fPts[1] = to_point((p0 + wp1) * scale);
1288    dst[0].fPts[2] = dst[1].fPts[0] = to_point(m);
1289    dst[1].fPts[1] = to_point((wp1 + p2) * scale);
1290    dst[1].fPts[2] = fPts[2];
1291
1292    dst[0].fW = dst[1].fW = newW;
1293}
1294
1295/*
1296 *  "High order approximation of conic sections by quadratic splines"
1297 *      by Michael Floater, 1993
1298 */
1299#define AS_QUAD_ERROR_SETUP                                         \
1300    SkScalar a = fW - 1;                                            \
1301    SkScalar k = a / (4 * (2 + a));                                 \
1302    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1303    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1304
1305void SkConic::computeAsQuadError(SkVector* err) const {
1306    AS_QUAD_ERROR_SETUP
1307    err->set(x, y);
1308}
1309
1310bool SkConic::asQuadTol(SkScalar tol) const {
1311    AS_QUAD_ERROR_SETUP
1312    return (x * x + y * y) <= tol * tol;
1313}
1314
1315// Limit the number of suggested quads to approximate a conic
1316#define kMaxConicToQuadPOW2     5
1317
1318int SkConic::computeQuadPOW2(SkScalar tol) const {
1319    if (tol < 0 || !SkScalarIsFinite(tol)) {
1320        return 0;
1321    }
1322
1323    AS_QUAD_ERROR_SETUP
1324
1325    SkScalar error = SkScalarSqrt(x * x + y * y);
1326    int pow2;
1327    for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
1328        if (error <= tol) {
1329            break;
1330        }
1331        error *= 0.25f;
1332    }
1333    // float version -- using ceil gives the same results as the above.
1334    if (false) {
1335        SkScalar err = SkScalarSqrt(x * x + y * y);
1336        if (err <= tol) {
1337            return 0;
1338        }
1339        SkScalar tol2 = tol * tol;
1340        if (tol2 == 0) {
1341            return kMaxConicToQuadPOW2;
1342        }
1343        SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
1344        int altPow2 = SkScalarCeilToInt(fpow2);
1345        if (altPow2 != pow2) {
1346            SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
1347        }
1348        pow2 = altPow2;
1349    }
1350    return pow2;
1351}
1352
1353static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1354    SkASSERT(level >= 0);
1355
1356    if (0 == level) {
1357        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1358        return pts + 2;
1359    } else {
1360        SkConic dst[2];
1361        src.chop(dst);
1362        --level;
1363        pts = subdivide(dst[0], pts, level);
1364        return subdivide(dst[1], pts, level);
1365    }
1366}
1367
1368int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1369    SkASSERT(pow2 >= 0);
1370    *pts = fPts[0];
1371    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1372    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1373    return 1 << pow2;
1374}
1375
1376bool SkConic::findXExtrema(SkScalar* t) const {
1377    return conic_find_extrema(&fPts[0].fX, fW, t);
1378}
1379
1380bool SkConic::findYExtrema(SkScalar* t) const {
1381    return conic_find_extrema(&fPts[0].fY, fW, t);
1382}
1383
1384bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1385    SkScalar t;
1386    if (this->findXExtrema(&t)) {
1387        this->chopAt(t, dst);
1388        // now clean-up the middle, since we know t was meant to be at
1389        // an X-extrema
1390        SkScalar value = dst[0].fPts[2].fX;
1391        dst[0].fPts[1].fX = value;
1392        dst[1].fPts[0].fX = value;
1393        dst[1].fPts[1].fX = value;
1394        return true;
1395    }
1396    return false;
1397}
1398
1399bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1400    SkScalar t;
1401    if (this->findYExtrema(&t)) {
1402        this->chopAt(t, dst);
1403        // now clean-up the middle, since we know t was meant to be at
1404        // an Y-extrema
1405        SkScalar value = dst[0].fPts[2].fY;
1406        dst[0].fPts[1].fY = value;
1407        dst[1].fPts[0].fY = value;
1408        dst[1].fPts[1].fY = value;
1409        return true;
1410    }
1411    return false;
1412}
1413
1414void SkConic::computeTightBounds(SkRect* bounds) const {
1415    SkPoint pts[4];
1416    pts[0] = fPts[0];
1417    pts[1] = fPts[2];
1418    int count = 2;
1419
1420    SkScalar t;
1421    if (this->findXExtrema(&t)) {
1422        this->evalAt(t, &pts[count++]);
1423    }
1424    if (this->findYExtrema(&t)) {
1425        this->evalAt(t, &pts[count++]);
1426    }
1427    bounds->set(pts, count);
1428}
1429
1430void SkConic::computeFastBounds(SkRect* bounds) const {
1431    bounds->set(fPts, 3);
1432}
1433
1434bool SkConic::findMaxCurvature(SkScalar* t) const {
1435    // TODO: Implement me
1436    return false;
1437}
1438
1439SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1440                             const SkMatrix& matrix) {
1441    if (!matrix.hasPerspective()) {
1442        return w;
1443    }
1444
1445    SkP3D src[3], dst[3];
1446
1447    ratquad_mapTo3D(pts, w, src);
1448
1449    matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1450
1451    // w' = sqrt(w1*w1/w0*w2)
1452    SkScalar w0 = dst[0].fZ;
1453    SkScalar w1 = dst[1].fZ;
1454    SkScalar w2 = dst[2].fZ;
1455    w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1456    return w;
1457}
1458
1459int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
1460                          const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
1461    // rotate by x,y so that uStart is (1.0)
1462    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1463    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1464
1465    SkScalar absY = SkScalarAbs(y);
1466
1467    // check for (effectively) coincident vectors
1468    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1469    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1470    if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) ||
1471                                                 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1472        return 0;
1473    }
1474
1475    if (dir == kCCW_SkRotationDirection) {
1476        y = -y;
1477    }
1478
1479    // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
1480    //      0 == [0  .. 90)
1481    //      1 == [90 ..180)
1482    //      2 == [180..270)
1483    //      3 == [270..360)
1484    //
1485    int quadrant = 0;
1486    if (0 == y) {
1487        quadrant = 2;        // 180
1488        SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1489    } else if (0 == x) {
1490        SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1491        quadrant = y > 0 ? 1 : 3; // 90 : 270
1492    } else {
1493        if (y < 0) {
1494            quadrant += 2;
1495        }
1496        if ((x < 0) != (y < 0)) {
1497            quadrant += 1;
1498        }
1499    }
1500
1501    const SkPoint quadrantPts[] = {
1502        { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
1503    };
1504    const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
1505
1506    int conicCount = quadrant;
1507    for (int i = 0; i < conicCount; ++i) {
1508        dst[i].set(&quadrantPts[i * 2], quadrantWeight);
1509    }
1510
1511    // Now compute any remaing (sub-90-degree) arc for the last conic
1512    const SkPoint finalP = { x, y };
1513    const SkPoint& lastQ = quadrantPts[quadrant * 2];  // will already be a unit-vector
1514    const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
1515    SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
1516
1517    if (dot < 1 - SK_ScalarNearlyZero) {
1518        SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
1519        // compute the bisector vector, and then rescale to be the off-curve point.
1520        // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
1521        // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
1522        // This is nice, since our computed weight is cos(theta/2) as well!
1523        //
1524        const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
1525        offCurve.setLength(SkScalarInvert(cosThetaOver2));
1526        dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
1527        conicCount += 1;
1528    }
1529
1530    // now handle counter-clockwise and the initial unitStart rotation
1531    SkMatrix    matrix;
1532    matrix.setSinCos(uStart.fY, uStart.fX);
1533    if (dir == kCCW_SkRotationDirection) {
1534        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1535    }
1536    if (userMatrix) {
1537        matrix.postConcat(*userMatrix);
1538    }
1539    for (int i = 0; i < conicCount; ++i) {
1540        matrix.mapPoints(dst[i].fPts, 3);
1541    }
1542    return conicCount;
1543}
1544