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comparison: gfx/skia/trunk/src/core/SkGeometry.cpp

gfx/skia/trunk/src/core/SkGeometry.cpp

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1 /*
2 * Copyright 2006 The Android Open Source Project
3 *
4 * Use of this source code is governed by a BSD-style license that can be
5 * found in the LICENSE file.
6 */
7
8 #include "SkGeometry.h"
9 #include "SkMatrix.h"
10
11 bool SkXRayCrossesLine(const SkXRay& pt,
12 const SkPoint pts[2],
13 bool* ambiguous) {
14 if (ambiguous) {
15 *ambiguous = false;
16 }
17 // Determine quick discards.
18 // Consider query line going exactly through point 0 to not
19 // intersect, for symmetry with SkXRayCrossesMonotonicCubic.
20 if (pt.fY == pts[0].fY) {
21 if (ambiguous) {
22 *ambiguous = true;
23 }
24 return false;
25 }
26 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY)
27 return false;
28 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY)
29 return false;
30 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX)
31 return false;
32 // Determine degenerate cases
33 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY))
34 return false;
35 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) {
36 // We've already determined the query point lies within the
37 // vertical range of the line segment.
38 if (pt.fX <= pts[0].fX) {
39 if (ambiguous) {
40 *ambiguous = (pt.fY == pts[1].fY);
41 }
42 return true;
43 }
44 return false;
45 }
46 // Ambiguity check
47 if (pt.fY == pts[1].fY) {
48 if (pt.fX <= pts[1].fX) {
49 if (ambiguous) {
50 *ambiguous = true;
51 }
52 return true;
53 }
54 return false;
55 }
56 // Full line segment evaluation
57 SkScalar delta_y = pts[1].fY - pts[0].fY;
58 SkScalar delta_x = pts[1].fX - pts[0].fX;
59 SkScalar slope = SkScalarDiv(delta_y, delta_x);
60 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX);
61 // Solve for x coordinate at y = pt.fY
62 SkScalar x = SkScalarDiv(pt.fY - b, slope);
63 return pt.fX <= x;
64 }
65
66 /** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
67 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
68 May also introduce overflow of fixed when we compute our setup.
69 */
70 // #define DIRECT_EVAL_OF_POLYNOMIALS
71
72 ////////////////////////////////////////////////////////////////////////
73
74 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
75 SkScalar ab = a - b;
76 SkScalar bc = b - c;
77 if (ab < 0) {
78 bc = -bc;
79 }
80 return ab == 0 || bc < 0;
81 }
82
83 ////////////////////////////////////////////////////////////////////////
84
85 static bool is_unit_interval(SkScalar x) {
86 return x > 0 && x < SK_Scalar1;
87 }
88
89 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
90 SkASSERT(ratio);
91
92 if (numer < 0) {
93 numer = -numer;
94 denom = -denom;
95 }
96
97 if (denom == 0 || numer == 0 || numer >= denom) {
98 return 0;
99 }
100
101 SkScalar r = SkScalarDiv(numer, denom);
102 if (SkScalarIsNaN(r)) {
103 return 0;
104 }
105 SkASSERT(r >= 0 && r < SK_Scalar1);
106 if (r == 0) { // catch underflow if numer <<<< denom
107 return 0;
108 }
109 *ratio = r;
110 return 1;
111 }
112
113 /** From Numerical Recipes in C.
114
115 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
116 x1 = Q / A
117 x2 = C / Q
118 */
119 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
120 SkASSERT(roots);
121
122 if (A == 0) {
123 return valid_unit_divide(-C, B, roots);
124 }
125
126 SkScalar* r = roots;
127
128 SkScalar R = B*B - 4*A*C;
129 if (R < 0 || SkScalarIsNaN(R)) { // complex roots
130 return 0;
131 }
132 R = SkScalarSqrt(R);
133
134 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
135 r += valid_unit_divide(Q, A, r);
136 r += valid_unit_divide(C, Q, r);
137 if (r - roots == 2) {
138 if (roots[0] > roots[1])
139 SkTSwap<SkScalar>(roots[0], roots[1]);
140 else if (roots[0] == roots[1]) // nearly-equal?
141 r -= 1; // skip the double root
142 }
143 return (int)(r - roots);
144 }
145
146 ///////////////////////////////////////////////////////////////////////////////
147 ///////////////////////////////////////////////////////////////////////////////
148
149 static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
150 SkASSERT(src);
151 SkASSERT(t >= 0 && t <= SK_Scalar1);
152
153 #ifdef DIRECT_EVAL_OF_POLYNOMIALS
154 SkScalar C = src[0];
155 SkScalar A = src[4] - 2 * src[2] + C;
156 SkScalar B = 2 * (src[2] - C);
157 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
158 #else
159 SkScalar ab = SkScalarInterp(src[0], src[2], t);
160 SkScalar bc = SkScalarInterp(src[2], src[4], t);
161 return SkScalarInterp(ab, bc, t);
162 #endif
163 }
164
165 static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
166 SkScalar A = src[4] - 2 * src[2] + src[0];
167 SkScalar B = src[2] - src[0];
168
169 return 2 * SkScalarMulAdd(A, t, B);
170 }
171
172 static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) {
173 SkScalar A = src[4] - 2 * src[2] + src[0];
174 SkScalar B = src[2] - src[0];
175 return A + 2 * B;
176 }
177
178 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt,
179 SkVector* tangent) {
180 SkASSERT(src);
181 SkASSERT(t >= 0 && t <= SK_Scalar1);
182
183 if (pt) {
184 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
185 }
186 if (tangent) {
187 tangent->set(eval_quad_derivative(&src[0].fX, t),
188 eval_quad_derivative(&src[0].fY, t));
189 }
190 }
191
192 void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) {
193 SkASSERT(src);
194
195 if (pt) {
196 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
197 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
198 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
199 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
200 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
201 }
202 if (tangent) {
203 tangent->set(eval_quad_derivative_at_half(&src[0].fX),
204 eval_quad_derivative_at_half(&src[0].fY));
205 }
206 }
207
208 static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) {
209 SkScalar ab = SkScalarInterp(src[0], src[2], t);
210 SkScalar bc = SkScalarInterp(src[2], src[4], t);
211
212 dst[0] = src[0];
213 dst[2] = ab;
214 dst[4] = SkScalarInterp(ab, bc, t);
215 dst[6] = bc;
216 dst[8] = src[4];
217 }
218
219 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
220 SkASSERT(t > 0 && t < SK_Scalar1);
221
222 interp_quad_coords(&src[0].fX, &dst[0].fX, t);
223 interp_quad_coords(&src[0].fY, &dst[0].fY, t);
224 }
225
226 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
227 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
228 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
229 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
230 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
231
232 dst[0] = src[0];
233 dst[1].set(x01, y01);
234 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
235 dst[3].set(x12, y12);
236 dst[4] = src[2];
237 }
238
239 /** Quad'(t) = At + B, where
240 A = 2(a - 2b + c)
241 B = 2(b - a)
242 Solve for t, only if it fits between 0 < t < 1
243 */
244 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
245 /* At + B == 0
246 t = -B / A
247 */
248 return valid_unit_divide(a - b, a - b - b + c, tValue);
249 }
250
251 static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
252 coords[2] = coords[6] = coords[4];
253 }
254
255 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
256 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
257 */
258 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
259 SkASSERT(src);
260 SkASSERT(dst);
261
262 SkScalar a = src[0].fY;
263 SkScalar b = src[1].fY;
264 SkScalar c = src[2].fY;
265
266 if (is_not_monotonic(a, b, c)) {
267 SkScalar tValue;
268 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
269 SkChopQuadAt(src, dst, tValue);
270 flatten_double_quad_extrema(&dst[0].fY);
271 return 1;
272 }
273 // if we get here, we need to force dst to be monotonic, even though
274 // we couldn't compute a unit_divide value (probably underflow).
275 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
276 }
277 dst[0].set(src[0].fX, a);
278 dst[1].set(src[1].fX, b);
279 dst[2].set(src[2].fX, c);
280 return 0;
281 }
282
283 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
284 stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
285 */
286 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
287 SkASSERT(src);
288 SkASSERT(dst);
289
290 SkScalar a = src[0].fX;
291 SkScalar b = src[1].fX;
292 SkScalar c = src[2].fX;
293
294 if (is_not_monotonic(a, b, c)) {
295 SkScalar tValue;
296 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
297 SkChopQuadAt(src, dst, tValue);
298 flatten_double_quad_extrema(&dst[0].fX);
299 return 1;
300 }
301 // if we get here, we need to force dst to be monotonic, even though
302 // we couldn't compute a unit_divide value (probably underflow).
303 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
304 }
305 dst[0].set(a, src[0].fY);
306 dst[1].set(b, src[1].fY);
307 dst[2].set(c, src[2].fY);
308 return 0;
309 }
310
311 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
312 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t
313 // F''(t) = 2 (a - 2b + c)
314 //
315 // A = 2 (b - a)
316 // B = 2 (a - 2b + c)
317 //
318 // Maximum curvature for a quadratic means solving
319 // Fx' Fx'' + Fy' Fy'' = 0
320 //
321 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
322 //
323 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
324 SkScalar Ax = src[1].fX - src[0].fX;
325 SkScalar Ay = src[1].fY - src[0].fY;
326 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
327 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
328 SkScalar t = 0; // 0 means don't chop
329
330 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
331 return t;
332 }
333
334 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
335 SkScalar t = SkFindQuadMaxCurvature(src);
336 if (t == 0) {
337 memcpy(dst, src, 3 * sizeof(SkPoint));
338 return 1;
339 } else {
340 SkChopQuadAt(src, dst, t);
341 return 2;
342 }
343 }
344
345 #define SK_ScalarTwoThirds (0.666666666f)
346
347 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
348 const SkScalar scale = SK_ScalarTwoThirds;
349 dst[0] = src[0];
350 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
351 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
352 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
353 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
354 dst[3] = src[2];
355 }
356
357 //////////////////////////////////////////////////////////////////////////////
358 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
359 //////////////////////////////////////////////////////////////////////////////
360
361 static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
362 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
363 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
364 coeff[2] = 3*(pt[2] - pt[0]);
365 coeff[3] = pt[0];
366 }
367
368 void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
369 SkASSERT(pts);
370
371 if (cx) {
372 get_cubic_coeff(&pts[0].fX, cx);
373 }
374 if (cy) {
375 get_cubic_coeff(&pts[0].fY, cy);
376 }
377 }
378
379 static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
380 SkASSERT(src);
381 SkASSERT(t >= 0 && t <= SK_Scalar1);
382
383 if (t == 0) {
384 return src[0];
385 }
386
387 #ifdef DIRECT_EVAL_OF_POLYNOMIALS
388 SkScalar D = src[0];
389 SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
390 SkScalar B = 3*(src[4] - src[2] - src[2] + D);
391 SkScalar C = 3*(src[2] - D);
392
393 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
394 #else
395 SkScalar ab = SkScalarInterp(src[0], src[2], t);
396 SkScalar bc = SkScalarInterp(src[2], src[4], t);
397 SkScalar cd = SkScalarInterp(src[4], src[6], t);
398 SkScalar abc = SkScalarInterp(ab, bc, t);
399 SkScalar bcd = SkScalarInterp(bc, cd, t);
400 return SkScalarInterp(abc, bcd, t);
401 #endif
402 }
403
404 /** return At^2 + Bt + C
405 */
406 static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
407 SkASSERT(t >= 0 && t <= SK_Scalar1);
408
409 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
410 }
411
412 static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
413 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
414 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
415 SkScalar C = src[2] - src[0];
416
417 return eval_quadratic(A, B, C, t);
418 }
419
420 static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
421 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
422 SkScalar B = src[4] - 2 * src[2] + src[0];
423
424 return SkScalarMulAdd(A, t, B);
425 }
426
427 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
428 SkVector* tangent, SkVector* curvature) {
429 SkASSERT(src);
430 SkASSERT(t >= 0 && t <= SK_Scalar1);
431
432 if (loc) {
433 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
434 }
435 if (tangent) {
436 tangent->set(eval_cubic_derivative(&src[0].fX, t),
437 eval_cubic_derivative(&src[0].fY, t));
438 }
439 if (curvature) {
440 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
441 eval_cubic_2ndDerivative(&src[0].fY, t));
442 }
443 }
444
445 /** Cubic'(t) = At^2 + Bt + C, where
446 A = 3(-a + 3(b - c) + d)
447 B = 6(a - 2b + c)
448 C = 3(b - a)
449 Solve for t, keeping only those that fit betwee 0 < t < 1
450 */
451 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
452 SkScalar tValues[2]) {
453 // we divide A,B,C by 3 to simplify
454 SkScalar A = d - a + 3*(b - c);
455 SkScalar B = 2*(a - b - b + c);
456 SkScalar C = b - a;
457
458 return SkFindUnitQuadRoots(A, B, C, tValues);
459 }
460
461 static void interp_cubic_coords(const SkScalar* src, SkScalar* dst,
462 SkScalar t) {
463 SkScalar ab = SkScalarInterp(src[0], src[2], t);
464 SkScalar bc = SkScalarInterp(src[2], src[4], t);
465 SkScalar cd = SkScalarInterp(src[4], src[6], t);
466 SkScalar abc = SkScalarInterp(ab, bc, t);
467 SkScalar bcd = SkScalarInterp(bc, cd, t);
468 SkScalar abcd = SkScalarInterp(abc, bcd, t);
469
470 dst[0] = src[0];
471 dst[2] = ab;
472 dst[4] = abc;
473 dst[6] = abcd;
474 dst[8] = bcd;
475 dst[10] = cd;
476 dst[12] = src[6];
477 }
478
479 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
480 SkASSERT(t > 0 && t < SK_Scalar1);
481
482 interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
483 interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
484 }
485
486 /* http://code.google.com/p/skia/issues/detail?id=32
487
488 This test code would fail when we didn't check the return result of
489 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
490 that after the first chop, the parameters to valid_unit_divide are equal
491 (thanks to finite float precision and rounding in the subtracts). Thus
492 even though the 2nd tValue looks < 1.0, after we renormalize it, we end
493 up with 1.0, hence the need to check and just return the last cubic as
494 a degenerate clump of 4 points in the sampe place.
495
496 static void test_cubic() {
497 SkPoint src[4] = {
498 { 556.25000, 523.03003 },
499 { 556.23999, 522.96002 },
500 { 556.21997, 522.89001 },
501 { 556.21997, 522.82001 }
502 };
503 SkPoint dst[10];
504 SkScalar tval[] = { 0.33333334f, 0.99999994f };
505 SkChopCubicAt(src, dst, tval, 2);
506 }
507 */
508
509 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
510 const SkScalar tValues[], int roots) {
511 #ifdef SK_DEBUG
512 {
513 for (int i = 0; i < roots - 1; i++)
514 {
515 SkASSERT(is_unit_interval(tValues[i]));
516 SkASSERT(is_unit_interval(tValues[i+1]));
517 SkASSERT(tValues[i] < tValues[i+1]);
518 }
519 }
520 #endif
521
522 if (dst) {
523 if (roots == 0) { // nothing to chop
524 memcpy(dst, src, 4*sizeof(SkPoint));
525 } else {
526 SkScalar t = tValues[0];
527 SkPoint tmp[4];
528
529 for (int i = 0; i < roots; i++) {
530 SkChopCubicAt(src, dst, t);
531 if (i == roots - 1) {
532 break;
533 }
534
535 dst += 3;
536 // have src point to the remaining cubic (after the chop)
537 memcpy(tmp, dst, 4 * sizeof(SkPoint));
538 src = tmp;
539
540 // watch out in case the renormalized t isn't in range
541 if (!valid_unit_divide(tValues[i+1] - tValues[i],
542 SK_Scalar1 - tValues[i], &t)) {
543 // if we can't, just create a degenerate cubic
544 dst[4] = dst[5] = dst[6] = src[3];
545 break;
546 }
547 }
548 }
549 }
550 }
551
552 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
553 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
554 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
555 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
556 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
557 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
558 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
559
560 SkScalar x012 = SkScalarAve(x01, x12);
561 SkScalar y012 = SkScalarAve(y01, y12);
562 SkScalar x123 = SkScalarAve(x12, x23);
563 SkScalar y123 = SkScalarAve(y12, y23);
564
565 dst[0] = src[0];
566 dst[1].set(x01, y01);
567 dst[2].set(x012, y012);
568 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
569 dst[4].set(x123, y123);
570 dst[5].set(x23, y23);
571 dst[6] = src[3];
572 }
573
574 static void flatten_double_cubic_extrema(SkScalar coords[14]) {
575 coords[4] = coords[8] = coords[6];
576 }
577
578 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
579 the resulting beziers are monotonic in Y. This is called by the scan
580 converter. Depending on what is returned, dst[] is treated as follows:
581 0 dst[0..3] is the original cubic
582 1 dst[0..3] and dst[3..6] are the two new cubics
583 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
584 If dst == null, it is ignored and only the count is returned.
585 */
586 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
587 SkScalar tValues[2];
588 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
589 src[3].fY, tValues);
590
591 SkChopCubicAt(src, dst, tValues, roots);
592 if (dst && roots > 0) {
593 // we do some cleanup to ensure our Y extrema are flat
594 flatten_double_cubic_extrema(&dst[0].fY);
595 if (roots == 2) {
596 flatten_double_cubic_extrema(&dst[3].fY);
597 }
598 }
599 return roots;
600 }
601
602 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
603 SkScalar tValues[2];
604 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
605 src[3].fX, tValues);
606
607 SkChopCubicAt(src, dst, tValues, roots);
608 if (dst && roots > 0) {
609 // we do some cleanup to ensure our Y extrema are flat
610 flatten_double_cubic_extrema(&dst[0].fX);
611 if (roots == 2) {
612 flatten_double_cubic_extrema(&dst[3].fX);
613 }
614 }
615 return roots;
616 }
617
618 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
619
620 Inflection means that curvature is zero.
621 Curvature is [F' x F''] / [F'^3]
622 So we solve F'x X F''y - F'y X F''y == 0
623 After some canceling of the cubic term, we get
624 A = b - a
625 B = c - 2b + a
626 C = d - 3c + 3b - a
627 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
628 */
629 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
630 SkScalar Ax = src[1].fX - src[0].fX;
631 SkScalar Ay = src[1].fY - src[0].fY;
632 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
633 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY;
634 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
635 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
636
637 return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
638 Ax*Cy - Ay*Cx,
639 Ax*By - Ay*Bx,
640 tValues);
641 }
642
643 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
644 SkScalar tValues[2];
645 int count = SkFindCubicInflections(src, tValues);
646
647 if (dst) {
648 if (count == 0) {
649 memcpy(dst, src, 4 * sizeof(SkPoint));
650 } else {
651 SkChopCubicAt(src, dst, tValues, count);
652 }
653 }
654 return count + 1;
655 }
656
657 template <typename T> void bubble_sort(T array[], int count) {
658 for (int i = count - 1; i > 0; --i)
659 for (int j = i; j > 0; --j)
660 if (array[j] < array[j-1])
661 {
662 T tmp(array[j]);
663 array[j] = array[j-1];
664 array[j-1] = tmp;
665 }
666 }
667
668 /**
669 * Given an array and count, remove all pair-wise duplicates from the array,
670 * keeping the existing sorting, and return the new count
671 */
672 static int collaps_duplicates(SkScalar array[], int count) {
673 for (int n = count; n > 1; --n) {
674 if (array[0] == array[1]) {
675 for (int i = 1; i < n; ++i) {
676 array[i - 1] = array[i];
677 }
678 count -= 1;
679 } else {
680 array += 1;
681 }
682 }
683 return count;
684 }
685
686 #ifdef SK_DEBUG
687
688 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array)
689
690 static void test_collaps_duplicates() {
691 static bool gOnce;
692 if (gOnce) { return; }
693 gOnce = true;
694 const SkScalar src0[] = { 0 };
695 const SkScalar src1[] = { 0, 0 };
696 const SkScalar src2[] = { 0, 1 };
697 const SkScalar src3[] = { 0, 0, 0 };
698 const SkScalar src4[] = { 0, 0, 1 };
699 const SkScalar src5[] = { 0, 1, 1 };
700 const SkScalar src6[] = { 0, 1, 2 };
701 const struct {
702 const SkScalar* fData;
703 int fCount;
704 int fCollapsedCount;
705 } data[] = {
706 { TEST_COLLAPS_ENTRY(src0), 1 },
707 { TEST_COLLAPS_ENTRY(src1), 1 },
708 { TEST_COLLAPS_ENTRY(src2), 2 },
709 { TEST_COLLAPS_ENTRY(src3), 1 },
710 { TEST_COLLAPS_ENTRY(src4), 2 },
711 { TEST_COLLAPS_ENTRY(src5), 2 },
712 { TEST_COLLAPS_ENTRY(src6), 3 },
713 };
714 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
715 SkScalar dst[3];
716 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
717 int count = collaps_duplicates(dst, data[i].fCount);
718 SkASSERT(data[i].fCollapsedCount == count);
719 for (int j = 1; j < count; ++j) {
720 SkASSERT(dst[j-1] < dst[j]);
721 }
722 }
723 }
724 #endif
725
726 static SkScalar SkScalarCubeRoot(SkScalar x) {
727 return SkScalarPow(x, 0.3333333f);
728 }
729
730 /* Solve coeff(t) == 0, returning the number of roots that
731 lie withing 0 < t < 1.
732 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
733
734 Eliminates repeated roots (so that all tValues are distinct, and are always
735 in increasing order.
736 */
737 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
738 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic
739 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
740 }
741
742 SkScalar a, b, c, Q, R;
743
744 {
745 SkASSERT(coeff[0] != 0);
746
747 SkScalar inva = SkScalarInvert(coeff[0]);
748 a = coeff[1] * inva;
749 b = coeff[2] * inva;
750 c = coeff[3] * inva;
751 }
752 Q = (a*a - b*3) / 9;
753 R = (2*a*a*a - 9*a*b + 27*c) / 54;
754
755 SkScalar Q3 = Q * Q * Q;
756 SkScalar R2MinusQ3 = R * R - Q3;
757 SkScalar adiv3 = a / 3;
758
759 SkScalar* roots = tValues;
760 SkScalar r;
761
762 if (R2MinusQ3 < 0) { // we have 3 real roots
763 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
764 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
765
766 r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
767 if (is_unit_interval(r)) {
768 *roots++ = r;
769 }
770 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
771 if (is_unit_interval(r)) {
772 *roots++ = r;
773 }
774 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
775 if (is_unit_interval(r)) {
776 *roots++ = r;
777 }
778 SkDEBUGCODE(test_collaps_duplicates();)
779
780 // now sort the roots
781 int count = (int)(roots - tValues);
782 SkASSERT((unsigned)count <= 3);
783 bubble_sort(tValues, count);
784 count = collaps_duplicates(tValues, count);
785 roots = tValues + count; // so we compute the proper count below
786 } else { // we have 1 real root
787 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
788 A = SkScalarCubeRoot(A);
789 if (R > 0) {
790 A = -A;
791 }
792 if (A != 0) {
793 A += Q / A;
794 }
795 r = A - adiv3;
796 if (is_unit_interval(r)) {
797 *roots++ = r;
798 }
799 }
800
801 return (int)(roots - tValues);
802 }
803
804 /* Looking for F' dot F'' == 0
805
806 A = b - a
807 B = c - 2b + a
808 C = d - 3c + 3b - a
809
810 F' = 3Ct^2 + 6Bt + 3A
811 F'' = 6Ct + 6B
812
813 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
814 */
815 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
816 SkScalar a = src[2] - src[0];
817 SkScalar b = src[4] - 2 * src[2] + src[0];
818 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0];
819
820 coeff[0] = c * c;
821 coeff[1] = 3 * b * c;
822 coeff[2] = 2 * b * b + c * a;
823 coeff[3] = a * b;
824 }
825
826 /* Looking for F' dot F'' == 0
827
828 A = b - a
829 B = c - 2b + a
830 C = d - 3c + 3b - a
831
832 F' = 3Ct^2 + 6Bt + 3A
833 F'' = 6Ct + 6B
834
835 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
836 */
837 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
838 SkScalar coeffX[4], coeffY[4];
839 int i;
840
841 formulate_F1DotF2(&src[0].fX, coeffX);
842 formulate_F1DotF2(&src[0].fY, coeffY);
843
844 for (i = 0; i < 4; i++) {
845 coeffX[i] += coeffY[i];
846 }
847
848 SkScalar t[3];
849 int count = solve_cubic_poly(coeffX, t);
850 int maxCount = 0;
851
852 // now remove extrema where the curvature is zero (mins)
853 // !!!! need a test for this !!!!
854 for (i = 0; i < count; i++) {
855 // if (not_min_curvature())
856 if (t[i] > 0 && t[i] < SK_Scalar1) {
857 tValues[maxCount++] = t[i];
858 }
859 }
860 return maxCount;
861 }
862
863 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
864 SkScalar tValues[3]) {
865 SkScalar t_storage[3];
866
867 if (tValues == NULL) {
868 tValues = t_storage;
869 }
870
871 int count = SkFindCubicMaxCurvature(src, tValues);
872
873 if (dst) {
874 if (count == 0) {
875 memcpy(dst, src, 4 * sizeof(SkPoint));
876 } else {
877 SkChopCubicAt(src, dst, tValues, count);
878 }
879 }
880 return count + 1;
881 }
882
883 bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4],
884 bool* ambiguous) {
885 if (ambiguous) {
886 *ambiguous = false;
887 }
888
889 // Find the minimum and maximum y of the extrema, which are the
890 // first and last points since this cubic is monotonic
891 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
892 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
893
894 if (pt.fY == cubic[0].fY
895 || pt.fY < min_y
896 || pt.fY > max_y) {
897 // The query line definitely does not cross the curve
898 if (ambiguous) {
899 *ambiguous = (pt.fY == cubic[0].fY);
900 }
901 return false;
902 }
903
904 bool pt_at_extremum = (pt.fY == cubic[3].fY);
905
906 SkScalar min_x =
907 SkMinScalar(
908 SkMinScalar(
909 SkMinScalar(cubic[0].fX, cubic[1].fX),
910 cubic[2].fX),
911 cubic[3].fX);
912 if (pt.fX < min_x) {
913 // The query line definitely crosses the curve
914 if (ambiguous) {
915 *ambiguous = pt_at_extremum;
916 }
917 return true;
918 }
919
920 SkScalar max_x =
921 SkMaxScalar(
922 SkMaxScalar(
923 SkMaxScalar(cubic[0].fX, cubic[1].fX),
924 cubic[2].fX),
925 cubic[3].fX);
926 if (pt.fX > max_x) {
927 // The query line definitely does not cross the curve
928 return false;
929 }
930
931 // Do a binary search to find the parameter value which makes y as
932 // close as possible to the query point. See whether the query
933 // line's origin is to the left of the associated x coordinate.
934
935 // kMaxIter is chosen as the number of mantissa bits for a float,
936 // since there's no way we are going to get more precision by
937 // iterating more times than that.
938 const int kMaxIter = 23;
939 SkPoint eval;
940 int iter = 0;
941 SkScalar upper_t;
942 SkScalar lower_t;
943 // Need to invert direction of t parameter if cubic goes up
944 // instead of down
945 if (cubic[3].fY > cubic[0].fY) {
946 upper_t = SK_Scalar1;
947 lower_t = 0;
948 } else {
949 upper_t = 0;
950 lower_t = SK_Scalar1;
951 }
952 do {
953 SkScalar t = SkScalarAve(upper_t, lower_t);
954 SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
955 if (pt.fY > eval.fY) {
956 lower_t = t;
957 } else {
958 upper_t = t;
959 }
960 } while (++iter < kMaxIter
961 && !SkScalarNearlyZero(eval.fY - pt.fY));
962 if (pt.fX <= eval.fX) {
963 if (ambiguous) {
964 *ambiguous = pt_at_extremum;
965 }
966 return true;
967 }
968 return false;
969 }
970
971 int SkNumXRayCrossingsForCubic(const SkXRay& pt,
972 const SkPoint cubic[4],
973 bool* ambiguous) {
974 int num_crossings = 0;
975 SkPoint monotonic_cubics[10];
976 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
977 if (ambiguous) {
978 *ambiguous = false;
979 }
980 bool locally_ambiguous;
981 if (SkXRayCrossesMonotonicCubic(pt,
982 &monotonic_cubics[0],
983 &locally_ambiguous))
984 ++num_crossings;
985 if (ambiguous) {
986 *ambiguous |= locally_ambiguous;
987 }
988 if (num_monotonic_cubics > 0)
989 if (SkXRayCrossesMonotonicCubic(pt,
990 &monotonic_cubics[3],
991 &locally_ambiguous))
992 ++num_crossings;
993 if (ambiguous) {
994 *ambiguous |= locally_ambiguous;
995 }
996 if (num_monotonic_cubics > 1)
997 if (SkXRayCrossesMonotonicCubic(pt,
998 &monotonic_cubics[6],
999 &locally_ambiguous))
1000 ++num_crossings;
1001 if (ambiguous) {
1002 *ambiguous |= locally_ambiguous;
1003 }
1004 return num_crossings;
1005 }
1006
1007 ///////////////////////////////////////////////////////////////////////////////
1008
1009 /* Find t value for quadratic [a, b, c] = d.
1010 Return 0 if there is no solution within [0, 1)
1011 */
1012 static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
1013 // At^2 + Bt + C = d
1014 SkScalar A = a - 2 * b + c;
1015 SkScalar B = 2 * (b - a);
1016 SkScalar C = a - d;
1017
1018 SkScalar roots[2];
1019 int count = SkFindUnitQuadRoots(A, B, C, roots);
1020
1021 SkASSERT(count <= 1);
1022 return count == 1 ? roots[0] : 0;
1023 }
1024
1025 /* given a quad-curve and a point (x,y), chop the quad at that point and place
1026 the new off-curve point and endpoint into 'dest'.
1027 Should only return false if the computed pos is the start of the curve
1028 (i.e. root == 0)
1029 */
1030 static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
1031 SkPoint* dest) {
1032 const SkScalar* base;
1033 SkScalar value;
1034
1035 if (SkScalarAbs(x) < SkScalarAbs(y)) {
1036 base = &quad[0].fX;
1037 value = x;
1038 } else {
1039 base = &quad[0].fY;
1040 value = y;
1041 }
1042
1043 // note: this returns 0 if it thinks value is out of range, meaning the
1044 // root might return something outside of [0, 1)
1045 SkScalar t = quad_solve(base[0], base[2], base[4], value);
1046
1047 if (t > 0) {
1048 SkPoint tmp[5];
1049 SkChopQuadAt(quad, tmp, t);
1050 dest[0] = tmp[1];
1051 dest[1].set(x, y);
1052 return true;
1053 } else {
1054 /* t == 0 means either the value triggered a root outside of [0, 1)
1055 For our purposes, we can ignore the <= 0 roots, but we want to
1056 catch the >= 1 roots (which given our caller, will basically mean
1057 a root of 1, give-or-take numerical instability). If we are in the
1058 >= 1 case, return the existing offCurve point.
1059
1060 The test below checks to see if we are close to the "end" of the
1061 curve (near base[4]). Rather than specifying a tolerance, I just
1062 check to see if value is on to the right/left of the middle point
1063 (depending on the direction/sign of the end points).
1064 */
1065 if ((base[0] < base[4] && value > base[2]) ||
1066 (base[0] > base[4] && value < base[2])) // should root have been 1
1067 {
1068 dest[0] = quad[1];
1069 dest[1].set(x, y);
1070 return true;
1071 }
1072 }
1073 return false;
1074 }
1075
1076 static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
1077 // The mid point of the quadratic arc approximation is half way between the two
1078 // control points. The float epsilon adjustment moves the on curve point out by
1079 // two bits, distributing the convex test error between the round rect
1080 // approximation and the convex cross product sign equality test.
1081 #define SK_MID_RRECT_OFFSET \
1082 (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
1083 { SK_Scalar1, 0 },
1084 { SK_Scalar1, SK_ScalarTanPIOver8 },
1085 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
1086 { SK_ScalarTanPIOver8, SK_Scalar1 },
1087
1088 { 0, SK_Scalar1 },
1089 { -SK_ScalarTanPIOver8, SK_Scalar1 },
1090 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
1091 { -SK_Scalar1, SK_ScalarTanPIOver8 },
1092
1093 { -SK_Scalar1, 0 },
1094 { -SK_Scalar1, -SK_ScalarTanPIOver8 },
1095 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
1096 { -SK_ScalarTanPIOver8, -SK_Scalar1 },
1097
1098 { 0, -SK_Scalar1 },
1099 { SK_ScalarTanPIOver8, -SK_Scalar1 },
1100 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
1101 { SK_Scalar1, -SK_ScalarTanPIOver8 },
1102
1103 { SK_Scalar1, 0 }
1104 #undef SK_MID_RRECT_OFFSET
1105 };
1106
1107 int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
1108 SkRotationDirection dir, const SkMatrix* userMatrix,
1109 SkPoint quadPoints[]) {
1110 // rotate by x,y so that uStart is (1.0)
1111 SkScalar x = SkPoint::DotProduct(uStart, uStop);
1112 SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1113
1114 SkScalar absX = SkScalarAbs(x);
1115 SkScalar absY = SkScalarAbs(y);
1116
1117 int pointCount;
1118
1119 // check for (effectively) coincident vectors
1120 // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1121 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1122 if (absY <= SK_ScalarNearlyZero && x > 0 &&
1123 ((y >= 0 && kCW_SkRotationDirection == dir) ||
1124 (y <= 0 && kCCW_SkRotationDirection == dir))) {
1125
1126 // just return the start-point
1127 quadPoints[0].set(SK_Scalar1, 0);
1128 pointCount = 1;
1129 } else {
1130 if (dir == kCCW_SkRotationDirection) {
1131 y = -y;
1132 }
1133 // what octant (quadratic curve) is [xy] in?
1134 int oct = 0;
1135 bool sameSign = true;
1136
1137 if (0 == y) {
1138 oct = 4; // 180
1139 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1140 } else if (0 == x) {
1141 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1142 oct = y > 0 ? 2 : 6; // 90 : 270
1143 } else {
1144 if (y < 0) {
1145 oct += 4;
1146 }
1147 if ((x < 0) != (y < 0)) {
1148 oct += 2;
1149 sameSign = false;
1150 }
1151 if ((absX < absY) == sameSign) {
1152 oct += 1;
1153 }
1154 }
1155
1156 int wholeCount = oct << 1;
1157 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
1158
1159 const SkPoint* arc = &gQuadCirclePts[wholeCount];
1160 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
1161 wholeCount += 2;
1162 }
1163 pointCount = wholeCount + 1;
1164 }
1165
1166 // now handle counter-clockwise and the initial unitStart rotation
1167 SkMatrix matrix;
1168 matrix.setSinCos(uStart.fY, uStart.fX);
1169 if (dir == kCCW_SkRotationDirection) {
1170 matrix.preScale(SK_Scalar1, -SK_Scalar1);
1171 }
1172 if (userMatrix) {
1173 matrix.postConcat(*userMatrix);
1174 }
1175 matrix.mapPoints(quadPoints, pointCount);
1176 return pointCount;
1177 }
1178
1179
1180 ///////////////////////////////////////////////////////////////////////////////
1181 //
1182 // NURB representation for conics. Helpful explanations at:
1183 //
1184 // http://citeseerx.ist.psu.edu/viewdoc/
1185 // download?doi=10.1.1.44.5740&rep=rep1&type=ps
1186 // and
1187 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
1188 //
1189 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
1190 // ------------------------------------------
1191 // ((1 - t)^2 + t^2 + 2 (1 - t) t w)
1192 //
1193 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
1194 // ------------------------------------------------
1195 // {t^2 (2 - 2 w), t (-2 + 2 w), 1}
1196 //
1197
1198 static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
1199 SkASSERT(src);
1200 SkASSERT(t >= 0 && t <= SK_Scalar1);
1201
1202 SkScalar src2w = SkScalarMul(src[2], w);
1203 SkScalar C = src[0];
1204 SkScalar A = src[4] - 2 * src2w + C;
1205 SkScalar B = 2 * (src2w - C);
1206 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1207
1208 B = 2 * (w - SK_Scalar1);
1209 C = SK_Scalar1;
1210 A = -B;
1211 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
1212
1213 return SkScalarDiv(numer, denom);
1214 }
1215
1216 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
1217 //
1218 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
1219 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
1220 // t^0 : -2 P0 w + 2 P1 w
1221 //
1222 // We disregard magnitude, so we can freely ignore the denominator of F', and
1223 // divide the numerator by 2
1224 //
1225 // coeff[0] for t^2
1226 // coeff[1] for t^1
1227 // coeff[2] for t^0
1228 //
1229 static void conic_deriv_coeff(const SkScalar src[],
1230 SkScalar w,
1231 SkScalar coeff[3]) {
1232 const SkScalar P20 = src[4] - src[0];
1233 const SkScalar P10 = src[2] - src[0];
1234 const SkScalar wP10 = w * P10;
1235 coeff[0] = w * P20 - P20;
1236 coeff[1] = P20 - 2 * wP10;
1237 coeff[2] = wP10;
1238 }
1239
1240 static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
1241 SkScalar coeff[3];
1242 conic_deriv_coeff(coord, w, coeff);
1243 return t * (t * coeff[0] + coeff[1]) + coeff[2];
1244 }
1245
1246 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
1247 SkScalar coeff[3];
1248 conic_deriv_coeff(src, w, coeff);
1249
1250 SkScalar tValues[2];
1251 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
1252 SkASSERT(0 == roots || 1 == roots);
1253
1254 if (1 == roots) {
1255 *t = tValues[0];
1256 return true;
1257 }
1258 return false;
1259 }
1260
1261 struct SkP3D {
1262 SkScalar fX, fY, fZ;
1263
1264 void set(SkScalar x, SkScalar y, SkScalar z) {
1265 fX = x; fY = y; fZ = z;
1266 }
1267
1268 void projectDown(SkPoint* dst) const {
1269 dst->set(fX / fZ, fY / fZ);
1270 }
1271 };
1272
1273 // We only interpolate one dimension at a time (the first, at +0, +3, +6).
1274 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
1275 SkScalar ab = SkScalarInterp(src[0], src[3], t);
1276 SkScalar bc = SkScalarInterp(src[3], src[6], t);
1277 dst[0] = ab;
1278 dst[3] = SkScalarInterp(ab, bc, t);
1279 dst[6] = bc;
1280 }
1281
1282 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
1283 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
1284 dst[1].set(src[1].fX * w, src[1].fY * w, w);
1285 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
1286 }
1287
1288 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1289 SkASSERT(t >= 0 && t <= SK_Scalar1);
1290
1291 if (pt) {
1292 pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
1293 conic_eval_pos(&fPts[0].fY, fW, t));
1294 }
1295 if (tangent) {
1296 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
1297 conic_eval_tan(&fPts[0].fY, fW, t));
1298 }
1299 }
1300
1301 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
1302 SkP3D tmp[3], tmp2[3];
1303
1304 ratquad_mapTo3D(fPts, fW, tmp);
1305
1306 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
1307 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
1308 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
1309
1310 dst[0].fPts[0] = fPts[0];
1311 tmp2[0].projectDown(&dst[0].fPts[1]);
1312 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
1313 tmp2[2].projectDown(&dst[1].fPts[1]);
1314 dst[1].fPts[2] = fPts[2];
1315
1316 // to put in "standard form", where w0 and w2 are both 1, we compute the
1317 // new w1 as sqrt(w1*w1/w0*w2)
1318 // or
1319 // w1 /= sqrt(w0*w2)
1320 //
1321 // However, in our case, we know that for dst[0]:
1322 // w0 == 1, and for dst[1], w2 == 1
1323 //
1324 SkScalar root = SkScalarSqrt(tmp2[1].fZ);
1325 dst[0].fW = tmp2[0].fZ / root;
1326 dst[1].fW = tmp2[2].fZ / root;
1327 }
1328
1329 static SkScalar subdivide_w_value(SkScalar w) {
1330 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1331 }
1332
1333 void SkConic::chop(SkConic dst[2]) const {
1334 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
1335 SkScalar p1x = fW * fPts[1].fX;
1336 SkScalar p1y = fW * fPts[1].fY;
1337 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
1338 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
1339
1340 dst[0].fPts[0] = fPts[0];
1341 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
1342 (fPts[0].fY + p1y) * scale);
1343 dst[0].fPts[2].set(mx, my);
1344
1345 dst[1].fPts[0].set(mx, my);
1346 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
1347 (p1y + fPts[2].fY) * scale);
1348 dst[1].fPts[2] = fPts[2];
1349
1350 dst[0].fW = dst[1].fW = subdivide_w_value(fW);
1351 }
1352
1353 /*
1354 * "High order approximation of conic sections by quadratic splines"
1355 * by Michael Floater, 1993
1356 */
1357 #define AS_QUAD_ERROR_SETUP \
1358 SkScalar a = fW - 1; \
1359 SkScalar k = a / (4 * (2 + a)); \
1360 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \
1361 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1362
1363 void SkConic::computeAsQuadError(SkVector* err) const {
1364 AS_QUAD_ERROR_SETUP
1365 err->set(x, y);
1366 }
1367
1368 bool SkConic::asQuadTol(SkScalar tol) const {
1369 AS_QUAD_ERROR_SETUP
1370 return (x * x + y * y) <= tol * tol;
1371 }
1372
1373 int SkConic::computeQuadPOW2(SkScalar tol) const {
1374 AS_QUAD_ERROR_SETUP
1375 SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
1376
1377 if (error <= 0) {
1378 return 0;
1379 }
1380 uint32_t ierr = (uint32_t)error;
1381 return (34 - SkCLZ(ierr)) >> 1;
1382 }
1383
1384 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1385 SkASSERT(level >= 0);
1386
1387 if (0 == level) {
1388 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1389 return pts + 2;
1390 } else {
1391 SkConic dst[2];
1392 src.chop(dst);
1393 --level;
1394 pts = subdivide(dst[0], pts, level);
1395 return subdivide(dst[1], pts, level);
1396 }
1397 }
1398
1399 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1400 SkASSERT(pow2 >= 0);
1401 *pts = fPts[0];
1402 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
1403 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
1404 return 1 << pow2;
1405 }
1406
1407 bool SkConic::findXExtrema(SkScalar* t) const {
1408 return conic_find_extrema(&fPts[0].fX, fW, t);
1409 }
1410
1411 bool SkConic::findYExtrema(SkScalar* t) const {
1412 return conic_find_extrema(&fPts[0].fY, fW, t);
1413 }
1414
1415 bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1416 SkScalar t;
1417 if (this->findXExtrema(&t)) {
1418 this->chopAt(t, dst);
1419 // now clean-up the middle, since we know t was meant to be at
1420 // an X-extrema
1421 SkScalar value = dst[0].fPts[2].fX;
1422 dst[0].fPts[1].fX = value;
1423 dst[1].fPts[0].fX = value;
1424 dst[1].fPts[1].fX = value;
1425 return true;
1426 }
1427 return false;
1428 }
1429
1430 bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1431 SkScalar t;
1432 if (this->findYExtrema(&t)) {
1433 this->chopAt(t, dst);
1434 // now clean-up the middle, since we know t was meant to be at
1435 // an Y-extrema
1436 SkScalar value = dst[0].fPts[2].fY;
1437 dst[0].fPts[1].fY = value;
1438 dst[1].fPts[0].fY = value;
1439 dst[1].fPts[1].fY = value;
1440 return true;
1441 }
1442 return false;
1443 }
1444
1445 void SkConic::computeTightBounds(SkRect* bounds) const {
1446 SkPoint pts[4];
1447 pts[0] = fPts[0];
1448 pts[1] = fPts[2];
1449 int count = 2;
1450
1451 SkScalar t;
1452 if (this->findXExtrema(&t)) {
1453 this->evalAt(t, &pts[count++]);
1454 }
1455 if (this->findYExtrema(&t)) {
1456 this->evalAt(t, &pts[count++]);
1457 }
1458 bounds->set(pts, count);
1459 }
1460
1461 void SkConic::computeFastBounds(SkRect* bounds) const {
1462 bounds->set(fPts, 3);
1463 }
1464
1465 bool SkConic::findMaxCurvature(SkScalar* t) const {
1466 // TODO: Implement me
1467 return false;
1468 }

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