```/* <![CDATA[ */
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    }
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);
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) {
185    }
186    if (tangent) {
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) {
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
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
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)) {
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
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)) {
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//
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]) {
336    if (t == 0) {
337        memcpy(dst, src, 3 * sizeof(SkPoint));
338        return 1;
339    } else {
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
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
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
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
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
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
657template <typename T> void bubble_sort(T array[], int count) {
658    for (int i = count - 1; i > 0; --i)
659        for (int j = i; j > 0; --j)
660            if (array[j] < array[j-1])
661            {
662                T   tmp(array[j]);
663                array[j] = array[j-1];
664                array[j-1] = tmp;
665            }
666}
667
668/**
669 *  Given an array and count, remove all pair-wise duplicates from the array,
670 *  keeping the existing sorting, and return the new count
671 */
672static int collaps_duplicates(SkScalar array[], int count) {
673    for (int n = count; n > 1; --n) {
674        if (array[0] == array[1]) {
675            for (int i = 1; i < n; ++i) {
676                array[i - 1] = array[i];
677            }
678            count -= 1;
679        } else {
680            array += 1;
681        }
682    }
683    return count;
684}
685
686#ifdef SK_DEBUG
687
688#define TEST_COLLAPS_ENTRY(array)   array, SK_ARRAY_COUNT(array)
689
690static void test_collaps_duplicates() {
691    static bool gOnce;
692    if (gOnce) { return; }
693    gOnce = true;
694    const SkScalar src0[] = { 0 };
695    const SkScalar src1[] = { 0, 0 };
696    const SkScalar src2[] = { 0, 1 };
697    const SkScalar src3[] = { 0, 0, 0 };
698    const SkScalar src4[] = { 0, 0, 1 };
699    const SkScalar src5[] = { 0, 1, 1 };
700    const SkScalar src6[] = { 0, 1, 2 };
701    const struct {
702        const SkScalar* fData;
703        int fCount;
704        int fCollapsedCount;
705    } data[] = {
706        { TEST_COLLAPS_ENTRY(src0), 1 },
707        { TEST_COLLAPS_ENTRY(src1), 1 },
708        { TEST_COLLAPS_ENTRY(src2), 2 },
709        { TEST_COLLAPS_ENTRY(src3), 1 },
710        { TEST_COLLAPS_ENTRY(src4), 2 },
711        { TEST_COLLAPS_ENTRY(src5), 2 },
712        { TEST_COLLAPS_ENTRY(src6), 3 },
713    };
714    for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
715        SkScalar dst[3];
716        memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
717        int count = collaps_duplicates(dst, data[i].fCount);
718        SkASSERT(data[i].fCollapsedCount == count);
719        for (int j = 1; j < count; ++j) {
720            SkASSERT(dst[j-1] < dst[j]);
721        }
722    }
723}
724#endif
725
726static SkScalar SkScalarCubeRoot(SkScalar x) {
727    return SkScalarPow(x, 0.3333333f);
728}
729
730/*  Solve coeff(t) == 0, returning the number of roots that
731    lie withing 0 < t < 1.
732    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
733
734    Eliminates repeated roots (so that all tValues are distinct, and are always
735    in increasing order.
736*/
737static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
738    if (SkScalarNearlyZero(coeff[0])) {  // we're just a quadratic
739        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
740    }
741
742    SkScalar a, b, c, Q, R;
743
744    {
745        SkASSERT(coeff[0] != 0);
746
747        SkScalar inva = SkScalarInvert(coeff[0]);
748        a = coeff[1] * inva;
749        b = coeff[2] * inva;
750        c = coeff[3] * inva;
751    }
752    Q = (a*a - b*3) / 9;
753    R = (2*a*a*a - 9*a*b + 27*c) / 54;
754
755    SkScalar Q3 = Q * Q * Q;
756    SkScalar R2MinusQ3 = R * R - Q3;
757    SkScalar adiv3 = a / 3;
758
759    SkScalar*   roots = tValues;
760    SkScalar    r;
761
762    if (R2MinusQ3 < 0) { // we have 3 real roots
763        SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
764        SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
765
766        r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
767        if (is_unit_interval(r)) {
768            *roots++ = r;
769        }
770        r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
771        if (is_unit_interval(r)) {
772            *roots++ = r;
773        }
774        r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
775        if (is_unit_interval(r)) {
776            *roots++ = r;
777        }
778        SkDEBUGCODE(test_collaps_duplicates();)
779
780        // now sort the roots
781        int count = (int)(roots - tValues);
782        SkASSERT((unsigned)count <= 3);
783        bubble_sort(tValues, count);
784        count = collaps_duplicates(tValues, count);
785        roots = tValues + count;    // so we compute the proper count below
786    } else {              // we have 1 real root
787        SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
788        A = SkScalarCubeRoot(A);
789        if (R > 0) {
790            A = -A;
791        }
792        if (A != 0) {
793            A += Q / A;
794        }
795        r = A - adiv3;
796        if (is_unit_interval(r)) {
797            *roots++ = r;
798        }
799    }
800
801    return (int)(roots - tValues);
802}
803
804/*  Looking for F' dot F'' == 0
805
806    A = b - a
807    B = c - 2b + a
808    C = d - 3c + 3b - a
809
810    F' = 3Ct^2 + 6Bt + 3A
811    F'' = 6Ct + 6B
812
813    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
814*/
815static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
816    SkScalar    a = src[2] - src[0];
817    SkScalar    b = src[4] - 2 * src[2] + src[0];
818    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
819
820    coeff[0] = c * c;
821    coeff[1] = 3 * b * c;
822    coeff[2] = 2 * b * b + c * a;
823    coeff[3] = a * b;
824}
825
826/*  Looking for F' dot F'' == 0
827
828    A = b - a
829    B = c - 2b + a
830    C = d - 3c + 3b - a
831
832    F' = 3Ct^2 + 6Bt + 3A
833    F'' = 6Ct + 6B
834
835    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
836*/
837int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
838    SkScalar coeffX[4], coeffY[4];
839    int      i;
840
841    formulate_F1DotF2(&src[0].fX, coeffX);
842    formulate_F1DotF2(&src[0].fY, coeffY);
843
844    for (i = 0; i < 4; i++) {
845        coeffX[i] += coeffY[i];
846    }
847
848    SkScalar    t[3];
849    int         count = solve_cubic_poly(coeffX, t);
850    int         maxCount = 0;
851
852    // now remove extrema where the curvature is zero (mins)
853    // !!!! need a test for this !!!!
854    for (i = 0; i < count; i++) {
855        // if (not_min_curvature())
856        if (t[i] > 0 && t[i] < SK_Scalar1) {
857            tValues[maxCount++] = t[i];
858        }
859    }
860    return maxCount;
861}
862
863int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
864                              SkScalar tValues[3]) {
865    SkScalar    t_storage[3];
866
867    if (tValues == NULL) {
868        tValues = t_storage;
869    }
870
871    int count = SkFindCubicMaxCurvature(src, tValues);
872
873    if (dst) {
874        if (count == 0) {
875            memcpy(dst, src, 4 * sizeof(SkPoint));
876        } else {
877            SkChopCubicAt(src, dst, tValues, count);
878        }
879    }
880    return count + 1;
881}
882
883bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4],
884                                 bool* ambiguous) {
885    if (ambiguous) {
886        *ambiguous = false;
887    }
888
889    // Find the minimum and maximum y of the extrema, which are the
890    // first and last points since this cubic is monotonic
891    SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
892    SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
893
894    if (pt.fY == cubic[0].fY
895        || pt.fY < min_y
896        || pt.fY > max_y) {
897        // The query line definitely does not cross the curve
898        if (ambiguous) {
899            *ambiguous = (pt.fY == cubic[0].fY);
900        }
901        return false;
902    }
903
904    bool pt_at_extremum = (pt.fY == cubic[3].fY);
905
906    SkScalar min_x =
907        SkMinScalar(
908            SkMinScalar(
909                SkMinScalar(cubic[0].fX, cubic[1].fX),
910                cubic[2].fX),
911            cubic[3].fX);
912    if (pt.fX < min_x) {
913        // The query line definitely crosses the curve
914        if (ambiguous) {
915            *ambiguous = pt_at_extremum;
916        }
917        return true;
918    }
919
920    SkScalar max_x =
921        SkMaxScalar(
922            SkMaxScalar(
923                SkMaxScalar(cubic[0].fX, cubic[1].fX),
924                cubic[2].fX),
925            cubic[3].fX);
926    if (pt.fX > max_x) {
927        // The query line definitely does not cross the curve
928        return false;
929    }
930
931    // Do a binary search to find the parameter value which makes y as
932    // close as possible to the query point. See whether the query
933    // line's origin is to the left of the associated x coordinate.
934
935    // kMaxIter is chosen as the number of mantissa bits for a float,
936    // since there's no way we are going to get more precision by
937    // iterating more times than that.
938    const int kMaxIter = 23;
939    SkPoint eval;
940    int iter = 0;
941    SkScalar upper_t;
942    SkScalar lower_t;
943    // Need to invert direction of t parameter if cubic goes up
945    if (cubic[3].fY > cubic[0].fY) {
946        upper_t = SK_Scalar1;
947        lower_t = 0;
948    } else {
949        upper_t = 0;
950        lower_t = SK_Scalar1;
951    }
952    do {
953        SkScalar t = SkScalarAve(upper_t, lower_t);
954        SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
955        if (pt.fY > eval.fY) {
956            lower_t = t;
957        } else {
958            upper_t = t;
959        }
960    } while (++iter < kMaxIter
961             && !SkScalarNearlyZero(eval.fY - pt.fY));
962    if (pt.fX <= eval.fX) {
963        if (ambiguous) {
964            *ambiguous = pt_at_extremum;
965        }
966        return true;
967    }
968    return false;
969}
970
971int SkNumXRayCrossingsForCubic(const SkXRay& pt,
972                               const SkPoint cubic[4],
973                               bool* ambiguous) {
974    int num_crossings = 0;
975    SkPoint monotonic_cubics[10];
976    int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
977    if (ambiguous) {
978        *ambiguous = false;
979    }
980    bool locally_ambiguous;
981    if (SkXRayCrossesMonotonicCubic(pt,
982                                    &monotonic_cubics[0],
983                                    &locally_ambiguous))
984        ++num_crossings;
985    if (ambiguous) {
986        *ambiguous |= locally_ambiguous;
987    }
988    if (num_monotonic_cubics > 0)
989        if (SkXRayCrossesMonotonicCubic(pt,
990                                        &monotonic_cubics[3],
991                                        &locally_ambiguous))
992            ++num_crossings;
993    if (ambiguous) {
994        *ambiguous |= locally_ambiguous;
995    }
996    if (num_monotonic_cubics > 1)
997        if (SkXRayCrossesMonotonicCubic(pt,
998                                        &monotonic_cubics[6],
999                                        &locally_ambiguous))
1000            ++num_crossings;
1001    if (ambiguous) {
1002        *ambiguous |= locally_ambiguous;
1003    }
1004    return num_crossings;
1005}
1006
1007///////////////////////////////////////////////////////////////////////////////
1008
1009/*  Find t value for quadratic [a, b, c] = d.
1010    Return 0 if there is no solution within [0, 1)
1011*/
1012static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
1013    // At^2 + Bt + C = d
1014    SkScalar A = a - 2 * b + c;
1015    SkScalar B = 2 * (b - a);
1016    SkScalar C = a - d;
1017
1018    SkScalar    roots[2];
1019    int         count = SkFindUnitQuadRoots(A, B, C, roots);
1020
1021    SkASSERT(count <= 1);
1022    return count == 1 ? roots[0] : 0;
1023}
1024
1025/*  given a quad-curve and a point (x,y), chop the quad at that point and place
1026    the new off-curve point and endpoint into 'dest'.
1027    Should only return false if the computed pos is the start of the curve
1028    (i.e. root == 0)
1029*/
1030static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
1031                                SkPoint* dest) {
1032    const SkScalar* base;
1033    SkScalar        value;
1034
1035    if (SkScalarAbs(x) < SkScalarAbs(y)) {
1037        value = x;
1038    } else {
1040        value = y;
1041    }
1042
1043    // note: this returns 0 if it thinks value is out of range, meaning the
1044    // root might return something outside of [0, 1)
1045    SkScalar t = quad_solve(base[0], base[2], base[4], value);
1046
1047    if (t > 0) {
1048        SkPoint tmp[5];
1050        dest[0] = tmp[1];
1051        dest[1].set(x, y);
1052        return true;
1053    } else {
1054        /*  t == 0 means either the value triggered a root outside of [0, 1)
1055            For our purposes, we can ignore the <= 0 roots, but we want to
1056            catch the >= 1 roots (which given our caller, will basically mean
1057            a root of 1, give-or-take numerical instability). If we are in the
1058            >= 1 case, return the existing offCurve point.
1059
1060            The test below checks to see if we are close to the "end" of the
1061            curve (near base[4]). Rather than specifying a tolerance, I just
1062            check to see if value is on to the right/left of the middle point
1063            (depending on the direction/sign of the end points).
1064        */
1065        if ((base[0] < base[4] && value > base[2]) ||
1066            (base[0] > base[4] && value < base[2]))   // should root have been 1
1067        {
1069            dest[1].set(x, y);
1070            return true;
1071        }
1072    }
1073    return false;
1074}
1075
1077// The mid point of the quadratic arc approximation is half way between the two
1078// control points. The float epsilon adjustment moves the on curve point out by
1079// two bits, distributing the convex test error between the round rect
1080// approximation and the convex cross product sign equality test.
1081#define SK_MID_RRECT_OFFSET \
1082    (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1083    { SK_Scalar1,            0                      },
1084    { SK_Scalar1,            SK_ScalarTanPIOver8    },
1085    { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
1086    { SK_ScalarTanPIOver8,   SK_Scalar1             },
1087
1088    { 0,                     SK_Scalar1             },
1089    { -SK_ScalarTanPIOver8,  SK_Scalar1             },
1090    { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
1091    { -SK_Scalar1,           SK_ScalarTanPIOver8    },
1092
1093    { -SK_Scalar1,           0                      },
1094    { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
1095    { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
1096    { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
1097
1098    { 0,                     -SK_Scalar1            },
1099    { SK_ScalarTanPIOver8,   -SK_Scalar1            },
1100    { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
1101    { SK_Scalar1,            -SK_ScalarTanPIOver8   },
1102
1103    { SK_Scalar1,            0                      }
1104#undef SK_MID_RRECT_OFFSET
1105};
1106
1107int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1108                   SkRotationDirection dir, const SkMatrix* userMatrix,
1110    // rotate by x,y so that uStart is (1.0)
1111    SkScalar x = SkPoint::DotProduct(uStart, uStop);
1112    SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1113
1114    SkScalar absX = SkScalarAbs(x);
1115    SkScalar absY = SkScalarAbs(y);
1116
1117    int pointCount;
1118
1119    // check for (effectively) coincident vectors
1120    // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1121    // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1122    if (absY <= SK_ScalarNearlyZero && x > 0 &&
1123        ((y >= 0 && kCW_SkRotationDirection == dir) ||
1124         (y <= 0 && kCCW_SkRotationDirection == dir))) {
1125
1126        // just return the start-point
1128        pointCount = 1;
1129    } else {
1130        if (dir == kCCW_SkRotationDirection) {
1131            y = -y;
1132        }
1133        // what octant (quadratic curve) is [xy] in?
1134        int oct = 0;
1135        bool sameSign = true;
1136
1137        if (0 == y) {
1138            oct = 4;        // 180
1139            SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1140        } else if (0 == x) {
1141            SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1142            oct = y > 0 ? 2 : 6; // 90 : 270
1143        } else {
1144            if (y < 0) {
1145                oct += 4;
1146            }
1147            if ((x < 0) != (y < 0)) {
1148                oct += 2;
1149                sameSign = false;
1150            }
1151            if ((absX < absY) == sameSign) {
1152                oct += 1;
1153            }
1154        }
1155
1156        int wholeCount = oct << 1;
1158
1159        const SkPoint* arc = &gQuadCirclePts[wholeCount];
1160        if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1161            wholeCount += 2;
1162        }
1163        pointCount = wholeCount + 1;
1164    }
1165
1166    // now handle counter-clockwise and the initial unitStart rotation
1167    SkMatrix    matrix;
1168    matrix.setSinCos(uStart.fY, uStart.fX);
1169    if (dir == kCCW_SkRotationDirection) {
1170        matrix.preScale(SK_Scalar1, -SK_Scalar1);
1171    }
1172    if (userMatrix) {
1173        matrix.postConcat(*userMatrix);
1174    }
1176    return pointCount;
1177}
1178
1179
1180///////////////////////////////////////////////////////////////////////////////
1181//
1182// NURB representation for conics.  Helpful explanations at:
1183//
1184// http://citeseerx.ist.psu.edu/viewdoc/
1186// and
1187// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1188//
1189// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1190//     ------------------------------------------
1191//         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1192//
1193//   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1194//     ------------------------------------------------
1195//             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1196//
1197
1198static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1199    SkASSERT(src);
1200    SkASSERT(t >= 0 && t <= SK_Scalar1);
1201
1202    SkScalar    src2w = SkScalarMul(src[2], w);
1203    SkScalar    C = src[0];
1204    SkScalar    A = src[4] - 2 * src2w + C;
1205    SkScalar    B = 2 * (src2w - C);
1207
1208    B = 2 * (w - SK_Scalar1);
1209    C = SK_Scalar1;
1210    A = -B;
1212
1213    return SkScalarDiv(numer, denom);
1214}
1215
1216// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1217//
1218//  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1219//  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1220//  t^0 : -2 P0 w + 2 P1 w
1221//
1222//  We disregard magnitude, so we can freely ignore the denominator of F', and
1223//  divide the numerator by 2
1224//
1225//    coeff[0] for t^2
1226//    coeff[1] for t^1
1227//    coeff[2] for t^0
1228//
1229static void conic_deriv_coeff(const SkScalar src[],
1230                              SkScalar w,
1231                              SkScalar coeff[3]) {
1232    const SkScalar P20 = src[4] - src[0];
1233    const SkScalar P10 = src[2] - src[0];
1234    const SkScalar wP10 = w * P10;
1235    coeff[0] = w * P20 - P20;
1236    coeff[1] = P20 - 2 * wP10;
1237    coeff[2] = wP10;
1238}
1239
1240static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1241    SkScalar coeff[3];
1242    conic_deriv_coeff(coord, w, coeff);
1243    return t * (t * coeff[0] + coeff[1]) + coeff[2];
1244}
1245
1246static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1247    SkScalar coeff[3];
1248    conic_deriv_coeff(src, w, coeff);
1249
1250    SkScalar tValues[2];
1251    int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1252    SkASSERT(0 == roots || 1 == roots);
1253
1254    if (1 == roots) {
1255        *t = tValues[0];
1256        return true;
1257    }
1258    return false;
1259}
1260
1261struct SkP3D {
1262    SkScalar fX, fY, fZ;
1263
1264    void set(SkScalar x, SkScalar y, SkScalar z) {
1265        fX = x; fY = y; fZ = z;
1266    }
1267
1268    void projectDown(SkPoint* dst) const {
1269        dst->set(fX / fZ, fY / fZ);
1270    }
1271};
1272
1273// We only interpolate one dimension at a time (the first, at +0, +3, +6).
1274static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1275    SkScalar ab = SkScalarInterp(src[0], src[3], t);
1276    SkScalar bc = SkScalarInterp(src[3], src[6], t);
1277    dst[0] = ab;
1278    dst[3] = SkScalarInterp(ab, bc, t);
1279    dst[6] = bc;
1280}
1281
1282static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1283    dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1284    dst[1].set(src[1].fX * w, src[1].fY * w, w);
1285    dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1286}
1287
1288void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1289    SkASSERT(t >= 0 && t <= SK_Scalar1);
1290
1291    if (pt) {
1292        pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1293                conic_eval_pos(&fPts[0].fY, fW, t));
1294    }
1295    if (tangent) {
1296        tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1297                     conic_eval_tan(&fPts[0].fY, fW, t));
1298    }
1299}
1300
1301void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1302    SkP3D tmp[3], tmp2[3];
1303
1305
1306    p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1307    p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1308    p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1309
1310    dst[0].fPts[0] = fPts[0];
1311    tmp2[0].projectDown(&dst[0].fPts[1]);
1312    tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1313    tmp2[2].projectDown(&dst[1].fPts[1]);
1314    dst[1].fPts[2] = fPts[2];
1315
1316    // to put in "standard form", where w0 and w2 are both 1, we compute the
1317    // new w1 as sqrt(w1*w1/w0*w2)
1318    // or
1319    // w1 /= sqrt(w0*w2)
1320    //
1321    // However, in our case, we know that for dst[0]:
1322    //     w0 == 1, and for dst[1], w2 == 1
1323    //
1324    SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1325    dst[0].fW = tmp2[0].fZ / root;
1326    dst[1].fW = tmp2[2].fZ / root;
1327}
1328
1329static SkScalar subdivide_w_value(SkScalar w) {
1330    return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1331}
1332
1333void SkConic::chop(SkConic dst[2]) const {
1334    SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
1335    SkScalar p1x = fW * fPts[1].fX;
1336    SkScalar p1y = fW * fPts[1].fY;
1337    SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
1338    SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
1339
1340    dst[0].fPts[0] = fPts[0];
1341    dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
1342                       (fPts[0].fY + p1y) * scale);
1343    dst[0].fPts[2].set(mx, my);
1344
1345    dst[1].fPts[0].set(mx, my);
1346    dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
1347                       (p1y + fPts[2].fY) * scale);
1348    dst[1].fPts[2] = fPts[2];
1349
1350    dst[0].fW = dst[1].fW = subdivide_w_value(fW);
1351}
1352
1353/*
1354 *  "High order approximation of conic sections by quadratic splines"
1355 *      by Michael Floater, 1993
1356 */
1358    SkScalar a = fW - 1;                                            \
1359    SkScalar k = a / (4 * (2 + a));                                 \
1360    SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
1361    SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1362
1365    err->set(x, y);
1366}
1367
1370    return (x * x + y * y) <= tol * tol;
1371}
1372
1375    SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
1376
1377    if (error <= 0) {
1378        return 0;
1379    }
1380    uint32_t ierr = (uint32_t)error;
1381    return (34 - SkCLZ(ierr)) >> 1;
1382}
1383
1384static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1385    SkASSERT(level >= 0);
1386
1387    if (0 == level) {
1388        memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1389        return pts + 2;
1390    } else {
1391        SkConic dst[2];
1392        src.chop(dst);
1393        --level;
1394        pts = subdivide(dst[0], pts, level);
1395        return subdivide(dst[1], pts, level);
1396    }
1397}
1398
1399int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1400    SkASSERT(pow2 >= 0);
1401    *pts = fPts[0];
1402    SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1403    SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1404    return 1 << pow2;
1405}
1406
1407bool SkConic::findXExtrema(SkScalar* t) const {
1408    return conic_find_extrema(&fPts[0].fX, fW, t);
1409}
1410
1411bool SkConic::findYExtrema(SkScalar* t) const {
1412    return conic_find_extrema(&fPts[0].fY, fW, t);
1413}
1414
1415bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1416    SkScalar t;
1417    if (this->findXExtrema(&t)) {
1418        this->chopAt(t, dst);
1419        // now clean-up the middle, since we know t was meant to be at
1420        // an X-extrema
1421        SkScalar value = dst[0].fPts[2].fX;
1422        dst[0].fPts[1].fX = value;
1423        dst[1].fPts[0].fX = value;
1424        dst[1].fPts[1].fX = value;
1425        return true;
1426    }
1427    return false;
1428}
1429
1430bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1431    SkScalar t;
1432    if (this->findYExtrema(&t)) {
1433        this->chopAt(t, dst);
1434        // now clean-up the middle, since we know t was meant to be at
1435        // an Y-extrema
1436        SkScalar value = dst[0].fPts[2].fY;
1437        dst[0].fPts[1].fY = value;
1438        dst[1].fPts[0].fY = value;
1439        dst[1].fPts[1].fY = value;
1440        return true;
1441    }
1442    return false;
1443}
1444
1445void SkConic::computeTightBounds(SkRect* bounds) const {
1446    SkPoint pts[4];
1447    pts[0] = fPts[0];
1448    pts[1] = fPts[2];
1449    int count = 2;
1450
1451    SkScalar t;
1452    if (this->findXExtrema(&t)) {
1453        this->evalAt(t, &pts[count++]);
1454    }
1455    if (this->findYExtrema(&t)) {
1456        this->evalAt(t, &pts[count++]);
1457    }
1458    bounds->set(pts, count);
1459}
1460
1461void SkConic::computeFastBounds(SkRect* bounds) const {
1462    bounds->set(fPts, 3);
1463}
1464
1465bool SkConic::findMaxCurvature(SkScalar* t) const {
1466    // TODO: Implement me
1467    return false;
1468}
1469```