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android / platform / external / chromium_org / third_party / skia / src / ceb7526c2f715f49a47e902a642fea7eb61153ec / . / core / SkGeometry.cpp
blob: f07ca2a10190fe99591c682d92a4fc6b31af1e7f [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);
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]207 SkScalar bc = SkScalarInterp(src[2], src[4], t);
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]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);
rmistry@google.com935e9f42012-08-23 18:09:54 +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];
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]327
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]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
rmistry@google.com935e9f42012-08-23 18:09:54 +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;
rmistry@google.com935e9f42012-08-23 18:09:54 +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);
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]363
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]364 SkScalar a = src[0].fX;
365 SkScalar b = src[1].fX;
366 SkScalar c = src[2].fX;
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]367
reed@android.com001bd972009-11-17 18:47:52 +0000[diff] [blame]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//
egdaniel@google.com63cc6e12013-07-12 20:15:34 +0000[diff] [blame]397float SkFindQuadMaxCurvature(const SkPoint src[3]) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]398 SkScalar Ax = src[1].fX - src[0].fX;
399 SkScalar Ay = src[1].fY - src[0].fY;
400 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
401 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
402 SkScalar t = 0; // 0 means don't chop
403
404#ifdef SK_SCALAR_IS_FLOAT
405 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
406#else
407 // !!! should I use SkFloat here? seems like it
408 Sk64 numer, denom, tmp;
409
410 numer.setMul(Ax, -Bx);
411 tmp.setMul(Ay, -By);
412 numer.add(tmp);
413
414 if (numer.isPos()) // do nothing if numer <= 0
415 {
416 denom.setMul(Bx, Bx);
417 tmp.setMul(By, By);
418 denom.add(tmp);
419 SkASSERT(!denom.isNeg());
420 if (numer < denom)
421 {
422 t = numer.getFixedDiv(denom);
423 SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!)
424 if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability
425 t = 0; // ignore the chop
426 }
427 }
428#endif
egdaniel@google.com63cc6e12013-07-12 20:15:34 +0000[diff] [blame]429 return t;
430}
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]431
egdaniel@google.com63cc6e12013-07-12 20:15:34 +0000[diff] [blame]432int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5])
433{
434 SkScalar t = SkFindQuadMaxCurvature(src);
435 if (t == 0) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]436 memcpy(dst, src, 3 * sizeof(SkPoint));
437 return 1;
egdaniel@google.com63cc6e12013-07-12 20:15:34 +0000[diff] [blame]438 } else {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]439 SkChopQuadAt(src, dst, t);
440 return 2;
441 }
442}
443
reed@google.com007593e2011-07-27 13:54:36 +0000[diff] [blame]444#ifdef SK_SCALAR_IS_FLOAT
445 #define SK_ScalarTwoThirds (0.666666666f)
446#else
447 #define SK_ScalarTwoThirds ((SkFixed)(43691))
448#endif
449
450void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
451 const SkScalar scale = SK_ScalarTwoThirds;
452 dst[0] = src[0];
453 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
454 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
455 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
456 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
457 dst[3] = src[2];
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]458}
459
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]460////////////////////////////////////////////////////////////////////////////////////////
461///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
462////////////////////////////////////////////////////////////////////////////////////////
463
464static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4])
465{
466 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
467 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
468 coeff[2] = 3*(pt[2] - pt[0]);
469 coeff[3] = pt[0];
470}
471
472void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4])
473{
474 SkASSERT(pts);
475
476 if (cx)
477 get_cubic_coeff(&pts[0].fX, cx);
478 if (cy)
479 get_cubic_coeff(&pts[0].fY, cy);
480}
481
482static SkScalar eval_cubic(const SkScalar src[], SkScalar t)
483{
484 SkASSERT(src);
485 SkASSERT(t >= 0 && t <= SK_Scalar1);
486
487 if (t == 0)
488 return src[0];
489
490#ifdef DIRECT_EVAL_OF_POLYNOMIALS
491 SkScalar D = src[0];
492 SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
493 SkScalar B = 3*(src[4] - src[2] - src[2] + D);
494 SkScalar C = 3*(src[2] - D);
495
496 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
497#else
498 SkScalar ab = SkScalarInterp(src[0], src[2], t);
499 SkScalar bc = SkScalarInterp(src[2], src[4], t);
500 SkScalar cd = SkScalarInterp(src[4], src[6], t);
501 SkScalar abc = SkScalarInterp(ab, bc, t);
502 SkScalar bcd = SkScalarInterp(bc, cd, t);
503 return SkScalarInterp(abc, bcd, t);
504#endif
505}
506
507/** return At^2 + Bt + C
508*/
509static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t)
510{
511 SkASSERT(t >= 0 && t <= SK_Scalar1);
512
513 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
514}
515
516static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t)
517{
518 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
519 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
520 SkScalar C = src[2] - src[0];
521
522 return eval_quadratic(A, B, C, t);
523}
524
525static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t)
526{
527 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
528 SkScalar B = src[4] - 2 * src[2] + src[0];
529
530 return SkScalarMulAdd(A, t, B);
531}
532
533void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature)
534{
535 SkASSERT(src);
536 SkASSERT(t >= 0 && t <= SK_Scalar1);
537
538 if (loc)
539 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
540 if (tangent)
541 tangent->set(eval_cubic_derivative(&src[0].fX, t),
542 eval_cubic_derivative(&src[0].fY, t));
543 if (curvature)
544 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
545 eval_cubic_2ndDerivative(&src[0].fY, t));
546}
547
548/** Cubic'(t) = At^2 + Bt + C, where
549 A = 3(-a + 3(b - c) + d)
550 B = 6(a - 2b + c)
551 C = 3(b - a)
552 Solve for t, keeping only those that fit betwee 0 < t < 1
553*/
554int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2])
555{
556#ifdef SK_SCALAR_IS_FIXED
557 if (!is_not_monotonic(a, b, c, d))
558 return 0;
559#endif
560
561 // we divide A,B,C by 3 to simplify
562 SkScalar A = d - a + 3*(b - c);
563 SkScalar B = 2*(a - b - b + c);
564 SkScalar C = b - a;
565
566 return SkFindUnitQuadRoots(A, B, C, tValues);
567}
568
569static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
570{
571 SkScalar ab = SkScalarInterp(src[0], src[2], t);
572 SkScalar bc = SkScalarInterp(src[2], src[4], t);
573 SkScalar cd = SkScalarInterp(src[4], src[6], t);
574 SkScalar abc = SkScalarInterp(ab, bc, t);
575 SkScalar bcd = SkScalarInterp(bc, cd, t);
576 SkScalar abcd = SkScalarInterp(abc, bcd, t);
577
578 dst[0] = src[0];
579 dst[2] = ab;
580 dst[4] = abc;
581 dst[6] = abcd;
582 dst[8] = bcd;
583 dst[10] = cd;
584 dst[12] = src[6];
585}
586
587void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t)
588{
589 SkASSERT(t > 0 && t < SK_Scalar1);
590
591 interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
592 interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
593}
594
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]595/* http://code.google.com/p/skia/issues/detail?id=32
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]596
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]597 This test code would fail when we didn't check the return result of
598 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
599 that after the first chop, the parameters to valid_unit_divide are equal
600 (thanks to finite float precision and rounding in the subtracts). Thus
601 even though the 2nd tValue looks < 1.0, after we renormalize it, we end
602 up with 1.0, hence the need to check and just return the last cubic as
603 a degenerate clump of 4 points in the sampe place.
604
605 static void test_cubic() {
606 SkPoint src[4] = {
607 { 556.25000, 523.03003 },
608 { 556.23999, 522.96002 },
609 { 556.21997, 522.89001 },
610 { 556.21997, 522.82001 }
611 };
612 SkPoint dst[10];
613 SkScalar tval[] = { 0.33333334f, 0.99999994f };
614 SkChopCubicAt(src, dst, tval, 2);
615 }
616 */
617
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]618void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots)
619{
620#ifdef SK_DEBUG
621 {
622 for (int i = 0; i < roots - 1; i++)
623 {
624 SkASSERT(is_unit_interval(tValues[i]));
625 SkASSERT(is_unit_interval(tValues[i+1]));
626 SkASSERT(tValues[i] < tValues[i+1]);
627 }
628 }
629#endif
630
631 if (dst)
632 {
633 if (roots == 0) // nothing to chop
634 memcpy(dst, src, 4*sizeof(SkPoint));
635 else
636 {
637 SkScalar t = tValues[0];
638 SkPoint tmp[4];
639
640 for (int i = 0; i < roots; i++)
641 {
642 SkChopCubicAt(src, dst, t);
643 if (i == roots - 1)
644 break;
645
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]646 dst += 3;
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]647 // have src point to the remaining cubic (after the chop)
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]648 memcpy(tmp, dst, 4 * sizeof(SkPoint));
649 src = tmp;
reed@android.com17bdc092009-08-28 20:06:54 +0000[diff] [blame]650
651 // watch out in case the renormalized t isn't in range
652 if (!valid_unit_divide(tValues[i+1] - tValues[i],
653 SK_Scalar1 - tValues[i], &t)) {
654 // if we can't, just create a degenerate cubic
655 dst[4] = dst[5] = dst[6] = src[3];
656 break;
657 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]658 }
659 }
660 }
661}
662
663void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7])
664{
665 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
666 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
667 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
668 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
669 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
670 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
671
672 SkScalar x012 = SkScalarAve(x01, x12);
673 SkScalar y012 = SkScalarAve(y01, y12);
674 SkScalar x123 = SkScalarAve(x12, x23);
675 SkScalar y123 = SkScalarAve(y12, y23);
676
677 dst[0] = src[0];
678 dst[1].set(x01, y01);
679 dst[2].set(x012, y012);
680 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
681 dst[4].set(x123, y123);
682 dst[5].set(x23, y23);
683 dst[6] = src[3];
684}
685
686static void flatten_double_cubic_extrema(SkScalar coords[14])
687{
688 coords[4] = coords[8] = coords[6];
689}
690
691/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
692 the resulting beziers are monotonic in Y. This is called by the scan converter.
693 Depending on what is returned, dst[] is treated as follows
694 0 dst[0..3] is the original cubic
695 1 dst[0..3] and dst[3..6] are the two new cubics
696 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
697 If dst == null, it is ignored and only the count is returned.
698*/
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]699int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]700 SkScalar tValues[2];
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]701 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
702 src[3].fY, tValues);
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]703
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]704 SkChopCubicAt(src, dst, tValues, roots);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]705 if (dst && roots > 0) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]706 // we do some cleanup to ensure our Y extrema are flat
707 flatten_double_cubic_extrema(&dst[0].fY);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]708 if (roots == 2) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]709 flatten_double_cubic_extrema(&dst[3].fY);
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]710 }
711 }
712 return roots;
713}
714
715int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
716 SkScalar tValues[2];
717 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
718 src[3].fX, tValues);
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]719
reed@android.com68779c32009-11-18 13:47:40 +0000[diff] [blame]720 SkChopCubicAt(src, dst, tValues, roots);
721 if (dst && roots > 0) {
722 // we do some cleanup to ensure our Y extrema are flat
723 flatten_double_cubic_extrema(&dst[0].fX);
724 if (roots == 2) {
725 flatten_double_cubic_extrema(&dst[3].fX);
726 }
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]727 }
728 return roots;
729}
730
731/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
732
733 Inflection means that curvature is zero.
734 Curvature is [F' x F''] / [F'^3]
735 So we solve F'x X F''y - F'y X F''y == 0
736 After some canceling of the cubic term, we get
737 A = b - a
738 B = c - 2b + a
739 C = d - 3c + 3b - a
740 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
741*/
742int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[])
743{
744 SkScalar Ax = src[1].fX - src[0].fX;
745 SkScalar Ay = src[1].fY - src[0].fY;
746 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
747 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY;
748 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
749 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
750 int count;
751
752#ifdef SK_SCALAR_IS_FLOAT
753 count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues);
754#else
755 Sk64 A, B, C, tmp;
756
757 A.setMul(Bx, Cy);
758 tmp.setMul(By, Cx);
759 A.sub(tmp);
760
761 B.setMul(Ax, Cy);
762 tmp.setMul(Ay, Cx);
763 B.sub(tmp);
764
765 C.setMul(Ax, By);
766 tmp.setMul(Ay, Bx);
767 C.sub(tmp);
768
769 count = Sk64FindFixedQuadRoots(A, B, C, tValues);
770#endif
771
772 return count;
773}
774
775int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10])
776{
777 SkScalar tValues[2];
778 int count = SkFindCubicInflections(src, tValues);
779
780 if (dst)
781 {
782 if (count == 0)
783 memcpy(dst, src, 4 * sizeof(SkPoint));
784 else
785 SkChopCubicAt(src, dst, tValues, count);
786 }
787 return count + 1;
788}
789
790template <typename T> void bubble_sort(T array[], int count)
791{
792 for (int i = count - 1; i > 0; --i)
793 for (int j = i; j > 0; --j)
794 if (array[j] < array[j-1])
795 {
796 T tmp(array[j]);
797 array[j] = array[j-1];
798 array[j-1] = tmp;
799 }
800}
801
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]802// newton refinement
803#if 0
804static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root)
805{
806 // x1 = x0 - f(t) / f'(t)
807
808 SkFP T = SkScalarToFloat(root);
809 SkFP N, D;
810
811 // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2]
812 D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3);
813 D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2));
814 D = SkFPAdd(D, coeff[2]);
815
816 if (D == 0)
817 return root;
818
819 // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3]
820 N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]);
821 N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1]));
822 N = SkFPAdd(N, SkFPMul(T, coeff[2]));
823 N = SkFPAdd(N, coeff[3]);
824
825 if (N)
826 {
827 SkScalar delta = SkFPToScalar(SkFPDiv(N, D));
828
829 if (delta)
830 root -= delta;
831 }
832 return root;
833}
834#endif
835
reed@google.com7de50932012-02-29 20:59:24 +0000[diff] [blame]836/**
837 * Given an array and count, remove all pair-wise duplicates from the array,
838 * keeping the existing sorting, and return the new count
839 */
840static int collaps_duplicates(float array[], int count) {
reed@google.com7de50932012-02-29 20:59:24 +0000[diff] [blame]841 for (int n = count; n > 1; --n) {
842 if (array[0] == array[1]) {
843 for (int i = 1; i < n; ++i) {
844 array[i - 1] = array[i];
845 }
846 count -= 1;
847 } else {
848 array += 1;
849 }
850 }
851 return count;
852}
853
854#ifdef SK_DEBUG
855
856#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array)
857
858static void test_collaps_duplicates() {
859 static bool gOnce;
860 if (gOnce) { return; }
861 gOnce = true;
862 const float src0[] = { 0 };
863 const float src1[] = { 0, 0 };
864 const float src2[] = { 0, 1 };
865 const float src3[] = { 0, 0, 0 };
866 const float src4[] = { 0, 0, 1 };
867 const float src5[] = { 0, 1, 1 };
868 const float src6[] = { 0, 1, 2 };
869 const struct {
870 const float* fData;
871 int fCount;
872 int fCollapsedCount;
873 } data[] = {
874 { TEST_COLLAPS_ENTRY(src0), 1 },
875 { TEST_COLLAPS_ENTRY(src1), 1 },
876 { TEST_COLLAPS_ENTRY(src2), 2 },
877 { TEST_COLLAPS_ENTRY(src3), 1 },
878 { TEST_COLLAPS_ENTRY(src4), 2 },
879 { TEST_COLLAPS_ENTRY(src5), 2 },
880 { TEST_COLLAPS_ENTRY(src6), 3 },
881 };
882 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
883 float dst[3];
884 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
885 int count = collaps_duplicates(dst, data[i].fCount);
886 SkASSERT(data[i].fCollapsedCount == count);
887 for (int j = 1; j < count; ++j) {
888 SkASSERT(dst[j-1] < dst[j]);
889 }
890 }
891}
892#endif
893
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]894#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop
895#pragma warning ( disable : 4702 )
896#endif
897
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]898static SkScalar SkScalarCubeRoot(SkScalar x) {
899 return sk_float_pow(x, 0.3333333f);
900}
901
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]902/* Solve coeff(t) == 0, returning the number of roots that
903 lie withing 0 < t < 1.
904 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]905
reed@google.com7de50932012-02-29 20:59:24 +0000[diff] [blame]906 Eliminates repeated roots (so that all tValues are distinct, and are always
907 in increasing order.
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]908*/
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]909static int solve_cubic_polynomial(const SkScalar coeff[4], SkScalar tValues[3])
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]910{
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]911 if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic
912 {
913 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
914 }
915
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]916 SkScalar a, b, c, Q, R;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]917
918 {
919 SkASSERT(coeff[0] != 0);
920
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]921 SkScalar inva = SkScalarInvert(coeff[0]);
922 a = coeff[1] * inva;
923 b = coeff[2] * inva;
924 c = coeff[3] * inva;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]925 }
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]926 Q = (a*a - b*3) / 9;
927 R = (2*a*a*a - 9*a*b + 27*c) / 54;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]928
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]929 SkScalar Q3 = Q * Q * Q;
930 SkScalar R2MinusQ3 = R * R - Q3;
931 SkScalar adiv3 = a / 3;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]932
933 SkScalar* roots = tValues;
934 SkScalar r;
935
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]936 if (R2MinusQ3 < 0) // we have 3 real roots
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]937 {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]938 float theta = sk_float_acos(R / sk_float_sqrt(Q3));
939 float neg2RootQ = -2 * sk_float_sqrt(Q);
940
941 r = neg2RootQ * sk_float_cos(theta/3) - adiv3;
942 if (is_unit_interval(r))
943 *roots++ = r;
944
945 r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3;
946 if (is_unit_interval(r))
947 *roots++ = r;
948
949 r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3;
950 if (is_unit_interval(r))
951 *roots++ = r;
952
reed@google.com7de50932012-02-29 20:59:24 +0000[diff] [blame]953 SkDEBUGCODE(test_collaps_duplicates();)
954
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]955 // now sort the roots
reed@google.com7de50932012-02-29 20:59:24 +0000[diff] [blame]956 int count = (int)(roots - tValues);
957 SkASSERT((unsigned)count <= 3);
958 bubble_sort(tValues, count);
959 count = collaps_duplicates(tValues, count);
960 roots = tValues + count; // so we compute the proper count below
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]961 }
962 else // we have 1 real root
963 {
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]964 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
965 A = SkScalarCubeRoot(A);
966 if (R > 0)
967 A = -A;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]968
969 if (A != 0)
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]970 A += Q / A;
971 r = A - adiv3;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]972 if (is_unit_interval(r))
973 *roots++ = r;
974 }
975
976 return (int)(roots - tValues);
977}
978
979/* Looking for F' dot F'' == 0
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]980
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]981 A = b - a
982 B = c - 2b + a
983 C = d - 3c + 3b - a
984
985 F' = 3Ct^2 + 6Bt + 3A
986 F'' = 6Ct + 6B
987
988 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
989*/
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]990static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4])
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]991{
992 SkScalar a = src[2] - src[0];
993 SkScalar b = src[4] - 2 * src[2] + src[0];
994 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0];
995
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]996 coeff[0] = c * c;
997 coeff[1] = 3 * b * c;
998 coeff[2] = 2 * b * b + c * a;
999 coeff[3] = a * b;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1000}
1001
1002// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1
1003//#define kMinTValueForChopping (SK_Scalar1 / 256)
1004#define kMinTValueForChopping 0
1005
1006/* Looking for F' dot F'' == 0
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]1007
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1008 A = b - a
1009 B = c - 2b + a
1010 C = d - 3c + 3b - a
1011
1012 F' = 3Ct^2 + 6Bt + 3A
1013 F'' = 6Ct + 6B
1014
1015 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
1016*/
1017int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3])
1018{
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]1019 SkScalar coeffX[4], coeffY[4];
1020 int i;
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1021
1022 formulate_F1DotF2(&src[0].fX, coeffX);
1023 formulate_F1DotF2(&src[0].fY, coeffY);
1024
1025 for (i = 0; i < 4; i++)
reed@google.comceb75262013-12-16 14:17:40 +0000[diff] [blame^]1026 coeffX[i] += coeffY[i];
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1027
1028 SkScalar t[3];
1029 int count = solve_cubic_polynomial(coeffX, t);
1030 int maxCount = 0;
1031
1032 // now remove extrema where the curvature is zero (mins)
1033 // !!!! need a test for this !!!!
1034 for (i = 0; i < count; i++)
1035 {
1036 // if (not_min_curvature())
1037 if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping)
1038 tValues[maxCount++] = t[i];
1039 }
1040 return maxCount;
1041}
1042
1043int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3])
1044{
1045 SkScalar t_storage[3];
1046
1047 if (tValues == NULL)
1048 tValues = t_storage;
1049
1050 int count = SkFindCubicMaxCurvature(src, tValues);
1051
egdaniel@google.com63cc6e12013-07-12 20:15:34 +0000[diff] [blame]1052 if (dst) {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1053 if (count == 0)
1054 memcpy(dst, src, 4 * sizeof(SkPoint));
1055 else
1056 SkChopCubicAt(src, dst, tValues, count);
1057 }
1058 return count + 1;
1059}
1060
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1061bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
1062 if (ambiguous) {
1063 *ambiguous = false;
1064 }
1065
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1066 // Find the minimum and maximum y of the extrema, which are the
1067 // first and last points since this cubic is monotonic
1068 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
1069 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
1070
1071 if (pt.fY == cubic[0].fY
1072 || pt.fY < min_y
1073 || pt.fY > max_y) {
1074 // The query line definitely does not cross the curve
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1075 if (ambiguous) {
1076 *ambiguous = (pt.fY == cubic[0].fY);
1077 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1078 return false;
1079 }
1080
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1081 bool pt_at_extremum = (pt.fY == cubic[3].fY);
1082
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1083 SkScalar min_x =
1084 SkMinScalar(
1085 SkMinScalar(
1086 SkMinScalar(cubic[0].fX, cubic[1].fX),
1087 cubic[2].fX),
1088 cubic[3].fX);
1089 if (pt.fX < min_x) {
1090 // The query line definitely crosses the curve
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1091 if (ambiguous) {
1092 *ambiguous = pt_at_extremum;
1093 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1094 return true;
1095 }
1096
1097 SkScalar max_x =
1098 SkMaxScalar(
1099 SkMaxScalar(
1100 SkMaxScalar(cubic[0].fX, cubic[1].fX),
1101 cubic[2].fX),
1102 cubic[3].fX);
1103 if (pt.fX > max_x) {
1104 // The query line definitely does not cross the curve
1105 return false;
1106 }
1107
1108 // Do a binary search to find the parameter value which makes y as
1109 // close as possible to the query point. See whether the query
1110 // line's origin is to the left of the associated x coordinate.
1111
1112 // kMaxIter is chosen as the number of mantissa bits for a float,
1113 // since there's no way we are going to get more precision by
1114 // iterating more times than that.
1115 const int kMaxIter = 23;
1116 SkPoint eval;
1117 int iter = 0;
1118 SkScalar upper_t;
1119 SkScalar lower_t;
1120 // Need to invert direction of t parameter if cubic goes up
1121 // instead of down
1122 if (cubic[3].fY > cubic[0].fY) {
1123 upper_t = SK_Scalar1;
commit-bot@chromium.orgdfc928c2013-11-25 19:44:07 +0000[diff] [blame]1124 lower_t = 0;
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1125 } else {
commit-bot@chromium.orgdfc928c2013-11-25 19:44:07 +0000[diff] [blame]1126 upper_t = 0;
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1127 lower_t = SK_Scalar1;
1128 }
1129 do {
1130 SkScalar t = SkScalarAve(upper_t, lower_t);
1131 SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
1132 if (pt.fY > eval.fY) {
1133 lower_t = t;
1134 } else {
1135 upper_t = t;
1136 }
1137 } while (++iter < kMaxIter
1138 && !SkScalarNearlyZero(eval.fY - pt.fY));
1139 if (pt.fX <= eval.fX) {
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1140 if (ambiguous) {
1141 *ambiguous = pt_at_extremum;
1142 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1143 return true;
1144 }
1145 return false;
1146}
1147
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1148int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1149 int num_crossings = 0;
1150 SkPoint monotonic_cubics[10];
1151 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1152 if (ambiguous) {
1153 *ambiguous = false;
1154 }
1155 bool locally_ambiguous;
1156 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1157 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1158 if (ambiguous) {
1159 *ambiguous |= locally_ambiguous;
1160 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1161 if (num_monotonic_cubics > 0)
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1162 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1163 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1164 if (ambiguous) {
1165 *ambiguous |= locally_ambiguous;
1166 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1167 if (num_monotonic_cubics > 1)
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1168 if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous))
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1169 ++num_crossings;
kbr@chromium.orgc1b53332010-07-07 22:20:35 +0000[diff] [blame]1170 if (ambiguous) {
1171 *ambiguous |= locally_ambiguous;
1172 }
reed@android.com5b4541e2010-02-05 20:41:02 +0000[diff] [blame]1173 return num_crossings;
1174}
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1175////////////////////////////////////////////////////////////////////////////////
1176
1177/* Find t value for quadratic [a, b, c] = d.
1178 Return 0 if there is no solution within [0, 1)
1179*/
1180static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d)
1181{
1182 // At^2 + Bt + C = d
1183 SkScalar A = a - 2 * b + c;
1184 SkScalar B = 2 * (b - a);
1185 SkScalar C = a - d;
1186
1187 SkScalar roots[2];
1188 int count = SkFindUnitQuadRoots(A, B, C, roots);
1189
1190 SkASSERT(count <= 1);
1191 return count == 1 ? roots[0] : 0;
1192}
1193
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1194/* given a quad-curve and a point (x,y), chop the quad at that point and place
skia.committer@gmail.com937bf442013-09-28 07:01:33 +0000[diff] [blame]1195 the new off-curve point and endpoint into 'dest'.
skia.committer@gmail.com6f1f1592013-07-10 07:00:58 +0000[diff] [blame]1196 Should only return false if the computed pos is the start of the curve
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1197 (i.e. root == 0)
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1198*/
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1199static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest)
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1200{
1201 const SkScalar* base;
1202 SkScalar value;
1203
1204 if (SkScalarAbs(x) < SkScalarAbs(y)) {
1205 base = &quad[0].fX;
1206 value = x;
1207 } else {
1208 base = &quad[0].fY;
1209 value = y;
1210 }
1211
1212 // note: this returns 0 if it thinks value is out of range, meaning the
1213 // root might return something outside of [0, 1)
1214 SkScalar t = quad_solve(base[0], base[2], base[4], value);
1215
1216 if (t > 0)
1217 {
1218 SkPoint tmp[5];
1219 SkChopQuadAt(quad, tmp, t);
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1220 dest[0] = tmp[1];
robertphillips@google.coma9531cd2013-09-27 17:05:59 +0000[diff] [blame]1221 dest[1].set(x, y);
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1222 return true;
1223 } else {
1224 /* t == 0 means either the value triggered a root outside of [0, 1)
1225 For our purposes, we can ignore the <= 0 roots, but we want to
1226 catch the >= 1 roots (which given our caller, will basically mean
1227 a root of 1, give-or-take numerical instability). If we are in the
1228 >= 1 case, return the existing offCurve point.
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]1229
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1230 The test below checks to see if we are close to the "end" of the
1231 curve (near base[4]). Rather than specifying a tolerance, I just
1232 check to see if value is on to the right/left of the middle point
1233 (depending on the direction/sign of the end points).
1234 */
1235 if ((base[0] < base[4] && value > base[2]) ||
1236 (base[0] > base[4] && value < base[2])) // should root have been 1
1237 {
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1238 dest[0] = quad[1];
1239 dest[1].set(x, y);
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1240 return true;
1241 }
1242 }
1243 return false;
1244}
1245
1246static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
commit-bot@chromium.org942ed4e2013-11-01 15:24:55 +0000[diff] [blame]1247// The mid point of the quadratic arc approximation is half way between the two
1248// control points. The float epsilon adjustment moves the on curve point out by
1249// two bits, distributing the convex test error between the round rect approximation
1250// and the convex cross product sign equality test.
1251#define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1252 { SK_Scalar1, 0 },
1253 { SK_Scalar1, SK_ScalarTanPIOver8 },
1254 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
1255 { SK_ScalarTanPIOver8, SK_Scalar1 },
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1256
commit-bot@chromium.org942ed4e2013-11-01 15:24:55 +0000[diff] [blame]1257 { 0, SK_Scalar1 },
1258 { -SK_ScalarTanPIOver8, SK_Scalar1 },
1259 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
1260 { -SK_Scalar1, SK_ScalarTanPIOver8 },
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1261
commit-bot@chromium.org942ed4e2013-11-01 15:24:55 +0000[diff] [blame]1262 { -SK_Scalar1, 0 },
1263 { -SK_Scalar1, -SK_ScalarTanPIOver8 },
1264 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
1265 { -SK_ScalarTanPIOver8, -SK_Scalar1 },
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1266
commit-bot@chromium.org942ed4e2013-11-01 15:24:55 +0000[diff] [blame]1267 { 0, -SK_Scalar1 },
1268 { SK_ScalarTanPIOver8, -SK_Scalar1 },
1269 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
1270 { SK_Scalar1, -SK_ScalarTanPIOver8 },
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1271
commit-bot@chromium.org942ed4e2013-11-01 15:24:55 +0000[diff] [blame]1272 { SK_Scalar1, 0 }
1273#undef SK_MID_RRECT_OFFSET
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1274};
1275
1276int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1277 SkRotationDirection dir, const SkMatrix* userMatrix,
1278 SkPoint quadPoints[])
1279{
1280 // rotate by x,y so that uStart is (1.0)
1281 SkScalar x = SkPoint::DotProduct(uStart, uStop);
1282 SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1283
1284 SkScalar absX = SkScalarAbs(x);
1285 SkScalar absY = SkScalarAbs(y);
1286
1287 int pointCount;
1288
1289 // check for (effectively) coincident vectors
1290 // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1291 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1292 if (absY <= SK_ScalarNearlyZero && x > 0 &&
1293 ((y >= 0 && kCW_SkRotationDirection == dir) ||
1294 (y <= 0 && kCCW_SkRotationDirection == dir))) {
rmistry@google.com935e9f42012-08-23 18:09:54 +0000[diff] [blame]1295
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1296 // just return the start-point
1297 quadPoints[0].set(SK_Scalar1, 0);
1298 pointCount = 1;
1299 } else {
1300 if (dir == kCCW_SkRotationDirection)
1301 y = -y;
1302
1303 // what octant (quadratic curve) is [xy] in?
1304 int oct = 0;
1305 bool sameSign = true;
1306
1307 if (0 == y)
1308 {
1309 oct = 4; // 180
1310 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1311 }
1312 else if (0 == x)
1313 {
1314 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1315 if (y > 0)
1316 oct = 2; // 90
1317 else
1318 oct = 6; // 270
1319 }
1320 else
1321 {
1322 if (y < 0)
1323 oct += 4;
1324 if ((x < 0) != (y < 0))
1325 {
1326 oct += 2;
1327 sameSign = false;
1328 }
1329 if ((absX < absY) == sameSign)
1330 oct += 1;
1331 }
1332
1333 int wholeCount = oct << 1;
1334 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1335
1336 const SkPoint* arc = &gQuadCirclePts[wholeCount];
robertphillips@google.comdd4bd432013-07-09 15:03:59 +0000[diff] [blame]1337 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1]))
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1338 {
reed@android.combcd4d5a2008-12-17 15:59:43 +0000[diff] [blame]1339 wholeCount += 2;
1340 }
1341 pointCount = wholeCount + 1;
1342 }
1343
1344 // now handle counter-clockwise and the initial unitStart rotation
1345 SkMatrix matrix;
1346 matrix.setSinCos(uStart.fY, uStart.fX);
1347 if (dir == kCCW_SkRotationDirection) {
1348 matrix.preScale(SK_Scalar1, -SK_Scalar1);
1349 }
1350 if (userMatrix) {
1351 matrix.postConcat(*userMatrix);
1352 }
1353 matrix.mapPoints(quadPoints, pointCount);
1354 return pointCount;
1355}
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1356
1357///////////////////////////////////////////////////////////////////////////////
1358
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1359// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1360// ------------------------------------------
1361// ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1362//
1363// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1364// ------------------------------------------------
1365// {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1366//
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1367
1368// Take the parametric specification for the conic (either X or Y) and return
1369// in coeff[] the coefficients for the simple quadratic polynomial
1370// coeff[0] for t^2
1371// coeff[1] for t
1372// coeff[2] for constant term
1373//
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1374static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1375 SkASSERT(src);
1376 SkASSERT(t >= 0 && t <= SK_Scalar1);
skia.committer@gmail.comcd487462013-04-17 07:00:56 +0000[diff] [blame]1377
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1378 SkScalar src2w = SkScalarMul(src[2], w);
1379 SkScalar C = src[0];
1380 SkScalar A = src[4] - 2 * src2w + C;
1381 SkScalar B = 2 * (src2w - C);
1382 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
skia.committer@gmail.comcd487462013-04-17 07:00:56 +0000[diff] [blame]1383
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1384 B = 2 * (w - SK_Scalar1);
1385 C = SK_Scalar1;
1386 A = -B;
1387 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
skia.committer@gmail.comcd487462013-04-17 07:00:56 +0000[diff] [blame]1388
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1389 return SkScalarDiv(numer, denom);
1390}
1391
1392// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1393//
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1394// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1395// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1396// t^0 : -2 P0 w + 2 P1 w
1397//
1398// We disregard magnitude, so we can freely ignore the denominator of F', and
1399// divide the numerator by 2
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1400//
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1401// coeff[0] for t^2
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1402// coeff[1] for t^1
1403// coeff[2] for t^0
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1404//
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1405static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) {
1406 const SkScalar P20 = src[4] - src[0];
1407 const SkScalar P10 = src[2] - src[0];
1408 const SkScalar wP10 = w * P10;
1409 coeff[0] = w * P20 - P20;
1410 coeff[1] = P20 - 2 * wP10;
1411 coeff[2] = wP10;
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1412}
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1413
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1414static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1415 SkScalar coeff[3];
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1416 conic_deriv_coeff(coord, w, coeff);
1417 return t * (t * coeff[0] + coeff[1]) + coeff[2];
1418}
1419
1420static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1421 SkScalar coeff[3];
1422 conic_deriv_coeff(src, w, coeff);
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1423
1424 SkScalar tValues[2];
1425 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1426 SkASSERT(0 == roots || 1 == roots);
skia.committer@gmail.comcd487462013-04-17 07:00:56 +0000[diff] [blame]1427
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1428 if (1 == roots) {
1429 *t = tValues[0];
1430 return true;
1431 }
1432 return false;
1433}
reed@google.com256f3102013-04-16 21:07:27 +0000[diff] [blame]1434
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1435struct SkP3D {
1436 SkScalar fX, fY, fZ;
skia.committer@gmail.coma43230c2013-04-13 07:01:15 +0000[diff] [blame]1437
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1438 void set(SkScalar x, SkScalar y, SkScalar z) {
1439 fX = x; fY = y; fZ = z;
1440 }
skia.committer@gmail.coma43230c2013-04-13 07:01:15 +0000[diff] [blame]1441
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1442 void projectDown(SkPoint* dst) const {
1443 dst->set(fX / fZ, fY / fZ);
1444 }
1445};
1446
1447// we just return the middle 3 points, since the first and last are dups of src
1448//
1449static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) {
1450 SkScalar ab = SkScalarInterp(src[0], src[3], t);
1451 SkScalar bc = SkScalarInterp(src[3], src[6], t);
1452 dst[0] = ab;
1453 dst[3] = SkScalarInterp(ab, bc, t);
1454 dst[6] = bc;
1455}
1456
1457static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1458 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1459 dst[1].set(src[1].fX * w, src[1].fY * w, w);
1460 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1461}
1462
reed@google.com4e1502a2013-05-07 20:42:35 +0000[diff] [blame]1463void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1464 SkASSERT(t >= 0 && t <= SK_Scalar1);
skia.committer@gmail.coma43230c2013-04-13 07:01:15 +0000[diff] [blame]1465
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1466 if (pt) {
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1467 pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1468 conic_eval_pos(&fPts[0].fY, fW, t));
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1469 }
reed@google.com4e1502a2013-05-07 20:42:35 +0000[diff] [blame]1470 if (tangent) {
1471 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1472 conic_eval_tan(&fPts[0].fY, fW, t));
1473 }
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1474}
1475
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1476void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1477 SkP3D tmp[3], tmp2[3];
1478
1479 ratquad_mapTo3D(fPts, fW, tmp);
skia.committer@gmail.coma43230c2013-04-13 07:01:15 +0000[diff] [blame]1480
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1481 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1482 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1483 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
skia.committer@gmail.coma43230c2013-04-13 07:01:15 +0000[diff] [blame]1484
reed@google.com0fef2ef2013-04-12 21:55:26 +0000[diff] [blame]1485 dst[0].fPts[0] = fPts[0];
1486 tmp2[0].projectDown(&dst[0].fPts[1]);
1487 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1488 tmp2[2].projectDown(&dst[1].fPts[1]);
1489 dst[1].fPts[2] = fPts[2];
1490
mike@reedtribe.org464a1532013-04-13 10:51:51 +0000[diff] [blame]1491 // to put in "standard form", where w0 and w2 are both 1, we compute the
1492 // new w1 as sqrt(w1*w1/w0*w2)
1493 // or
1494 // w1 /= sqrt(w0*w2)
1495 //
1496 // However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1
1497 //
1498 SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1499 dst[0].fW = tmp2[0].fZ / root;
1500 dst[1].fW = tmp2[2].fZ / root;
reed@google.com79975882013-04-12 19:11:10 +0000[diff] [blame]1501}
mike@reedtribe.org91068392013-04-14 02:40:50 +0000[diff] [blame]1502
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1503static SkScalar subdivide_w_value(SkScalar w) {
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1504 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1505}
1506
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1507void SkConic::chop(SkConic dst[2]) const {
mike@reedtribe.org91068392013-04-14 02:40:50 +0000[diff] [blame]1508 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
1509 SkScalar p1x = fW * fPts[1].fX;
1510 SkScalar p1y = fW * fPts[1].fY;
1511 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
1512 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
1513
1514 dst[0].fPts[0] = fPts[0];
1515 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
1516 (fPts[0].fY + p1y) * scale);
1517 dst[0].fPts[2].set(mx, my);
1518
1519 dst[1].fPts[0].set(mx, my);
1520 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
1521 (p1y + fPts[2].fY) * scale);
1522 dst[1].fPts[2] = fPts[2];
1523
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1524 dst[0].fW = dst[1].fW = subdivide_w_value(fW);
mike@reedtribe.org91068392013-04-14 02:40:50 +0000[diff] [blame]1525}
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1526
mike@reedtribe.orgdc5176e2013-04-27 18:23:16 +0000[diff] [blame]1527/*
1528 * "High order approximation of conic sections by quadratic splines"
1529 * by Michael Floater, 1993
1530 */
mike@reedtribe.org3661c442013-04-30 02:14:58 +0000[diff] [blame]1531#define AS_QUAD_ERROR_SETUP \
1532 SkScalar a = fW - 1; \
1533 SkScalar k = a / (4 * (2 + a)); \
1534 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \
1535 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1536
1537void SkConic::computeAsQuadError(SkVector* err) const {
1538 AS_QUAD_ERROR_SETUP
1539 err->set(x, y);
1540}
1541
1542bool SkConic::asQuadTol(SkScalar tol) const {
1543 AS_QUAD_ERROR_SETUP
1544 return (x * x + y * y) <= tol * tol;
mike@reedtribe.orgdc5176e2013-04-27 18:23:16 +0000[diff] [blame]1545}
1546
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1547int SkConic::computeQuadPOW2(SkScalar tol) const {
mike@reedtribe.org3661c442013-04-30 02:14:58 +0000[diff] [blame]1548 AS_QUAD_ERROR_SETUP
1549 SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
1550
1551 if (error <= 0) {
mike@reedtribe.orgdc5176e2013-04-27 18:23:16 +0000[diff] [blame]1552 return 0;
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1553 }
mike@reedtribe.orgdc5176e2013-04-27 18:23:16 +0000[diff] [blame]1554 uint32_t ierr = (uint32_t)error;
mike@reedtribe.org3661c442013-04-30 02:14:58 +0000[diff] [blame]1555 return (34 - SkCLZ(ierr)) >> 1;
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1556}
1557
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1558static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1559 SkASSERT(level >= 0);
mike@reedtribe.org3661c442013-04-30 02:14:58 +0000[diff] [blame]1560
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1561 if (0 == level) {
1562 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1563 return pts + 2;
1564 } else {
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1565 SkConic dst[2];
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1566 src.chop(dst);
1567 --level;
1568 pts = subdivide(dst[0], pts, level);
1569 return subdivide(dst[1], pts, level);
1570 }
1571}
1572
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1573int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
mike@reedtribe.org3661c442013-04-30 02:14:58 +0000[diff] [blame]1574 SkASSERT(pow2 >= 0);
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1575 *pts = fPts[0];
reed@google.com901641b2013-04-15 15:23:38 +0000[diff] [blame]1576 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
mike@reedtribe.org6c125bd2013-04-15 15:20:52 +0000[diff] [blame]1577 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1578 return 1 << pow2;
1579}
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1580
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1581bool SkConic::findXExtrema(SkScalar* t) const {
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1582 return conic_find_extrema(&fPts[0].fX, fW, t);
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1583}
1584
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1585bool SkConic::findYExtrema(SkScalar* t) const {
mike@reedtribe.org3f0cf542013-05-08 01:55:49 +0000[diff] [blame]1586 return conic_find_extrema(&fPts[0].fY, fW, t);
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1587}
1588
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1589bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1590 SkScalar t;
1591 if (this->findXExtrema(&t)) {
1592 this->chopAt(t, dst);
1593 // now clean-up the middle, since we know t was meant to be at
1594 // an X-extrema
1595 SkScalar value = dst[0].fPts[2].fX;
1596 dst[0].fPts[1].fX = value;
1597 dst[1].fPts[0].fX = value;
1598 dst[1].fPts[1].fX = value;
1599 return true;
1600 }
1601 return false;
1602}
1603
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1604bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
mike@reedtribe.org472a49c2013-04-17 01:21:01 +0000[diff] [blame]1605 SkScalar t;
1606 if (this->findYExtrema(&t)) {
1607 this->chopAt(t, dst);
1608 // now clean-up the middle, since we know t was meant to be at
1609 // an Y-extrema
1610 SkScalar value = dst[0].fPts[2].fY;
1611 dst[0].fPts[1].fY = value;
1612 dst[1].fPts[0].fY = value;
1613 dst[1].fPts[1].fY = value;
1614 return true;
1615 }
1616 return false;
1617}
1618
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1619void SkConic::computeTightBounds(SkRect* bounds) const {
mike@reedtribe.orgc7325912013-04-17 02:25:33 +0000[diff] [blame]1620 SkPoint pts[4];
1621 pts[0] = fPts[0];
1622 pts[1] = fPts[2];
1623 int count = 2;
1624
1625 SkScalar t;
1626 if (this->findXExtrema(&t)) {
1627 this->evalAt(t, &pts[count++]);
1628 }
1629 if (this->findYExtrema(&t)) {
1630 this->evalAt(t, &pts[count++]);
1631 }
1632 bounds->set(pts, count);
1633}
1634
mike@reedtribe.org27a0a562013-04-26 00:58:29 +0000[diff] [blame]1635void SkConic::computeFastBounds(SkRect* bounds) const {
mike@reedtribe.orgc7325912013-04-17 02:25:33 +0000[diff] [blame]1636 bounds->set(fPts, 3);
1637}