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android / platform / external / chromium_org / third_party / skia / src / 685cfc0ee13d7c355ae2f4f3d225ad45e945763f / . / core / SkGeometry.cpp
blob: 5308d56dc0efc69b524d732f08e76a78a51afc8f [file] [log] [blame]
epoger@google.com685cfc02011-07-28 14:26:00 +0000[diff] [blame^]1
2/*
3 * Copyright 2006 The Android Open Source Project
4 *
5 * Use of this source code is governed by a BSD-style license that can be
6 * found in the LICENSE file.
7 */
8
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]9
10#include "SkGeometry.h"
11#include "Sk64.h"
12#include "SkMatrix.h"
13
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]14bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) {
15 if (ambiguous) {
16 *ambiguous = false;
17 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]18 // Determine quick discards.
19 // Consider query line going exactly through point 0 to not
20 // intersect, for symmetry with SkXRayCrossesMonotonicCubic.
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]21 if (pt.fY == pts[0].fY) {
22 if (ambiguous) {
23 *ambiguous = true;
24 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]25 return false;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]26 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]27 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY)
28 return false;
29 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY)
30 return false;
31 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX)
32 return false;
33 // Determine degenerate cases
34 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY))
35 return false;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]36 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) {
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]37 // We've already determined the query point lies within the
38 // vertical range of the line segment.
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]39 if (pt.fX <= pts[0].fX) {
40 if (ambiguous) {
41 *ambiguous = (pt.fY == pts[1].fY);
42 }
43 return true;
44 }
45 return false;
46 }
47 // Ambiguity check
48 if (pt.fY == pts[1].fY) {
49 if (pt.fX <= pts[1].fX) {
50 if (ambiguous) {
51 *ambiguous = true;
52 }
53 return true;
54 }
55 return false;
56 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]57 // Full line segment evaluation
58 SkScalar delta_y = pts[1].fY - pts[0].fY;
59 SkScalar delta_x = pts[1].fX - pts[0].fX;
60 SkScalar slope = SkScalarDiv(delta_y, delta_x);
61 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX);
62 // Solve for x coordinate at y = pt.fY
63 SkScalar x = SkScalarDiv(pt.fY - b, slope);
64 return pt.fX <= x;
65}
66
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]67/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
68 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
69 May also introduce overflow of fixed when we compute our setup.
70*/
71#ifdef SK_SCALAR_IS_FIXED
72 #define DIRECT_EVAL_OF_POLYNOMIALS
73#endif
74
75////////////////////////////////////////////////////////////////////////
76
77#ifdef SK_SCALAR_IS_FIXED
78 static int is_not_monotonic(int a, int b, int c, int d)
79 {
80 return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31;
81 }
82
83 static int is_not_monotonic(int a, int b, int c)
84 {
85 return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31;
86 }
87#else
88 static int is_not_monotonic(float a, float b, float c)
89 {
90 float ab = a - b;
91 float bc = b - c;
92 if (ab < 0)
93 bc = -bc;
94 return ab == 0 || bc < 0;
95 }
96#endif
97
98////////////////////////////////////////////////////////////////////////
99
100static bool is_unit_interval(SkScalar x)
101{
102 return x > 0 && x < SK_Scalar1;
103}
104
105static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio)
106{
107 SkASSERT(ratio);
108
109 if (numer < 0)
110 {
111 numer = -numer;
112 denom = -denom;
113 }
114
115 if (denom == 0 || numer == 0 || numer >= denom)
116 return 0;
117
118 SkScalar r = SkScalarDiv(numer, denom);
reed@android.com7c83e1c2010-03-08 17:44:42 +0000[diff] [blame]119 if (SkScalarIsNaN(r)) {
120 return 0;
121 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]122 SkASSERT(r >= 0 && r < SK_Scalar1);
123 if (r == 0) // catch underflow if numer <<<< denom
124 return 0;
125 *ratio = r;
126 return 1;
127}
128
129/** From Numerical Recipes in C.
130
131 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
132 x1 = Q / A
133 x2 = C / Q
134*/
135int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2])
136{
137 SkASSERT(roots);
138
139 if (A == 0)
140 return valid_unit_divide(-C, B, roots);
141
142 SkScalar* r = roots;
143
144#ifdef SK_SCALAR_IS_FLOAT
145 float R = B*B - 4*A*C;
reed@android.com7c83e1c2010-03-08 17:44:42 +0000[diff] [blame]146 if (R < 0 || SkScalarIsNaN(R)) { // complex roots
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]147 return 0;
reed@android.com7c83e1c2010-03-08 17:44:42 +0000[diff] [blame]148 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]149 R = sk_float_sqrt(R);
150#else
151 Sk64 RR, tmp;
152
153 RR.setMul(B,B);
154 tmp.setMul(A,C);
155 tmp.shiftLeft(2);
156 RR.sub(tmp);
157 if (RR.isNeg())
158 return 0;
159 SkFixed R = RR.getSqrt();
160#endif
161
162 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
163 r += valid_unit_divide(Q, A, r);
164 r += valid_unit_divide(C, Q, r);
165 if (r - roots == 2)
166 {
167 if (roots[0] > roots[1])
168 SkTSwap<SkScalar>(roots[0], roots[1]);
169 else if (roots[0] == roots[1]) // nearly-equal?
170 r -= 1; // skip the double root
171 }
172 return (int)(r - roots);
173}
174
175#ifdef SK_SCALAR_IS_FIXED
176/** Trim A/B/C down so that they are all <= 32bits
177 and then call SkFindUnitQuadRoots()
178*/
179static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2])
180{
181 int na = A.shiftToMake32();
182 int nb = B.shiftToMake32();
183 int nc = C.shiftToMake32();
184
185 int shift = SkMax32(na, SkMax32(nb, nc));
186 SkASSERT(shift >= 0);
187
188 return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots);
189}
190#endif
191
192/////////////////////////////////////////////////////////////////////////////////////
193/////////////////////////////////////////////////////////////////////////////////////
194
195static SkScalar eval_quad(const SkScalar src[], SkScalar t)
196{
197 SkASSERT(src);
198 SkASSERT(t >= 0 && t <= SK_Scalar1);
199
200#ifdef DIRECT_EVAL_OF_POLYNOMIALS
201 SkScalar C = src[0];
202 SkScalar A = src[4] - 2 * src[2] + C;
203 SkScalar B = 2 * (src[2] - C);
204 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
205#else
206 SkScalar ab = SkScalarInterp(src[0], src[2], t);
207 SkScalar bc = SkScalarInterp(src[2], src[4], t);
208 return SkScalarInterp(ab, bc, t);
209#endif
210}
211
212static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t)
213{
214 SkScalar A = src[4] - 2 * src[2] + src[0];
215 SkScalar B = src[2] - src[0];
216
217 return 2 * SkScalarMulAdd(A, t, B);
218}
219
220static SkScalar eval_quad_derivative_at_half(const SkScalar src[])
221{
222 SkScalar A = src[4] - 2 * src[2] + src[0];
223 SkScalar B = src[2] - src[0];
224 return A + 2 * B;
225}
226
227void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent)
228{
229 SkASSERT(src);
230 SkASSERT(t >= 0 && t <= SK_Scalar1);
231
232 if (pt)
233 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
234 if (tangent)
235 tangent->set(eval_quad_derivative(&src[0].fX, t),
236 eval_quad_derivative(&src[0].fY, t));
237}
238
239void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent)
240{
241 SkASSERT(src);
242
243 if (pt)
244 {
245 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
246 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
247 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
248 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
249 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
250 }
251 if (tangent)
252 tangent->set(eval_quad_derivative_at_half(&src[0].fX),
253 eval_quad_derivative_at_half(&src[0].fY));
254}
255
256static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
257{
258 SkScalar ab = SkScalarInterp(src[0], src[2], t);
259 SkScalar bc = SkScalarInterp(src[2], src[4], t);
260
261 dst[0] = src[0];
262 dst[2] = ab;
263 dst[4] = SkScalarInterp(ab, bc, t);
264 dst[6] = bc;
265 dst[8] = src[4];
266}
267
268void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t)
269{
270 SkASSERT(t > 0 && t < SK_Scalar1);
271
272 interp_quad_coords(&src[0].fX, &dst[0].fX, t);
273 interp_quad_coords(&src[0].fY, &dst[0].fY, t);
274}
275
276void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5])
277{
278 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
279 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
280 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
281 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
282
283 dst[0] = src[0];
284 dst[1].set(x01, y01);
285 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
286 dst[3].set(x12, y12);
287 dst[4] = src[2];
288}
289
290/** Quad'(t) = At + B, where
291 A = 2(a - 2b + c)
292 B = 2(b - a)
293 Solve for t, only if it fits between 0 < t < 1
294*/
295int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1])
296{
297 /* At + B == 0
298 t = -B / A
299 */
300#ifdef SK_SCALAR_IS_FIXED
301 return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue);
302#else
303 return valid_unit_divide(a - b, a - b - b + c, tValue);
304#endif
305}
306
reed@android.come5dd6cd2009-01-15 14:38:33 +0000[diff] [blame]307static inline void flatten_double_quad_extrema(SkScalar coords[14])
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]308{
309 coords[2] = coords[6] = coords[4];
310}
311
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]312/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]313 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
314 */
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]315int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5])
316{
317 SkASSERT(src);
318 SkASSERT(dst);
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]319
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]320#if 0
321 static bool once = true;
322 if (once)
323 {
324 once = false;
325 SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 };
326 SkPoint d[6];
327
328 int n = SkChopQuadAtYExtrema(s, d);
329 SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY);
330 }
331#endif
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]332
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]333 SkScalar a = src[0].fY;
334 SkScalar b = src[1].fY;
335 SkScalar c = src[2].fY;
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]336
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]337 if (is_not_monotonic(a, b, c))
338 {
339 SkScalar tValue;
340 if (valid_unit_divide(a - b, a - b - b + c, &tValue))
341 {
342 SkChopQuadAt(src, dst, tValue);
343 flatten_double_quad_extrema(&dst[0].fY);
344 return 1;
345 }
346 // if we get here, we need to force dst to be monotonic, even though
347 // we couldn't compute a unit_divide value (probably underflow).
348 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
349 }
350 dst[0].set(src[0].fX, a);
351 dst[1].set(src[1].fX, b);
352 dst[2].set(src[2].fX, c);
353 return 0;
354}
355
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]356/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
357 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
358 */
359int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5])
360{
361 SkASSERT(src);
362 SkASSERT(dst);
363
364 SkScalar a = src[0].fX;
365 SkScalar b = src[1].fX;
366 SkScalar c = src[2].fX;
367
368 if (is_not_monotonic(a, b, c)) {
369 SkScalar tValue;
370 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
371 SkChopQuadAt(src, dst, tValue);
372 flatten_double_quad_extrema(&dst[0].fX);
373 return 1;
374 }
375 // if we get here, we need to force dst to be monotonic, even though
376 // we couldn't compute a unit_divide value (probably underflow).
377 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
378 }
379 dst[0].set(a, src[0].fY);
380 dst[1].set(b, src[1].fY);
381 dst[2].set(c, src[2].fY);
382 return 0;
383}
384
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]385// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
386// F'(t) = 2 (b - a) + 2 (a - 2b + c) t
387// F''(t) = 2 (a - 2b + c)
388//
389// A = 2 (b - a)
390// B = 2 (a - 2b + c)
391//
392// Maximum curvature for a quadratic means solving
393// Fx' Fx'' + Fy' Fy'' = 0
394//
395// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
396//
397int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5])
398{
399 SkScalar Ax = src[1].fX - src[0].fX;
400 SkScalar Ay = src[1].fY - src[0].fY;
401 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
402 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
403 SkScalar t = 0; // 0 means don't chop
404
405#ifdef SK_SCALAR_IS_FLOAT
406 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
407#else
408 // !!! should I use SkFloat here? seems like it
409 Sk64 numer, denom, tmp;
410
411 numer.setMul(Ax, -Bx);
412 tmp.setMul(Ay, -By);
413 numer.add(tmp);
414
415 if (numer.isPos()) // do nothing if numer <= 0
416 {
417 denom.setMul(Bx, Bx);
418 tmp.setMul(By, By);
419 denom.add(tmp);
420 SkASSERT(!denom.isNeg());
421 if (numer < denom)
422 {
423 t = numer.getFixedDiv(denom);
424 SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!)
425 if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability
426 t = 0; // ignore the chop
427 }
428 }
429#endif
430
431 if (t == 0)
432 {
433 memcpy(dst, src, 3 * sizeof(SkPoint));
434 return 1;
435 }
436 else
437 {
438 SkChopQuadAt(src, dst, t);
439 return 2;
440 }
441}
442
reed@google.com007593e2011-07-27 13:54:36 +0000[diff] [blame]443#ifdef SK_SCALAR_IS_FLOAT
444 #define SK_ScalarTwoThirds (0.666666666f)
445#else
446 #define SK_ScalarTwoThirds ((SkFixed)(43691))
447#endif
448
449void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
450 const SkScalar scale = SK_ScalarTwoThirds;
451 dst[0] = src[0];
452 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
453 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
454 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
455 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
456 dst[3] = src[2];
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]457}
458
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]459////////////////////////////////////////////////////////////////////////////////////////
460///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
461////////////////////////////////////////////////////////////////////////////////////////
462
463static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4])
464{
465 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
466 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
467 coeff[2] = 3*(pt[2] - pt[0]);
468 coeff[3] = pt[0];
469}
470
471void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4])
472{
473 SkASSERT(pts);
474
475 if (cx)
476 get_cubic_coeff(&pts[0].fX, cx);
477 if (cy)
478 get_cubic_coeff(&pts[0].fY, cy);
479}
480
481static SkScalar eval_cubic(const SkScalar src[], SkScalar t)
482{
483 SkASSERT(src);
484 SkASSERT(t >= 0 && t <= SK_Scalar1);
485
486 if (t == 0)
487 return src[0];
488
489#ifdef DIRECT_EVAL_OF_POLYNOMIALS
490 SkScalar D = src[0];
491 SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
492 SkScalar B = 3*(src[4] - src[2] - src[2] + D);
493 SkScalar C = 3*(src[2] - D);
494
495 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
496#else
497 SkScalar ab = SkScalarInterp(src[0], src[2], t);
498 SkScalar bc = SkScalarInterp(src[2], src[4], t);
499 SkScalar cd = SkScalarInterp(src[4], src[6], t);
500 SkScalar abc = SkScalarInterp(ab, bc, t);
501 SkScalar bcd = SkScalarInterp(bc, cd, t);
502 return SkScalarInterp(abc, bcd, t);
503#endif
504}
505
506/** return At^2 + Bt + C
507*/
508static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t)
509{
510 SkASSERT(t >= 0 && t <= SK_Scalar1);
511
512 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
513}
514
515static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t)
516{
517 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
518 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
519 SkScalar C = src[2] - src[0];
520
521 return eval_quadratic(A, B, C, t);
522}
523
524static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t)
525{
526 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
527 SkScalar B = src[4] - 2 * src[2] + src[0];
528
529 return SkScalarMulAdd(A, t, B);
530}
531
532void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature)
533{
534 SkASSERT(src);
535 SkASSERT(t >= 0 && t <= SK_Scalar1);
536
537 if (loc)
538 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
539 if (tangent)
540 tangent->set(eval_cubic_derivative(&src[0].fX, t),
541 eval_cubic_derivative(&src[0].fY, t));
542 if (curvature)
543 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
544 eval_cubic_2ndDerivative(&src[0].fY, t));
545}
546
547/** Cubic'(t) = At^2 + Bt + C, where
548 A = 3(-a + 3(b - c) + d)
549 B = 6(a - 2b + c)
550 C = 3(b - a)
551 Solve for t, keeping only those that fit betwee 0 < t < 1
552*/
553int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2])
554{
555#ifdef SK_SCALAR_IS_FIXED
556 if (!is_not_monotonic(a, b, c, d))
557 return 0;
558#endif
559
560 // we divide A,B,C by 3 to simplify
561 SkScalar A = d - a + 3*(b - c);
562 SkScalar B = 2*(a - b - b + c);
563 SkScalar C = b - a;
564
565 return SkFindUnitQuadRoots(A, B, C, tValues);
566}
567
568static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
569{
570 SkScalar ab = SkScalarInterp(src[0], src[2], t);
571 SkScalar bc = SkScalarInterp(src[2], src[4], t);
572 SkScalar cd = SkScalarInterp(src[4], src[6], t);
573 SkScalar abc = SkScalarInterp(ab, bc, t);
574 SkScalar bcd = SkScalarInterp(bc, cd, t);
575 SkScalar abcd = SkScalarInterp(abc, bcd, t);
576
577 dst[0] = src[0];
578 dst[2] = ab;
579 dst[4] = abc;
580 dst[6] = abcd;
581 dst[8] = bcd;
582 dst[10] = cd;
583 dst[12] = src[6];
584}
585
586void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t)
587{
588 SkASSERT(t > 0 && t < SK_Scalar1);
589
590 interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
591 interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
592}
593
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]594/* http://code.google.com/p/skia/issues/detail?id=32
595
596 This test code would fail when we didn't check the return result of
597 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
598 that after the first chop, the parameters to valid_unit_divide are equal
599 (thanks to finite float precision and rounding in the subtracts). Thus
600 even though the 2nd tValue looks < 1.0, after we renormalize it, we end
601 up with 1.0, hence the need to check and just return the last cubic as
602 a degenerate clump of 4 points in the sampe place.
603
604 static void test_cubic() {
605 SkPoint src[4] = {
606 { 556.25000, 523.03003 },
607 { 556.23999, 522.96002 },
608 { 556.21997, 522.89001 },
609 { 556.21997, 522.82001 }
610 };
611 SkPoint dst[10];
612 SkScalar tval[] = { 0.33333334f, 0.99999994f };
613 SkChopCubicAt(src, dst, tval, 2);
614 }
615 */
616
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]617void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots)
618{
619#ifdef SK_DEBUG
620 {
621 for (int i = 0; i < roots - 1; i++)
622 {
623 SkASSERT(is_unit_interval(tValues[i]));
624 SkASSERT(is_unit_interval(tValues[i+1]));
625 SkASSERT(tValues[i] < tValues[i+1]);
626 }
627 }
628#endif
629
630 if (dst)
631 {
632 if (roots == 0) // nothing to chop
633 memcpy(dst, src, 4*sizeof(SkPoint));
634 else
635 {
636 SkScalar t = tValues[0];
637 SkPoint tmp[4];
638
639 for (int i = 0; i < roots; i++)
640 {
641 SkChopCubicAt(src, dst, t);
642 if (i == roots - 1)
643 break;
644
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]645 dst += 3;
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]646 // have src point to the remaining cubic (after the chop)
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]647 memcpy(tmp, dst, 4 * sizeof(SkPoint));
648 src = tmp;
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]649
650 // watch out in case the renormalized t isn't in range
651 if (!valid_unit_divide(tValues[i+1] - tValues[i],
652 SK_Scalar1 - tValues[i], &t)) {
653 // if we can't, just create a degenerate cubic
654 dst[4] = dst[5] = dst[6] = src[3];
655 break;
656 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]657 }
658 }
659 }
660}
661
662void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7])
663{
664 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
665 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
666 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
667 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
668 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
669 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
670
671 SkScalar x012 = SkScalarAve(x01, x12);
672 SkScalar y012 = SkScalarAve(y01, y12);
673 SkScalar x123 = SkScalarAve(x12, x23);
674 SkScalar y123 = SkScalarAve(y12, y23);
675
676 dst[0] = src[0];
677 dst[1].set(x01, y01);
678 dst[2].set(x012, y012);
679 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
680 dst[4].set(x123, y123);
681 dst[5].set(x23, y23);
682 dst[6] = src[3];
683}
684
685static void flatten_double_cubic_extrema(SkScalar coords[14])
686{
687 coords[4] = coords[8] = coords[6];
688}
689
690/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
691 the resulting beziers are monotonic in Y. This is called by the scan converter.
692 Depending on what is returned, dst[] is treated as follows
693 0 dst[0..3] is the original cubic
694 1 dst[0..3] and dst[3..6] are the two new cubics
695 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
696 If dst == null, it is ignored and only the count is returned.
697*/
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]698int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]699 SkScalar tValues[2];
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]700 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
701 src[3].fY, tValues);
702
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]703 SkChopCubicAt(src, dst, tValues, roots);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]704 if (dst && roots > 0) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]705 // we do some cleanup to ensure our Y extrema are flat
706 flatten_double_cubic_extrema(&dst[0].fY);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]707 if (roots == 2) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]708 flatten_double_cubic_extrema(&dst[3].fY);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]709 }
710 }
711 return roots;
712}
713
714int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
715 SkScalar tValues[2];
716 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
717 src[3].fX, tValues);
718
719 SkChopCubicAt(src, dst, tValues, roots);
720 if (dst && roots > 0) {
721 // we do some cleanup to ensure our Y extrema are flat
722 flatten_double_cubic_extrema(&dst[0].fX);
723 if (roots == 2) {
724 flatten_double_cubic_extrema(&dst[3].fX);
725 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]726 }
727 return roots;
728}
729
730/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
731
732 Inflection means that curvature is zero.
733 Curvature is [F' x F''] / [F'^3]
734 So we solve F'x X F''y - F'y X F''y == 0
735 After some canceling of the cubic term, we get
736 A = b - a
737 B = c - 2b + a
738 C = d - 3c + 3b - a
739 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
740*/
741int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[])
742{
743 SkScalar Ax = src[1].fX - src[0].fX;
744 SkScalar Ay = src[1].fY - src[0].fY;
745 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
746 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY;
747 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
748 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
749 int count;
750
751#ifdef SK_SCALAR_IS_FLOAT
752 count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues);
753#else
754 Sk64 A, B, C, tmp;
755
756 A.setMul(Bx, Cy);
757 tmp.setMul(By, Cx);
758 A.sub(tmp);
759
760 B.setMul(Ax, Cy);
761 tmp.setMul(Ay, Cx);
762 B.sub(tmp);
763
764 C.setMul(Ax, By);
765 tmp.setMul(Ay, Bx);
766 C.sub(tmp);
767
768 count = Sk64FindFixedQuadRoots(A, B, C, tValues);
769#endif
770
771 return count;
772}
773
774int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10])
775{
776 SkScalar tValues[2];
777 int count = SkFindCubicInflections(src, tValues);
778
779 if (dst)
780 {
781 if (count == 0)
782 memcpy(dst, src, 4 * sizeof(SkPoint));
783 else
784 SkChopCubicAt(src, dst, tValues, count);
785 }
786 return count + 1;
787}
788
789template <typename T> void bubble_sort(T array[], int count)
790{
791 for (int i = count - 1; i > 0; --i)
792 for (int j = i; j > 0; --j)
793 if (array[j] < array[j-1])
794 {
795 T tmp(array[j]);
796 array[j] = array[j-1];
797 array[j-1] = tmp;
798 }
799}
800
801#include "SkFP.h"
802
803// newton refinement
804#if 0
805static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root)
806{
807 // x1 = x0 - f(t) / f'(t)
808
809 SkFP T = SkScalarToFloat(root);
810 SkFP N, D;
811
812 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2]
813 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3);
814 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2));
815 D = SkFPAdd(D, coeff[2]);
816
817 if (D == 0)
818 return root;
819
820 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3]
821 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]);
822 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1]));
823 N = SkFPAdd(N, SkFPMul(T, coeff[2]));
824 N = SkFPAdd(N, coeff[3]);
825
826 if (N)
827 {
828 SkScalar delta = SkFPToScalar(SkFPDiv(N, D));
829
830 if (delta)
831 root -= delta;
832 }
833 return root;
834}
835#endif
836
837#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop
838#pragma warning ( disable : 4702 )
839#endif
840
841/* Solve coeff(t) == 0, returning the number of roots that
842 lie withing 0 < t < 1.
843 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
844*/
845static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3])
846{
847#ifndef SK_SCALAR_IS_FLOAT
848 return 0; // this is not yet implemented for software float
849#endif
850
851 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic
852 {
853 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
854 }
855
856 SkFP a, b, c, Q, R;
857
858 {
859 SkASSERT(coeff[0] != 0);
860
861 SkFP inva = SkFPInvert(coeff[0]);
862 a = SkFPMul(coeff[1], inva);
863 b = SkFPMul(coeff[2], inva);
864 c = SkFPMul(coeff[3], inva);
865 }
866 Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9);
867// R = (2*a*a*a - 9*a*b + 27*c) / 54;
868 R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2);
869 R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9));
870 R = SkFPAdd(R, SkFPMulInt(c, 27));
871 R = SkFPDivInt(R, 54);
872
873 SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q);
874 SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3);
875 SkFP adiv3 = SkFPDivInt(a, 3);
876
877 SkScalar* roots = tValues;
878 SkScalar r;
879
880 if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots
881 {
882#ifdef SK_SCALAR_IS_FLOAT
883 float theta = sk_float_acos(R / sk_float_sqrt(Q3));
884 float neg2RootQ = -2 * sk_float_sqrt(Q);
885
886 r = neg2RootQ * sk_float_cos(theta/3) - adiv3;
887 if (is_unit_interval(r))
888 *roots++ = r;
889
890 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3;
891 if (is_unit_interval(r))
892 *roots++ = r;
893
894 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3;
895 if (is_unit_interval(r))
896 *roots++ = r;
897
898 // now sort the roots
899 bubble_sort(tValues, (int)(roots - tValues));
900#endif
901 }
902 else // we have 1 real root
903 {
904 SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3));
905 A = SkFPCubeRoot(A);
906 if (SkFPGT(R, 0))
907 A = SkFPNeg(A);
908
909 if (A != 0)
910 A = SkFPAdd(A, SkFPDiv(Q, A));
911 r = SkFPToScalar(SkFPSub(A, adiv3));
912 if (is_unit_interval(r))
913 *roots++ = r;
914 }
915
916 return (int)(roots - tValues);
917}
918
919/* Looking for F' dot F'' == 0
920
921 A = b - a
922 B = c - 2b + a
923 C = d - 3c + 3b - a
924
925 F' = 3Ct^2 + 6Bt + 3A
926 F'' = 6Ct + 6B
927
928 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
929*/
930static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4])
931{
932 SkScalar a = src[2] - src[0];
933 SkScalar b = src[4] - 2 * src[2] + src[0];
934 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0];
935
936 SkFP A = SkScalarToFP(a);
937 SkFP B = SkScalarToFP(b);
938 SkFP C = SkScalarToFP(c);
939
940 coeff[0] = SkFPMul(C, C);
941 coeff[1] = SkFPMulInt(SkFPMul(B, C), 3);
942 coeff[2] = SkFPMulInt(SkFPMul(B, B), 2);
943 coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A));
944 coeff[3] = SkFPMul(A, B);
945}
946
947// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1
948//#define kMinTValueForChopping (SK_Scalar1 / 256)
949#define kMinTValueForChopping 0
950
951/* Looking for F' dot F'' == 0
952
953 A = b - a
954 B = c - 2b + a
955 C = d - 3c + 3b - a
956
957 F' = 3Ct^2 + 6Bt + 3A
958 F'' = 6Ct + 6B
959
960 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
961*/
962int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3])
963{
964 SkFP coeffX[4], coeffY[4];
965 int i;
966
967 formulate_F1DotF2(&src[0].fX, coeffX);
968 formulate_F1DotF2(&src[0].fY, coeffY);
969
970 for (i = 0; i < 4; i++)
971 coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]);
972
973 SkScalar t[3];
974 int count = solve_cubic_polynomial(coeffX, t);
975 int maxCount = 0;
976
977 // now remove extrema where the curvature is zero (mins)
978 // !!!! need a test for this !!!!
979 for (i = 0; i < count; i++)
980 {
981 // if (not_min_curvature())
982 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping)
983 tValues[maxCount++] = t[i];
984 }
985 return maxCount;
986}
987
988int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3])
989{
990 SkScalar t_storage[3];
991
992 if (tValues == NULL)
993 tValues = t_storage;
994
995 int count = SkFindCubicMaxCurvature(src, tValues);
996
997 if (dst)
998 {
999 if (count == 0)
1000 memcpy(dst, src, 4 * sizeof(SkPoint));
1001 else
1002 SkChopCubicAt(src, dst, tValues, count);
1003 }
1004 return count + 1;
1005}
1006
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1007bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
1008 if (ambiguous) {
1009 *ambiguous = false;
1010 }
1011
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1012 // Find the minimum and maximum y of the extrema, which are the
1013 // first and last points since this cubic is monotonic
1014 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
1015 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
1016
1017 if (pt.fY == cubic[0].fY
1018 || pt.fY < min_y
1019 || pt.fY > max_y) {
1020 // The query line definitely does not cross the curve
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1021 if (ambiguous) {
1022 *ambiguous = (pt.fY == cubic[0].fY);
1023 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1024 return false;
1025 }
1026
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1027 bool pt_at_extremum = (pt.fY == cubic[3].fY);
1028
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1029 SkScalar min_x =
1030 SkMinScalar(
1031 SkMinScalar(
1032 SkMinScalar(cubic[0].fX, cubic[1].fX),
1033 cubic[2].fX),
1034 cubic[3].fX);
1035 if (pt.fX < min_x) {
1036 // The query line definitely crosses the curve
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1037 if (ambiguous) {
1038 *ambiguous = pt_at_extremum;
1039 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1040 return true;
1041 }
1042
1043 SkScalar max_x =
1044 SkMaxScalar(
1045 SkMaxScalar(
1046 SkMaxScalar(cubic[0].fX, cubic[1].fX),
1047 cubic[2].fX),
1048 cubic[3].fX);
1049 if (pt.fX > max_x) {
1050 // The query line definitely does not cross the curve
1051 return false;
1052 }
1053
1054 // Do a binary search to find the parameter value which makes y as
1055 // close as possible to the query point. See whether the query
1056 // line's origin is to the left of the associated x coordinate.
1057
1058 // kMaxIter is chosen as the number of mantissa bits for a float,
1059 // since there's no way we are going to get more precision by
1060 // iterating more times than that.
1061 const int kMaxIter = 23;
1062 SkPoint eval;
1063 int iter = 0;
1064 SkScalar upper_t;
1065 SkScalar lower_t;
1066 // Need to invert direction of t parameter if cubic goes up
1067 // instead of down
1068 if (cubic[3].fY > cubic[0].fY) {
1069 upper_t = SK_Scalar1;
1070 lower_t = SkFloatToScalar(0);
1071 } else {
1072 upper_t = SkFloatToScalar(0);
1073 lower_t = SK_Scalar1;
1074 }
1075 do {
1076 SkScalar t = SkScalarAve(upper_t, lower_t);
1077 SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
1078 if (pt.fY > eval.fY) {
1079 lower_t = t;
1080 } else {
1081 upper_t = t;
1082 }
1083 } while (++iter < kMaxIter
1084 && !SkScalarNearlyZero(eval.fY - pt.fY));
1085 if (pt.fX <= eval.fX) {
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1086 if (ambiguous) {
1087 *ambiguous = pt_at_extremum;
1088 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1089 return true;
1090 }
1091 return false;
1092}
1093
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1094int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1095 int num_crossings = 0;
1096 SkPoint monotonic_cubics[10];
1097 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1098 if (ambiguous) {
1099 *ambiguous = false;
1100 }
1101 bool locally_ambiguous;
1102 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1103 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1104 if (ambiguous) {
1105 *ambiguous |= locally_ambiguous;
1106 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1107 if (num_monotonic_cubics > 0)
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1108 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1109 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1110 if (ambiguous) {
1111 *ambiguous |= locally_ambiguous;
1112 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1113 if (num_monotonic_cubics > 1)
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1114 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1115 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1116 if (ambiguous) {
1117 *ambiguous |= locally_ambiguous;
1118 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1119 return num_crossings;
1120}
1121
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1122////////////////////////////////////////////////////////////////////////////////
1123
1124/* Find t value for quadratic [a, b, c] = d.
1125 Return 0 if there is no solution within [0, 1)
1126*/
1127static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d)
1128{
1129 // At^2 + Bt + C = d
1130 SkScalar A = a - 2 * b + c;
1131 SkScalar B = 2 * (b - a);
1132 SkScalar C = a - d;
1133
1134 SkScalar roots[2];
1135 int count = SkFindUnitQuadRoots(A, B, C, roots);
1136
1137 SkASSERT(count <= 1);
1138 return count == 1 ? roots[0] : 0;
1139}
1140
1141/* given a quad-curve and a point (x,y), chop the quad at that point and return
1142 the new quad's offCurve point. Should only return false if the computed pos
1143 is the start of the curve (i.e. root == 0)
1144*/
1145static bool quad_pt2OffCurve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* offCurve)
1146{
1147 const SkScalar* base;
1148 SkScalar value;
1149
1150 if (SkScalarAbs(x) < SkScalarAbs(y)) {
1151 base = &quad[0].fX;
1152 value = x;
1153 } else {
1154 base = &quad[0].fY;
1155 value = y;
1156 }
1157
1158 // note: this returns 0 if it thinks value is out of range, meaning the
1159 // root might return something outside of [0, 1)
1160 SkScalar t = quad_solve(base[0], base[2], base[4], value);
1161
1162 if (t > 0)
1163 {
1164 SkPoint tmp[5];
1165 SkChopQuadAt(quad, tmp, t);
1166 *offCurve = tmp[1];
1167 return true;
1168 } else {
1169 /* t == 0 means either the value triggered a root outside of [0, 1)
1170 For our purposes, we can ignore the <= 0 roots, but we want to
1171 catch the >= 1 roots (which given our caller, will basically mean
1172 a root of 1, give-or-take numerical instability). If we are in the
1173 >= 1 case, return the existing offCurve point.
1174
1175 The test below checks to see if we are close to the "end" of the
1176 curve (near base[4]). Rather than specifying a tolerance, I just
1177 check to see if value is on to the right/left of the middle point
1178 (depending on the direction/sign of the end points).
1179 */
1180 if ((base[0] < base[4] && value > base[2]) ||
1181 (base[0] > base[4] && value < base[2])) // should root have been 1
1182 {
1183 *offCurve = quad[1];
1184 return true;
1185 }
1186 }
1187 return false;
1188}
1189
1190static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
1191 { SK_Scalar1, 0 },
1192 { SK_Scalar1, SK_ScalarTanPIOver8 },
1193 { SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 },
1194 { SK_ScalarTanPIOver8, SK_Scalar1 },
1195
1196 { 0, SK_Scalar1 },
1197 { -SK_ScalarTanPIOver8, SK_Scalar1 },
1198 { -SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 },
1199 { -SK_Scalar1, SK_ScalarTanPIOver8 },
1200
1201 { -SK_Scalar1, 0 },
1202 { -SK_Scalar1, -SK_ScalarTanPIOver8 },
1203 { -SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 },
1204 { -SK_ScalarTanPIOver8, -SK_Scalar1 },
1205
1206 { 0, -SK_Scalar1 },
1207 { SK_ScalarTanPIOver8, -SK_Scalar1 },
1208 { SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 },
1209 { SK_Scalar1, -SK_ScalarTanPIOver8 },
1210
1211 { SK_Scalar1, 0 }
1212};
1213
1214int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1215 SkRotationDirection dir, const SkMatrix* userMatrix,
1216 SkPoint quadPoints[])
1217{
1218 // rotate by x,y so that uStart is (1.0)
1219 SkScalar x = SkPoint::DotProduct(uStart, uStop);
1220 SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1221
1222 SkScalar absX = SkScalarAbs(x);
1223 SkScalar absY = SkScalarAbs(y);
1224
1225 int pointCount;
1226
1227 // check for (effectively) coincident vectors
1228 // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1229 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1230 if (absY <= SK_ScalarNearlyZero && x > 0 &&
1231 ((y >= 0 && kCW_SkRotationDirection == dir) ||
1232 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1233
1234 // just return the start-point
1235 quadPoints[0].set(SK_Scalar1, 0);
1236 pointCount = 1;
1237 } else {
1238 if (dir == kCCW_SkRotationDirection)
1239 y = -y;
1240
1241 // what octant (quadratic curve) is [xy] in?
1242 int oct = 0;
1243 bool sameSign = true;
1244
1245 if (0 == y)
1246 {
1247 oct = 4; // 180
1248 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1249 }
1250 else if (0 == x)
1251 {
1252 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1253 if (y > 0)
1254 oct = 2; // 90
1255 else
1256 oct = 6; // 270
1257 }
1258 else
1259 {
1260 if (y < 0)
1261 oct += 4;
1262 if ((x < 0) != (y < 0))
1263 {
1264 oct += 2;
1265 sameSign = false;
1266 }
1267 if ((absX < absY) == sameSign)
1268 oct += 1;
1269 }
1270
1271 int wholeCount = oct << 1;
1272 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1273
1274 const SkPoint* arc = &gQuadCirclePts[wholeCount];
1275 if (quad_pt2OffCurve(arc, x, y, &quadPoints[wholeCount + 1]))
1276 {
1277 quadPoints[wholeCount + 2].set(x, y);
1278 wholeCount += 2;
1279 }
1280 pointCount = wholeCount + 1;
1281 }
1282
1283 // now handle counter-clockwise and the initial unitStart rotation
1284 SkMatrix matrix;
1285 matrix.setSinCos(uStart.fY, uStart.fX);
1286 if (dir == kCCW_SkRotationDirection) {
1287 matrix.preScale(SK_Scalar1, -SK_Scalar1);
1288 }
1289 if (userMatrix) {
1290 matrix.postConcat(*userMatrix);
1291 }
1292 matrix.mapPoints(quadPoints, pointCount);
1293 return pointCount;
1294}
1295