Line data Source code
1 : /*
2 : * Copyright 2006 The Android Open Source Project
3 : *
4 : * Use of this source code is governed by a BSD-style license that can be
5 : * found in the LICENSE file.
6 : */
7 :
8 : #include "SkGeometry.h"
9 : #include "SkMatrix.h"
10 : #include "SkNx.h"
11 :
12 48 : static SkVector to_vector(const Sk2s& x) {
13 : SkVector vector;
14 : x.store(&vector);
15 48 : return vector;
16 : }
17 :
18 : ////////////////////////////////////////////////////////////////////////
19 :
20 16 : static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
21 16 : SkScalar ab = a - b;
22 16 : SkScalar bc = b - c;
23 16 : if (ab < 0) {
24 4 : bc = -bc;
25 : }
26 16 : return ab == 0 || bc < 0;
27 : }
28 :
29 : ////////////////////////////////////////////////////////////////////////
30 :
31 0 : static bool is_unit_interval(SkScalar x) {
32 0 : return x > 0 && x < SK_Scalar1;
33 : }
34 :
35 3038 : static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
36 3038 : SkASSERT(ratio);
37 :
38 3038 : if (numer < 0) {
39 1309 : numer = -numer;
40 1309 : denom = -denom;
41 : }
42 :
43 3038 : if (denom == 0 || numer == 0 || numer >= denom) {
44 2772 : return 0;
45 : }
46 :
47 266 : SkScalar r = numer / denom;
48 266 : if (SkScalarIsNaN(r)) {
49 0 : return 0;
50 : }
51 266 : SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
52 266 : if (r == 0) { // catch underflow if numer <<<< denom
53 0 : return 0;
54 : }
55 266 : *ratio = r;
56 266 : return 1;
57 : }
58 :
59 : /** From Numerical Recipes in C.
60 :
61 : Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
62 : x1 = Q / A
63 : x2 = C / Q
64 : */
65 1636 : int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
66 1636 : SkASSERT(roots);
67 :
68 1636 : if (A == 0) {
69 32 : return valid_unit_divide(-C, B, roots);
70 : }
71 :
72 1604 : SkScalar* r = roots;
73 :
74 1604 : SkScalar R = B*B - 4*A*C;
75 1604 : if (R < 0 || !SkScalarIsFinite(R)) { // complex roots
76 : // if R is infinite, it's possible that it may still produce
77 : // useful results if the operation was repeated in doubles
78 : // the flipside is determining if the more precise answer
79 : // isn't useful because surrounding machinery (e.g., subtracting
80 : // the axis offset from C) already discards the extra precision
81 : // more investigation and unit tests required...
82 105 : return 0;
83 : }
84 1499 : R = SkScalarSqrt(R);
85 :
86 1499 : SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
87 1499 : r += valid_unit_divide(Q, A, r);
88 1499 : r += valid_unit_divide(C, Q, r);
89 1499 : if (r - roots == 2) {
90 40 : if (roots[0] > roots[1])
91 40 : SkTSwap<SkScalar>(roots[0], roots[1]);
92 0 : else if (roots[0] == roots[1]) // nearly-equal?
93 0 : r -= 1; // skip the double root
94 : }
95 1499 : return (int)(r - roots);
96 : }
97 :
98 : ///////////////////////////////////////////////////////////////////////////////
99 : ///////////////////////////////////////////////////////////////////////////////
100 :
101 0 : void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) {
102 0 : SkASSERT(src);
103 0 : SkASSERT(t >= 0 && t <= SK_Scalar1);
104 :
105 0 : if (pt) {
106 0 : *pt = SkEvalQuadAt(src, t);
107 : }
108 0 : if (tangent) {
109 0 : *tangent = SkEvalQuadTangentAt(src, t);
110 : }
111 0 : }
112 :
113 16 : SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) {
114 16 : return to_point(SkQuadCoeff(src).eval(t));
115 : }
116 :
117 0 : SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) {
118 : // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a
119 : // zero tangent vector when t is 0 or 1, and the control point is equal
120 : // to the end point. In this case, use the quad end points to compute the tangent.
121 0 : if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) {
122 0 : return src[2] - src[0];
123 : }
124 0 : SkASSERT(src);
125 0 : SkASSERT(t >= 0 && t <= SK_Scalar1);
126 :
127 0 : Sk2s P0 = from_point(src[0]);
128 0 : Sk2s P1 = from_point(src[1]);
129 0 : Sk2s P2 = from_point(src[2]);
130 :
131 0 : Sk2s B = P1 - P0;
132 0 : Sk2s A = P2 - P1 - B;
133 0 : Sk2s T = A * Sk2s(t) + B;
134 :
135 0 : return to_vector(T + T);
136 : }
137 :
138 144 : static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) {
139 432 : return v0 + (v1 - v0) * t;
140 : }
141 :
142 0 : void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
143 0 : SkASSERT(t > 0 && t < SK_Scalar1);
144 :
145 0 : Sk2s p0 = from_point(src[0]);
146 0 : Sk2s p1 = from_point(src[1]);
147 0 : Sk2s p2 = from_point(src[2]);
148 : Sk2s tt(t);
149 :
150 0 : Sk2s p01 = interp(p0, p1, tt);
151 0 : Sk2s p12 = interp(p1, p2, tt);
152 :
153 0 : dst[0] = to_point(p0);
154 0 : dst[1] = to_point(p01);
155 0 : dst[2] = to_point(interp(p01, p12, tt));
156 0 : dst[3] = to_point(p12);
157 0 : dst[4] = to_point(p2);
158 0 : }
159 :
160 0 : void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
161 0 : SkChopQuadAt(src, dst, 0.5f);
162 0 : }
163 :
164 : /** Quad'(t) = At + B, where
165 : A = 2(a - 2b + c)
166 : B = 2(b - a)
167 : Solve for t, only if it fits between 0 < t < 1
168 : */
169 0 : int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
170 : /* At + B == 0
171 : t = -B / A
172 : */
173 0 : return valid_unit_divide(a - b, a - b - b + c, tValue);
174 : }
175 :
176 0 : static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
177 0 : coords[2] = coords[6] = coords[4];
178 0 : }
179 :
180 : /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
181 : stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
182 : */
183 16 : int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
184 16 : SkASSERT(src);
185 16 : SkASSERT(dst);
186 :
187 16 : SkScalar a = src[0].fY;
188 16 : SkScalar b = src[1].fY;
189 16 : SkScalar c = src[2].fY;
190 :
191 16 : if (is_not_monotonic(a, b, c)) {
192 : SkScalar tValue;
193 8 : if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
194 0 : SkChopQuadAt(src, dst, tValue);
195 0 : flatten_double_quad_extrema(&dst[0].fY);
196 0 : return 1;
197 : }
198 : // if we get here, we need to force dst to be monotonic, even though
199 : // we couldn't compute a unit_divide value (probably underflow).
200 8 : b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
201 : }
202 16 : dst[0].set(src[0].fX, a);
203 16 : dst[1].set(src[1].fX, b);
204 16 : dst[2].set(src[2].fX, c);
205 16 : return 0;
206 : }
207 :
208 : /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
209 : stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
210 : */
211 0 : int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
212 0 : SkASSERT(src);
213 0 : SkASSERT(dst);
214 :
215 0 : SkScalar a = src[0].fX;
216 0 : SkScalar b = src[1].fX;
217 0 : SkScalar c = src[2].fX;
218 :
219 0 : if (is_not_monotonic(a, b, c)) {
220 : SkScalar tValue;
221 0 : if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
222 0 : SkChopQuadAt(src, dst, tValue);
223 0 : flatten_double_quad_extrema(&dst[0].fX);
224 0 : return 1;
225 : }
226 : // if we get here, we need to force dst to be monotonic, even though
227 : // we couldn't compute a unit_divide value (probably underflow).
228 0 : b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
229 : }
230 0 : dst[0].set(a, src[0].fY);
231 0 : dst[1].set(b, src[1].fY);
232 0 : dst[2].set(c, src[2].fY);
233 0 : return 0;
234 : }
235 :
236 : // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
237 : // F'(t) = 2 (b - a) + 2 (a - 2b + c) t
238 : // F''(t) = 2 (a - 2b + c)
239 : //
240 : // A = 2 (b - a)
241 : // B = 2 (a - 2b + c)
242 : //
243 : // Maximum curvature for a quadratic means solving
244 : // Fx' Fx'' + Fy' Fy'' = 0
245 : //
246 : // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
247 : //
248 0 : SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
249 0 : SkScalar Ax = src[1].fX - src[0].fX;
250 0 : SkScalar Ay = src[1].fY - src[0].fY;
251 0 : SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
252 0 : SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
253 0 : SkScalar t = 0; // 0 means don't chop
254 :
255 0 : (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
256 0 : return t;
257 : }
258 :
259 0 : int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
260 0 : SkScalar t = SkFindQuadMaxCurvature(src);
261 0 : if (t == 0) {
262 0 : memcpy(dst, src, 3 * sizeof(SkPoint));
263 0 : return 1;
264 : } else {
265 0 : SkChopQuadAt(src, dst, t);
266 0 : return 2;
267 : }
268 : }
269 :
270 0 : void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
271 : Sk2s scale(SkDoubleToScalar(2.0 / 3.0));
272 0 : Sk2s s0 = from_point(src[0]);
273 0 : Sk2s s1 = from_point(src[1]);
274 0 : Sk2s s2 = from_point(src[2]);
275 :
276 0 : dst[0] = src[0];
277 0 : dst[1] = to_point(s0 + (s1 - s0) * scale);
278 0 : dst[2] = to_point(s2 + (s1 - s2) * scale);
279 0 : dst[3] = src[2];
280 0 : }
281 :
282 : //////////////////////////////////////////////////////////////////////////////
283 : ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
284 : //////////////////////////////////////////////////////////////////////////////
285 :
286 48 : static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) {
287 48 : SkQuadCoeff coeff;
288 48 : Sk2s P0 = from_point(src[0]);
289 48 : Sk2s P1 = from_point(src[1]);
290 48 : Sk2s P2 = from_point(src[2]);
291 48 : Sk2s P3 = from_point(src[3]);
292 :
293 192 : coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0;
294 144 : coeff.fB = times_2(P2 - times_2(P1) + P0);
295 48 : coeff.fC = P1 - P0;
296 48 : return to_vector(coeff.eval(t));
297 : }
298 :
299 0 : static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) {
300 0 : Sk2s P0 = from_point(src[0]);
301 0 : Sk2s P1 = from_point(src[1]);
302 0 : Sk2s P2 = from_point(src[2]);
303 0 : Sk2s P3 = from_point(src[3]);
304 0 : Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0;
305 0 : Sk2s B = P2 - times_2(P1) + P0;
306 :
307 0 : return to_vector(A * Sk2s(t) + B);
308 : }
309 :
310 290 : void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
311 : SkVector* tangent, SkVector* curvature) {
312 290 : SkASSERT(src);
313 290 : SkASSERT(t >= 0 && t <= SK_Scalar1);
314 :
315 290 : if (loc) {
316 290 : *loc = to_point(SkCubicCoeff(src).eval(t));
317 : }
318 290 : if (tangent) {
319 : // The derivative equation returns a zero tangent vector when t is 0 or 1, and the
320 : // adjacent control point is equal to the end point. In this case, use the
321 : // next control point or the end points to compute the tangent.
322 48 : if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) {
323 0 : if (t == 0) {
324 0 : *tangent = src[2] - src[0];
325 : } else {
326 0 : *tangent = src[3] - src[1];
327 : }
328 0 : if (!tangent->fX && !tangent->fY) {
329 0 : *tangent = src[3] - src[0];
330 : }
331 : } else {
332 48 : *tangent = eval_cubic_derivative(src, t);
333 : }
334 : }
335 290 : if (curvature) {
336 0 : *curvature = eval_cubic_2ndDerivative(src, t);
337 : }
338 290 : }
339 :
340 : /** Cubic'(t) = At^2 + Bt + C, where
341 : A = 3(-a + 3(b - c) + d)
342 : B = 6(a - 2b + c)
343 : C = 3(b - a)
344 : Solve for t, keeping only those that fit betwee 0 < t < 1
345 : */
346 1628 : int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
347 : SkScalar tValues[2]) {
348 : // we divide A,B,C by 3 to simplify
349 1628 : SkScalar A = d - a + 3*(b - c);
350 1628 : SkScalar B = 2*(a - b - b + c);
351 1628 : SkScalar C = b - a;
352 :
353 1628 : return SkFindUnitQuadRoots(A, B, C, tValues);
354 : }
355 :
356 24 : void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
357 24 : SkASSERT(t > 0 && t < SK_Scalar1);
358 :
359 24 : Sk2s p0 = from_point(src[0]);
360 24 : Sk2s p1 = from_point(src[1]);
361 24 : Sk2s p2 = from_point(src[2]);
362 24 : Sk2s p3 = from_point(src[3]);
363 : Sk2s tt(t);
364 :
365 24 : Sk2s ab = interp(p0, p1, tt);
366 24 : Sk2s bc = interp(p1, p2, tt);
367 24 : Sk2s cd = interp(p2, p3, tt);
368 24 : Sk2s abc = interp(ab, bc, tt);
369 24 : Sk2s bcd = interp(bc, cd, tt);
370 24 : Sk2s abcd = interp(abc, bcd, tt);
371 :
372 24 : dst[0] = src[0];
373 24 : dst[1] = to_point(ab);
374 24 : dst[2] = to_point(abc);
375 24 : dst[3] = to_point(abcd);
376 24 : dst[4] = to_point(bcd);
377 24 : dst[5] = to_point(cd);
378 24 : dst[6] = src[3];
379 24 : }
380 :
381 : /* http://code.google.com/p/skia/issues/detail?id=32
382 :
383 : This test code would fail when we didn't check the return result of
384 : valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
385 : that after the first chop, the parameters to valid_unit_divide are equal
386 : (thanks to finite float precision and rounding in the subtracts). Thus
387 : even though the 2nd tValue looks < 1.0, after we renormalize it, we end
388 : up with 1.0, hence the need to check and just return the last cubic as
389 : a degenerate clump of 4 points in the sampe place.
390 :
391 : static void test_cubic() {
392 : SkPoint src[4] = {
393 : { 556.25000, 523.03003 },
394 : { 556.23999, 522.96002 },
395 : { 556.21997, 522.89001 },
396 : { 556.21997, 522.82001 }
397 : };
398 : SkPoint dst[10];
399 : SkScalar tval[] = { 0.33333334f, 0.99999994f };
400 : SkChopCubicAt(src, dst, tval, 2);
401 : }
402 : */
403 :
404 550 : void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
405 : const SkScalar tValues[], int roots) {
406 : #ifdef SK_DEBUG
407 : {
408 550 : for (int i = 0; i < roots - 1; i++)
409 : {
410 0 : SkASSERT(is_unit_interval(tValues[i]));
411 0 : SkASSERT(is_unit_interval(tValues[i+1]));
412 0 : SkASSERT(tValues[i] < tValues[i+1]);
413 : }
414 : }
415 : #endif
416 :
417 550 : if (dst) {
418 550 : if (roots == 0) { // nothing to chop
419 526 : memcpy(dst, src, 4*sizeof(SkPoint));
420 : } else {
421 24 : SkScalar t = tValues[0];
422 : SkPoint tmp[4];
423 :
424 24 : for (int i = 0; i < roots; i++) {
425 24 : SkChopCubicAt(src, dst, t);
426 24 : if (i == roots - 1) {
427 24 : break;
428 : }
429 :
430 0 : dst += 3;
431 : // have src point to the remaining cubic (after the chop)
432 0 : memcpy(tmp, dst, 4 * sizeof(SkPoint));
433 0 : src = tmp;
434 :
435 : // watch out in case the renormalized t isn't in range
436 0 : if (!valid_unit_divide(tValues[i+1] - tValues[i],
437 0 : SK_Scalar1 - tValues[i], &t)) {
438 : // if we can't, just create a degenerate cubic
439 0 : dst[4] = dst[5] = dst[6] = src[3];
440 0 : break;
441 : }
442 : }
443 : }
444 : }
445 550 : }
446 :
447 0 : void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
448 0 : SkChopCubicAt(src, dst, 0.5f);
449 0 : }
450 :
451 24 : static void flatten_double_cubic_extrema(SkScalar coords[14]) {
452 24 : coords[4] = coords[8] = coords[6];
453 24 : }
454 :
455 : /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
456 : the resulting beziers are monotonic in Y. This is called by the scan
457 : converter. Depending on what is returned, dst[] is treated as follows:
458 : 0 dst[0..3] is the original cubic
459 : 1 dst[0..3] and dst[3..6] are the two new cubics
460 : 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
461 : If dst == null, it is ignored and only the count is returned.
462 : */
463 413 : int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
464 : SkScalar tValues[2];
465 413 : int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
466 826 : src[3].fY, tValues);
467 :
468 413 : SkChopCubicAt(src, dst, tValues, roots);
469 413 : if (dst && roots > 0) {
470 : // we do some cleanup to ensure our Y extrema are flat
471 14 : flatten_double_cubic_extrema(&dst[0].fY);
472 14 : if (roots == 2) {
473 0 : flatten_double_cubic_extrema(&dst[3].fY);
474 : }
475 : }
476 413 : return roots;
477 : }
478 :
479 137 : int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
480 : SkScalar tValues[2];
481 137 : int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
482 274 : src[3].fX, tValues);
483 :
484 137 : SkChopCubicAt(src, dst, tValues, roots);
485 137 : if (dst && roots > 0) {
486 : // we do some cleanup to ensure our Y extrema are flat
487 10 : flatten_double_cubic_extrema(&dst[0].fX);
488 10 : if (roots == 2) {
489 0 : flatten_double_cubic_extrema(&dst[3].fX);
490 : }
491 : }
492 137 : return roots;
493 : }
494 :
495 : /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
496 :
497 : Inflection means that curvature is zero.
498 : Curvature is [F' x F''] / [F'^3]
499 : So we solve F'x X F''y - F'y X F''y == 0
500 : After some canceling of the cubic term, we get
501 : A = b - a
502 : B = c - 2b + a
503 : C = d - 3c + 3b - a
504 : (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
505 : */
506 8 : int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
507 8 : SkScalar Ax = src[1].fX - src[0].fX;
508 8 : SkScalar Ay = src[1].fY - src[0].fY;
509 8 : SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
510 8 : SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY;
511 8 : SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
512 8 : SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
513 :
514 16 : return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
515 8 : Ax*Cy - Ay*Cx,
516 8 : Ax*By - Ay*Bx,
517 8 : tValues);
518 : }
519 :
520 0 : int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
521 : SkScalar tValues[2];
522 0 : int count = SkFindCubicInflections(src, tValues);
523 :
524 0 : if (dst) {
525 0 : if (count == 0) {
526 0 : memcpy(dst, src, 4 * sizeof(SkPoint));
527 : } else {
528 0 : SkChopCubicAt(src, dst, tValues, count);
529 : }
530 : }
531 0 : return count + 1;
532 : }
533 :
534 : // See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
535 : // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
536 : // Classification:
537 : // discr(I) > 0 Serpentine
538 : // discr(I) = 0 Cusp
539 : // discr(I) < 0 Loop
540 : // d0 = d1 = 0 Quadratic
541 : // d0 = d1 = d2 = 0 Line
542 : // p0 = p1 = p2 = p3 Point
543 0 : static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
544 0 : if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
545 0 : return kPoint_SkCubicType;
546 : }
547 0 : const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
548 0 : if (discr > SK_ScalarNearlyZero) {
549 0 : return kSerpentine_SkCubicType;
550 0 : } else if (discr < -SK_ScalarNearlyZero) {
551 0 : return kLoop_SkCubicType;
552 : } else {
553 0 : if (SkScalarAbs(d[0]) < SK_ScalarNearlyZero && SkScalarAbs(d[1]) < SK_ScalarNearlyZero) {
554 0 : return ((SkScalarAbs(d[2]) < SK_ScalarNearlyZero) ? kLine_SkCubicType
555 0 : : kQuadratic_SkCubicType);
556 : } else {
557 0 : return kCusp_SkCubicType;
558 : }
559 : }
560 : }
561 :
562 : // Assumes the third component of points is 1.
563 : // Calcs p0 . (p1 x p2)
564 0 : static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
565 0 : const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
566 0 : const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
567 0 : const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
568 0 : return (xComp + yComp + wComp);
569 : }
570 :
571 : // Calc coefficients of I(s,t) where roots of I are inflection points of curve
572 : // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
573 : // d0 = a1 - 2*a2+3*a3
574 : // d1 = -a2 + 3*a3
575 : // d2 = 3*a3
576 : // a1 = p0 . (p3 x p2)
577 : // a2 = p1 . (p0 x p3)
578 : // a3 = p2 . (p1 x p0)
579 : // Places the values of d1, d2, d3 in array d passed in
580 0 : static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
581 0 : SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
582 0 : SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
583 0 : SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
584 :
585 : // need to scale a's or values in later calculations will grow to high
586 0 : SkScalar max = SkScalarAbs(a1);
587 0 : max = SkMaxScalar(max, SkScalarAbs(a2));
588 0 : max = SkMaxScalar(max, SkScalarAbs(a3));
589 0 : max = 1.f/max;
590 0 : a1 = a1 * max;
591 0 : a2 = a2 * max;
592 0 : a3 = a3 * max;
593 :
594 0 : d[2] = 3.f * a3;
595 0 : d[1] = d[2] - a2;
596 0 : d[0] = d[1] - a2 + a1;
597 0 : }
598 :
599 0 : SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
600 0 : calc_cubic_inflection_func(src, d);
601 0 : return classify_cubic(src, d);
602 : }
603 :
604 0 : template <typename T> void bubble_sort(T array[], int count) {
605 0 : for (int i = count - 1; i > 0; --i)
606 0 : for (int j = i; j > 0; --j)
607 0 : if (array[j] < array[j-1])
608 : {
609 0 : T tmp(array[j]);
610 0 : array[j] = array[j-1];
611 0 : array[j-1] = tmp;
612 : }
613 0 : }
614 :
615 : /**
616 : * Given an array and count, remove all pair-wise duplicates from the array,
617 : * keeping the existing sorting, and return the new count
618 : */
619 0 : static int collaps_duplicates(SkScalar array[], int count) {
620 0 : for (int n = count; n > 1; --n) {
621 0 : if (array[0] == array[1]) {
622 0 : for (int i = 1; i < n; ++i) {
623 0 : array[i - 1] = array[i];
624 : }
625 0 : count -= 1;
626 : } else {
627 0 : array += 1;
628 : }
629 : }
630 0 : return count;
631 : }
632 :
633 : #ifdef SK_DEBUG
634 :
635 : #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array)
636 :
637 0 : static void test_collaps_duplicates() {
638 : static bool gOnce;
639 0 : if (gOnce) { return; }
640 0 : gOnce = true;
641 0 : const SkScalar src0[] = { 0 };
642 0 : const SkScalar src1[] = { 0, 0 };
643 0 : const SkScalar src2[] = { 0, 1 };
644 0 : const SkScalar src3[] = { 0, 0, 0 };
645 0 : const SkScalar src4[] = { 0, 0, 1 };
646 0 : const SkScalar src5[] = { 0, 1, 1 };
647 0 : const SkScalar src6[] = { 0, 1, 2 };
648 : const struct {
649 : const SkScalar* fData;
650 : int fCount;
651 : int fCollapsedCount;
652 : } data[] = {
653 : { TEST_COLLAPS_ENTRY(src0), 1 },
654 : { TEST_COLLAPS_ENTRY(src1), 1 },
655 : { TEST_COLLAPS_ENTRY(src2), 2 },
656 : { TEST_COLLAPS_ENTRY(src3), 1 },
657 : { TEST_COLLAPS_ENTRY(src4), 2 },
658 : { TEST_COLLAPS_ENTRY(src5), 2 },
659 : { TEST_COLLAPS_ENTRY(src6), 3 },
660 0 : };
661 0 : for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
662 : SkScalar dst[3];
663 0 : memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
664 0 : int count = collaps_duplicates(dst, data[i].fCount);
665 0 : SkASSERT(data[i].fCollapsedCount == count);
666 0 : for (int j = 1; j < count; ++j) {
667 0 : SkASSERT(dst[j-1] < dst[j]);
668 : }
669 : }
670 : }
671 : #endif
672 :
673 0 : static SkScalar SkScalarCubeRoot(SkScalar x) {
674 0 : return SkScalarPow(x, 0.3333333f);
675 : }
676 :
677 : /* Solve coeff(t) == 0, returning the number of roots that
678 : lie withing 0 < t < 1.
679 : coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
680 :
681 : Eliminates repeated roots (so that all tValues are distinct, and are always
682 : in increasing order.
683 : */
684 0 : static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
685 0 : if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic
686 0 : return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
687 : }
688 :
689 : SkScalar a, b, c, Q, R;
690 :
691 : {
692 0 : SkASSERT(coeff[0] != 0);
693 :
694 0 : SkScalar inva = SkScalarInvert(coeff[0]);
695 0 : a = coeff[1] * inva;
696 0 : b = coeff[2] * inva;
697 0 : c = coeff[3] * inva;
698 : }
699 0 : Q = (a*a - b*3) / 9;
700 0 : R = (2*a*a*a - 9*a*b + 27*c) / 54;
701 :
702 0 : SkScalar Q3 = Q * Q * Q;
703 0 : SkScalar R2MinusQ3 = R * R - Q3;
704 0 : SkScalar adiv3 = a / 3;
705 :
706 0 : SkScalar* roots = tValues;
707 : SkScalar r;
708 :
709 0 : if (R2MinusQ3 < 0) { // we have 3 real roots
710 : // the divide/root can, due to finite precisions, be slightly outside of -1...1
711 0 : SkScalar theta = SkScalarACos(SkScalarPin(R / SkScalarSqrt(Q3), -1, 1));
712 0 : SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
713 :
714 0 : r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
715 0 : if (is_unit_interval(r)) {
716 0 : *roots++ = r;
717 : }
718 0 : r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
719 0 : if (is_unit_interval(r)) {
720 0 : *roots++ = r;
721 : }
722 0 : r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
723 0 : if (is_unit_interval(r)) {
724 0 : *roots++ = r;
725 : }
726 0 : SkDEBUGCODE(test_collaps_duplicates();)
727 :
728 : // now sort the roots
729 0 : int count = (int)(roots - tValues);
730 0 : SkASSERT((unsigned)count <= 3);
731 0 : bubble_sort(tValues, count);
732 0 : count = collaps_duplicates(tValues, count);
733 0 : roots = tValues + count; // so we compute the proper count below
734 : } else { // we have 1 real root
735 0 : SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
736 0 : A = SkScalarCubeRoot(A);
737 0 : if (R > 0) {
738 0 : A = -A;
739 : }
740 0 : if (A != 0) {
741 0 : A += Q / A;
742 : }
743 0 : r = A - adiv3;
744 0 : if (is_unit_interval(r)) {
745 0 : *roots++ = r;
746 : }
747 : }
748 :
749 0 : return (int)(roots - tValues);
750 : }
751 :
752 : /* Looking for F' dot F'' == 0
753 :
754 : A = b - a
755 : B = c - 2b + a
756 : C = d - 3c + 3b - a
757 :
758 : F' = 3Ct^2 + 6Bt + 3A
759 : F'' = 6Ct + 6B
760 :
761 : F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
762 : */
763 0 : static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
764 0 : SkScalar a = src[2] - src[0];
765 0 : SkScalar b = src[4] - 2 * src[2] + src[0];
766 0 : SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0];
767 :
768 0 : coeff[0] = c * c;
769 0 : coeff[1] = 3 * b * c;
770 0 : coeff[2] = 2 * b * b + c * a;
771 0 : coeff[3] = a * b;
772 0 : }
773 :
774 : /* Looking for F' dot F'' == 0
775 :
776 : A = b - a
777 : B = c - 2b + a
778 : C = d - 3c + 3b - a
779 :
780 : F' = 3Ct^2 + 6Bt + 3A
781 : F'' = 6Ct + 6B
782 :
783 : F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
784 : */
785 0 : int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
786 : SkScalar coeffX[4], coeffY[4];
787 : int i;
788 :
789 0 : formulate_F1DotF2(&src[0].fX, coeffX);
790 0 : formulate_F1DotF2(&src[0].fY, coeffY);
791 :
792 0 : for (i = 0; i < 4; i++) {
793 0 : coeffX[i] += coeffY[i];
794 : }
795 :
796 : SkScalar t[3];
797 0 : int count = solve_cubic_poly(coeffX, t);
798 0 : int maxCount = 0;
799 :
800 : // now remove extrema where the curvature is zero (mins)
801 : // !!!! need a test for this !!!!
802 0 : for (i = 0; i < count; i++) {
803 : // if (not_min_curvature())
804 0 : if (t[i] > 0 && t[i] < SK_Scalar1) {
805 0 : tValues[maxCount++] = t[i];
806 : }
807 : }
808 0 : return maxCount;
809 : }
810 :
811 0 : int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
812 : SkScalar tValues[3]) {
813 : SkScalar t_storage[3];
814 :
815 0 : if (tValues == nullptr) {
816 0 : tValues = t_storage;
817 : }
818 :
819 0 : int count = SkFindCubicMaxCurvature(src, tValues);
820 :
821 0 : if (dst) {
822 0 : if (count == 0) {
823 0 : memcpy(dst, src, 4 * sizeof(SkPoint));
824 : } else {
825 0 : SkChopCubicAt(src, dst, tValues, count);
826 : }
827 : }
828 0 : return count + 1;
829 : }
830 :
831 : #include "../pathops/SkPathOpsCubic.h"
832 :
833 : typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const;
834 :
835 33 : static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7],
836 : InterceptProc method) {
837 : SkDCubic cubic;
838 : double roots[3];
839 33 : int count = (cubic.set(src).*method)(intercept, roots);
840 33 : if (count > 0) {
841 33 : SkDCubicPair pair = cubic.chopAt(roots[0]);
842 264 : for (int i = 0; i < 7; ++i) {
843 231 : dst[i] = pair.pts[i].asSkPoint();
844 : }
845 33 : return true;
846 : }
847 0 : return false;
848 : }
849 :
850 19 : bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) {
851 19 : return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect);
852 : }
853 :
854 14 : bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) {
855 14 : return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect);
856 : }
857 :
858 : ///////////////////////////////////////////////////////////////////////////////
859 : //
860 : // NURB representation for conics. Helpful explanations at:
861 : //
862 : // http://citeseerx.ist.psu.edu/viewdoc/
863 : // download?doi=10.1.1.44.5740&rep=rep1&type=ps
864 : // and
865 : // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
866 : //
867 : // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
868 : // ------------------------------------------
869 : // ((1 - t)^2 + t^2 + 2 (1 - t) t w)
870 : //
871 : // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
872 : // ------------------------------------------------
873 : // {t^2 (2 - 2 w), t (-2 + 2 w), 1}
874 : //
875 :
876 : // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
877 : //
878 : // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
879 : // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
880 : // t^0 : -2 P0 w + 2 P1 w
881 : //
882 : // We disregard magnitude, so we can freely ignore the denominator of F', and
883 : // divide the numerator by 2
884 : //
885 : // coeff[0] for t^2
886 : // coeff[1] for t^1
887 : // coeff[2] for t^0
888 : //
889 0 : static void conic_deriv_coeff(const SkScalar src[],
890 : SkScalar w,
891 : SkScalar coeff[3]) {
892 0 : const SkScalar P20 = src[4] - src[0];
893 0 : const SkScalar P10 = src[2] - src[0];
894 0 : const SkScalar wP10 = w * P10;
895 0 : coeff[0] = w * P20 - P20;
896 0 : coeff[1] = P20 - 2 * wP10;
897 0 : coeff[2] = wP10;
898 0 : }
899 :
900 0 : static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
901 : SkScalar coeff[3];
902 0 : conic_deriv_coeff(src, w, coeff);
903 :
904 : SkScalar tValues[2];
905 0 : int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
906 0 : SkASSERT(0 == roots || 1 == roots);
907 :
908 0 : if (1 == roots) {
909 0 : *t = tValues[0];
910 0 : return true;
911 : }
912 0 : return false;
913 : }
914 :
915 : struct SkP3D {
916 : SkScalar fX, fY, fZ;
917 :
918 0 : void set(SkScalar x, SkScalar y, SkScalar z) {
919 0 : fX = x; fY = y; fZ = z;
920 0 : }
921 :
922 0 : void projectDown(SkPoint* dst) const {
923 0 : dst->set(fX / fZ, fY / fZ);
924 0 : }
925 : };
926 :
927 : // We only interpolate one dimension at a time (the first, at +0, +3, +6).
928 0 : static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
929 0 : SkScalar ab = SkScalarInterp(src[0], src[3], t);
930 0 : SkScalar bc = SkScalarInterp(src[3], src[6], t);
931 0 : dst[0] = ab;
932 0 : dst[3] = SkScalarInterp(ab, bc, t);
933 0 : dst[6] = bc;
934 0 : }
935 :
936 0 : static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
937 0 : dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
938 0 : dst[1].set(src[1].fX * w, src[1].fY * w, w);
939 0 : dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
940 0 : }
941 :
942 : // return false if infinity or NaN is generated; caller must check
943 0 : bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
944 : SkP3D tmp[3], tmp2[3];
945 :
946 0 : ratquad_mapTo3D(fPts, fW, tmp);
947 :
948 0 : p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
949 0 : p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
950 0 : p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
951 :
952 0 : dst[0].fPts[0] = fPts[0];
953 0 : tmp2[0].projectDown(&dst[0].fPts[1]);
954 0 : tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
955 0 : tmp2[2].projectDown(&dst[1].fPts[1]);
956 0 : dst[1].fPts[2] = fPts[2];
957 :
958 : // to put in "standard form", where w0 and w2 are both 1, we compute the
959 : // new w1 as sqrt(w1*w1/w0*w2)
960 : // or
961 : // w1 /= sqrt(w0*w2)
962 : //
963 : // However, in our case, we know that for dst[0]:
964 : // w0 == 1, and for dst[1], w2 == 1
965 : //
966 0 : SkScalar root = SkScalarSqrt(tmp2[1].fZ);
967 0 : dst[0].fW = tmp2[0].fZ / root;
968 0 : dst[1].fW = tmp2[2].fZ / root;
969 : SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7);
970 : SkASSERT(0 == offsetof(SkConic, fPts[0].fX));
971 0 : return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2);
972 : }
973 :
974 0 : void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const {
975 0 : if (0 == t1 || 1 == t2) {
976 0 : if (0 == t1 && 1 == t2) {
977 0 : *dst = *this;
978 0 : return;
979 : } else {
980 0 : SkConic pair[2];
981 0 : if (this->chopAt(t1 ? t1 : t2, pair)) {
982 0 : *dst = pair[SkToBool(t1)];
983 0 : return;
984 : }
985 : }
986 : }
987 0 : SkConicCoeff coeff(*this);
988 : Sk2s tt1(t1);
989 0 : Sk2s aXY = coeff.fNumer.eval(tt1);
990 0 : Sk2s aZZ = coeff.fDenom.eval(tt1);
991 0 : Sk2s midTT((t1 + t2) / 2);
992 0 : Sk2s dXY = coeff.fNumer.eval(midTT);
993 0 : Sk2s dZZ = coeff.fDenom.eval(midTT);
994 : Sk2s tt2(t2);
995 0 : Sk2s cXY = coeff.fNumer.eval(tt2);
996 0 : Sk2s cZZ = coeff.fDenom.eval(tt2);
997 0 : Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f);
998 0 : Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f);
999 0 : dst->fPts[0] = to_point(aXY / aZZ);
1000 0 : dst->fPts[1] = to_point(bXY / bZZ);
1001 0 : dst->fPts[2] = to_point(cXY / cZZ);
1002 0 : Sk2s ww = bZZ / (aZZ * cZZ).sqrt();
1003 0 : dst->fW = ww[0];
1004 : }
1005 :
1006 0 : SkPoint SkConic::evalAt(SkScalar t) const {
1007 0 : return to_point(SkConicCoeff(*this).eval(t));
1008 : }
1009 :
1010 0 : SkVector SkConic::evalTangentAt(SkScalar t) const {
1011 : // The derivative equation returns a zero tangent vector when t is 0 or 1,
1012 : // and the control point is equal to the end point.
1013 : // In this case, use the conic endpoints to compute the tangent.
1014 0 : if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) {
1015 0 : return fPts[2] - fPts[0];
1016 : }
1017 0 : Sk2s p0 = from_point(fPts[0]);
1018 0 : Sk2s p1 = from_point(fPts[1]);
1019 0 : Sk2s p2 = from_point(fPts[2]);
1020 0 : Sk2s ww(fW);
1021 :
1022 0 : Sk2s p20 = p2 - p0;
1023 0 : Sk2s p10 = p1 - p0;
1024 :
1025 0 : Sk2s C = ww * p10;
1026 0 : Sk2s A = ww * p20 - p20;
1027 0 : Sk2s B = p20 - C - C;
1028 :
1029 0 : return to_vector(SkQuadCoeff(A, B, C).eval(t));
1030 : }
1031 :
1032 0 : void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
1033 0 : SkASSERT(t >= 0 && t <= SK_Scalar1);
1034 :
1035 0 : if (pt) {
1036 0 : *pt = this->evalAt(t);
1037 : }
1038 0 : if (tangent) {
1039 0 : *tangent = this->evalTangentAt(t);
1040 : }
1041 0 : }
1042 :
1043 0 : static SkScalar subdivide_w_value(SkScalar w) {
1044 0 : return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
1045 : }
1046 :
1047 0 : void SkConic::chop(SkConic * SK_RESTRICT dst) const {
1048 0 : Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW));
1049 0 : SkScalar newW = subdivide_w_value(fW);
1050 :
1051 0 : Sk2s p0 = from_point(fPts[0]);
1052 0 : Sk2s p1 = from_point(fPts[1]);
1053 0 : Sk2s p2 = from_point(fPts[2]);
1054 0 : Sk2s ww(fW);
1055 :
1056 0 : Sk2s wp1 = ww * p1;
1057 0 : Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f);
1058 :
1059 0 : dst[0].fPts[0] = fPts[0];
1060 0 : dst[0].fPts[1] = to_point((p0 + wp1) * scale);
1061 0 : dst[0].fPts[2] = dst[1].fPts[0] = to_point(m);
1062 0 : dst[1].fPts[1] = to_point((wp1 + p2) * scale);
1063 0 : dst[1].fPts[2] = fPts[2];
1064 :
1065 0 : dst[0].fW = dst[1].fW = newW;
1066 0 : }
1067 :
1068 : /*
1069 : * "High order approximation of conic sections by quadratic splines"
1070 : * by Michael Floater, 1993
1071 : */
1072 : #define AS_QUAD_ERROR_SETUP \
1073 : SkScalar a = fW - 1; \
1074 : SkScalar k = a / (4 * (2 + a)); \
1075 : SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \
1076 : SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
1077 :
1078 0 : void SkConic::computeAsQuadError(SkVector* err) const {
1079 0 : AS_QUAD_ERROR_SETUP
1080 0 : err->set(x, y);
1081 0 : }
1082 :
1083 0 : bool SkConic::asQuadTol(SkScalar tol) const {
1084 0 : AS_QUAD_ERROR_SETUP
1085 0 : return (x * x + y * y) <= tol * tol;
1086 : }
1087 :
1088 : // Limit the number of suggested quads to approximate a conic
1089 : #define kMaxConicToQuadPOW2 5
1090 :
1091 0 : int SkConic::computeQuadPOW2(SkScalar tol) const {
1092 0 : if (tol < 0 || !SkScalarIsFinite(tol)) {
1093 0 : return 0;
1094 : }
1095 :
1096 0 : AS_QUAD_ERROR_SETUP
1097 :
1098 0 : SkScalar error = SkScalarSqrt(x * x + y * y);
1099 : int pow2;
1100 0 : for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
1101 0 : if (error <= tol) {
1102 0 : break;
1103 : }
1104 0 : error *= 0.25f;
1105 : }
1106 : // float version -- using ceil gives the same results as the above.
1107 : if (false) {
1108 : SkScalar err = SkScalarSqrt(x * x + y * y);
1109 : if (err <= tol) {
1110 : return 0;
1111 : }
1112 : SkScalar tol2 = tol * tol;
1113 : if (tol2 == 0) {
1114 : return kMaxConicToQuadPOW2;
1115 : }
1116 : SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
1117 : int altPow2 = SkScalarCeilToInt(fpow2);
1118 : if (altPow2 != pow2) {
1119 : SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
1120 : }
1121 : pow2 = altPow2;
1122 : }
1123 0 : return pow2;
1124 : }
1125 :
1126 : // This was originally developed and tested for pathops: see SkOpTypes.h
1127 : // returns true if (a <= b <= c) || (a >= b >= c)
1128 0 : static bool between(SkScalar a, SkScalar b, SkScalar c) {
1129 0 : return (a - b) * (c - b) <= 0;
1130 : }
1131 :
1132 0 : static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
1133 0 : SkASSERT(level >= 0);
1134 :
1135 0 : if (0 == level) {
1136 0 : memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
1137 0 : return pts + 2;
1138 : } else {
1139 0 : SkConic dst[2];
1140 0 : src.chop(dst);
1141 0 : const SkScalar startY = src.fPts[0].fY;
1142 0 : const SkScalar endY = src.fPts[2].fY;
1143 0 : if (between(startY, src.fPts[1].fY, endY)) {
1144 : // If the input is monotonic and the output is not, the scan converter hangs.
1145 : // Ensure that the chopped conics maintain their y-order.
1146 0 : SkScalar midY = dst[0].fPts[2].fY;
1147 0 : if (!between(startY, midY, endY)) {
1148 : // If the computed midpoint is outside the ends, move it to the closer one.
1149 0 : SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY;
1150 0 : dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY;
1151 : }
1152 0 : if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) {
1153 : // If the 1st control is not between the start and end, put it at the start.
1154 : // This also reduces the quad to a line.
1155 0 : dst[0].fPts[1].fY = startY;
1156 : }
1157 0 : if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) {
1158 : // If the 2nd control is not between the start and end, put it at the end.
1159 : // This also reduces the quad to a line.
1160 0 : dst[1].fPts[1].fY = endY;
1161 : }
1162 : // Verify that all five points are in order.
1163 0 : SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY));
1164 0 : SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY));
1165 0 : SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY));
1166 : }
1167 0 : --level;
1168 0 : pts = subdivide(dst[0], pts, level);
1169 0 : return subdivide(dst[1], pts, level);
1170 : }
1171 : }
1172 :
1173 0 : int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
1174 0 : SkASSERT(pow2 >= 0);
1175 0 : *pts = fPts[0];
1176 : SkDEBUGCODE(SkPoint* endPts);
1177 0 : if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ...
1178 0 : SkConic dst[2];
1179 0 : this->chop(dst);
1180 : // check to see if the first chop generates a pair of lines
1181 0 : if (dst[0].fPts[1].equalsWithinTolerance(dst[0].fPts[2])
1182 0 : && dst[1].fPts[0].equalsWithinTolerance(dst[1].fPts[1])) {
1183 0 : pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines
1184 0 : pts[4] = dst[1].fPts[2];
1185 0 : pow2 = 1;
1186 0 : SkDEBUGCODE(endPts = &pts[5]);
1187 0 : goto commonFinitePtCheck;
1188 : }
1189 : }
1190 0 : SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2);
1191 : commonFinitePtCheck:
1192 0 : const int quadCount = 1 << pow2;
1193 0 : const int ptCount = 2 * quadCount + 1;
1194 0 : SkASSERT(endPts - pts == ptCount);
1195 0 : if (!SkPointsAreFinite(pts, ptCount)) {
1196 : // if we generated a non-finite, pin ourselves to the middle of the hull,
1197 : // as our first and last are already on the first/last pts of the hull.
1198 0 : for (int i = 1; i < ptCount - 1; ++i) {
1199 0 : pts[i] = fPts[1];
1200 : }
1201 : }
1202 0 : return 1 << pow2;
1203 : }
1204 :
1205 0 : bool SkConic::findXExtrema(SkScalar* t) const {
1206 0 : return conic_find_extrema(&fPts[0].fX, fW, t);
1207 : }
1208 :
1209 0 : bool SkConic::findYExtrema(SkScalar* t) const {
1210 0 : return conic_find_extrema(&fPts[0].fY, fW, t);
1211 : }
1212 :
1213 0 : bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
1214 : SkScalar t;
1215 0 : if (this->findXExtrema(&t)) {
1216 0 : if (!this->chopAt(t, dst)) {
1217 : // if chop can't return finite values, don't chop
1218 0 : return false;
1219 : }
1220 : // now clean-up the middle, since we know t was meant to be at
1221 : // an X-extrema
1222 0 : SkScalar value = dst[0].fPts[2].fX;
1223 0 : dst[0].fPts[1].fX = value;
1224 0 : dst[1].fPts[0].fX = value;
1225 0 : dst[1].fPts[1].fX = value;
1226 0 : return true;
1227 : }
1228 0 : return false;
1229 : }
1230 :
1231 0 : bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
1232 : SkScalar t;
1233 0 : if (this->findYExtrema(&t)) {
1234 0 : if (!this->chopAt(t, dst)) {
1235 : // if chop can't return finite values, don't chop
1236 0 : return false;
1237 : }
1238 : // now clean-up the middle, since we know t was meant to be at
1239 : // an Y-extrema
1240 0 : SkScalar value = dst[0].fPts[2].fY;
1241 0 : dst[0].fPts[1].fY = value;
1242 0 : dst[1].fPts[0].fY = value;
1243 0 : dst[1].fPts[1].fY = value;
1244 0 : return true;
1245 : }
1246 0 : return false;
1247 : }
1248 :
1249 0 : void SkConic::computeTightBounds(SkRect* bounds) const {
1250 : SkPoint pts[4];
1251 0 : pts[0] = fPts[0];
1252 0 : pts[1] = fPts[2];
1253 0 : int count = 2;
1254 :
1255 : SkScalar t;
1256 0 : if (this->findXExtrema(&t)) {
1257 0 : this->evalAt(t, &pts[count++]);
1258 : }
1259 0 : if (this->findYExtrema(&t)) {
1260 0 : this->evalAt(t, &pts[count++]);
1261 : }
1262 0 : bounds->set(pts, count);
1263 0 : }
1264 :
1265 0 : void SkConic::computeFastBounds(SkRect* bounds) const {
1266 0 : bounds->set(fPts, 3);
1267 0 : }
1268 :
1269 : #if 0 // unimplemented
1270 : bool SkConic::findMaxCurvature(SkScalar* t) const {
1271 : // TODO: Implement me
1272 : return false;
1273 : }
1274 : #endif
1275 :
1276 0 : SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
1277 : const SkMatrix& matrix) {
1278 0 : if (!matrix.hasPerspective()) {
1279 0 : return w;
1280 : }
1281 :
1282 : SkP3D src[3], dst[3];
1283 :
1284 0 : ratquad_mapTo3D(pts, w, src);
1285 :
1286 0 : matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
1287 :
1288 : // w' = sqrt(w1*w1/w0*w2)
1289 0 : SkScalar w0 = dst[0].fZ;
1290 0 : SkScalar w1 = dst[1].fZ;
1291 0 : SkScalar w2 = dst[2].fZ;
1292 0 : w = SkScalarSqrt((w1 * w1) / (w0 * w2));
1293 0 : return w;
1294 : }
1295 :
1296 0 : int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
1297 : const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
1298 : // rotate by x,y so that uStart is (1.0)
1299 0 : SkScalar x = SkPoint::DotProduct(uStart, uStop);
1300 0 : SkScalar y = SkPoint::CrossProduct(uStart, uStop);
1301 :
1302 0 : SkScalar absY = SkScalarAbs(y);
1303 :
1304 : // check for (effectively) coincident vectors
1305 : // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
1306 : // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
1307 0 : if (absY <= SK_ScalarNearlyZero && x > 0 && ((y >= 0 && kCW_SkRotationDirection == dir) ||
1308 0 : (y <= 0 && kCCW_SkRotationDirection == dir))) {
1309 0 : return 0;
1310 : }
1311 :
1312 0 : if (dir == kCCW_SkRotationDirection) {
1313 0 : y = -y;
1314 : }
1315 :
1316 : // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
1317 : // 0 == [0 .. 90)
1318 : // 1 == [90 ..180)
1319 : // 2 == [180..270)
1320 : // 3 == [270..360)
1321 : //
1322 0 : int quadrant = 0;
1323 0 : if (0 == y) {
1324 0 : quadrant = 2; // 180
1325 0 : SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
1326 0 : } else if (0 == x) {
1327 0 : SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
1328 0 : quadrant = y > 0 ? 1 : 3; // 90 : 270
1329 : } else {
1330 0 : if (y < 0) {
1331 0 : quadrant += 2;
1332 : }
1333 0 : if ((x < 0) != (y < 0)) {
1334 0 : quadrant += 1;
1335 : }
1336 : }
1337 :
1338 : const SkPoint quadrantPts[] = {
1339 : { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
1340 0 : };
1341 0 : const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
1342 :
1343 0 : int conicCount = quadrant;
1344 0 : for (int i = 0; i < conicCount; ++i) {
1345 0 : dst[i].set(&quadrantPts[i * 2], quadrantWeight);
1346 : }
1347 :
1348 : // Now compute any remaing (sub-90-degree) arc for the last conic
1349 0 : const SkPoint finalP = { x, y };
1350 0 : const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector
1351 0 : const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
1352 0 : if (!SkScalarIsFinite(dot)) {
1353 0 : return 0;
1354 : }
1355 0 : SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
1356 :
1357 0 : if (dot < 1) {
1358 0 : SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
1359 : // compute the bisector vector, and then rescale to be the off-curve point.
1360 : // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
1361 : // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
1362 : // This is nice, since our computed weight is cos(theta/2) as well!
1363 : //
1364 0 : const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
1365 0 : offCurve.setLength(SkScalarInvert(cosThetaOver2));
1366 0 : if (!lastQ.equalsWithinTolerance(offCurve)) {
1367 0 : dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
1368 0 : conicCount += 1;
1369 : }
1370 : }
1371 :
1372 : // now handle counter-clockwise and the initial unitStart rotation
1373 : SkMatrix matrix;
1374 0 : matrix.setSinCos(uStart.fY, uStart.fX);
1375 0 : if (dir == kCCW_SkRotationDirection) {
1376 0 : matrix.preScale(SK_Scalar1, -SK_Scalar1);
1377 : }
1378 0 : if (userMatrix) {
1379 0 : matrix.postConcat(*userMatrix);
1380 : }
1381 0 : for (int i = 0; i < conicCount; ++i) {
1382 0 : matrix.mapPoints(dst[i].fPts, 3);
1383 : }
1384 0 : return conicCount;
1385 : }
|
