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 #ifndef SkGeometry_DEFINED
9 #define SkGeometry_DEFINED
10
11 #include "SkMatrix.h"
12 #include "SkNx.h"
13
from_point(const SkPoint & point)14 static inline Sk2s from_point(const SkPoint& point) {
15 return Sk2s::Load(&point);
16 }
17
to_point(const Sk2s & x)18 static inline SkPoint to_point(const Sk2s& x) {
19 SkPoint point;
20 x.store(&point);
21 return point;
22 }
23
times_2(const Sk2s & value)24 static Sk2s times_2(const Sk2s& value) {
25 return value + value;
26 }
27
28 /** Given a quadratic equation Ax^2 + Bx + C = 0, return 0, 1, 2 roots for the
29 equation.
30 */
31 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]);
32
33 ///////////////////////////////////////////////////////////////////////////////
34
35 SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t);
36 SkPoint SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t);
37
38 /** Set pt to the point on the src quadratic specified by t. t must be
39 0 <= t <= 1.0
40 */
41 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent = nullptr);
42
43 /** Given a src quadratic bezier, chop it at the specified t value,
44 where 0 < t < 1, and return the two new quadratics in dst:
45 dst[0..2] and dst[2..4]
46 */
47 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t);
48
49 /** Given a src quadratic bezier, chop it at the specified t == 1/2,
50 The new quads are returned in dst[0..2] and dst[2..4]
51 */
52 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]);
53
54 /** Given the 3 coefficients for a quadratic bezier (either X or Y values), look
55 for extrema, and return the number of t-values that are found that represent
56 these extrema. If the quadratic has no extrema betwee (0..1) exclusive, the
57 function returns 0.
58 Returned count tValues[]
59 0 ignored
60 1 0 < tValues[0] < 1
61 */
62 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValues[1]);
63
64 /** Given 3 points on a quadratic bezier, chop it into 1, 2 beziers such that
65 the resulting beziers are monotonic in Y. This is called by the scan converter.
66 Depending on what is returned, dst[] is treated as follows
67 0 dst[0..2] is the original quad
68 1 dst[0..2] and dst[2..4] are the two new quads
69 */
70 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]);
71 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]);
72
73 /** Given 3 points on a quadratic bezier, if the point of maximum
74 curvature exists on the segment, returns the t value for this
75 point along the curve. Otherwise it will return a value of 0.
76 */
77 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]);
78
79 /** Given 3 points on a quadratic bezier, divide it into 2 quadratics
80 if the point of maximum curvature exists on the quad segment.
81 Depending on what is returned, dst[] is treated as follows
82 1 dst[0..2] is the original quad
83 2 dst[0..2] and dst[2..4] are the two new quads
84 If dst == null, it is ignored and only the count is returned.
85 */
86 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]);
87
88 /** Given 3 points on a quadratic bezier, use degree elevation to
89 convert it into the cubic fitting the same curve. The new cubic
90 curve is returned in dst[0..3].
91 */
92 SK_API void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]);
93
94 ///////////////////////////////////////////////////////////////////////////////
95
96 /** Set pt to the point on the src cubic specified by t. t must be
97 0 <= t <= 1.0
98 */
99 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* locOrNull,
100 SkVector* tangentOrNull, SkVector* curvatureOrNull);
101
102 /** Given a src cubic bezier, chop it at the specified t value,
103 where 0 < t < 1, and return the two new cubics in dst:
104 dst[0..3] and dst[3..6]
105 */
106 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t);
107
108 /** Given a src cubic bezier, chop it at the specified t values,
109 where 0 < t < 1, and return the new cubics in dst:
110 dst[0..3],dst[3..6],...,dst[3*t_count..3*(t_count+1)]
111 */
112 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar t[],
113 int t_count);
114
115 /** Given a src cubic bezier, chop it at the specified t == 1/2,
116 The new cubics are returned in dst[0..3] and dst[3..6]
117 */
118 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]);
119
120 /** Given the 4 coefficients for a cubic bezier (either X or Y values), look
121 for extrema, and return the number of t-values that are found that represent
122 these extrema. If the cubic has no extrema betwee (0..1) exclusive, the
123 function returns 0.
124 Returned count tValues[]
125 0 ignored
126 1 0 < tValues[0] < 1
127 2 0 < tValues[0] < tValues[1] < 1
128 */
129 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
130 SkScalar tValues[2]);
131
132 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
133 the resulting beziers are monotonic in Y. This is called by the scan converter.
134 Depending on what is returned, dst[] is treated as follows
135 0 dst[0..3] is the original cubic
136 1 dst[0..3] and dst[3..6] are the two new cubics
137 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
138 If dst == null, it is ignored and only the count is returned.
139 */
140 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]);
141 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]);
142
143 /** Given a cubic bezier, return 0, 1, or 2 t-values that represent the
144 inflection points.
145 */
146 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[2]);
147
148 /** Return 1 for no chop, 2 for having chopped the cubic at a single
149 inflection point, 3 for having chopped at 2 inflection points.
150 dst will hold the resulting 1, 2, or 3 cubics.
151 */
152 int SkChopCubicAtInflections(const SkPoint src[4], SkPoint dst[10]);
153
154 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]);
155 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
156 SkScalar tValues[3] = nullptr);
157
158 bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar y, SkPoint dst[7]);
159 bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar x, SkPoint dst[7]);
160
161 enum class SkCubicType {
162 kSerpentine,
163 kLoop,
164 kLocalCusp, // Cusp at a non-infinite parameter value with an inflection at t=infinity.
165 kCuspAtInfinity, // Cusp with a cusp at t=infinity and a local inflection.
166 kQuadratic,
167 kLineOrPoint
168 };
169
SkCubicIsDegenerate(SkCubicType type)170 static inline bool SkCubicIsDegenerate(SkCubicType type) {
171 switch (type) {
172 case SkCubicType::kSerpentine:
173 case SkCubicType::kLoop:
174 case SkCubicType::kLocalCusp:
175 case SkCubicType::kCuspAtInfinity:
176 return false;
177 case SkCubicType::kQuadratic:
178 case SkCubicType::kLineOrPoint:
179 return true;
180 }
181 SK_ABORT("Invalid SkCubicType");
182 return true;
183 }
184
SkCubicTypeName(SkCubicType type)185 static inline const char* SkCubicTypeName(SkCubicType type) {
186 switch (type) {
187 case SkCubicType::kSerpentine: return "kSerpentine";
188 case SkCubicType::kLoop: return "kLoop";
189 case SkCubicType::kLocalCusp: return "kLocalCusp";
190 case SkCubicType::kCuspAtInfinity: return "kCuspAtInfinity";
191 case SkCubicType::kQuadratic: return "kQuadratic";
192 case SkCubicType::kLineOrPoint: return "kLineOrPoint";
193 }
194 SK_ABORT("Invalid SkCubicType");
195 return "";
196 }
197
198 /** Returns the cubic classification.
199
200 t[],s[] are set to the two homogeneous parameter values at which points the lines L & M
201 intersect with K, sorted from smallest to largest and oriented so positive values of the
202 implicit are on the "left" side. For a serpentine curve they are the inflection points. For a
203 loop they are the double point. For a local cusp, they are both equal and denote the cusp point.
204 For a cusp at an infinite parameter value, one will be the local inflection point and the other
205 +inf (t,s = 1,0). If the curve is degenerate (i.e. quadratic or linear) they are both set to a
206 parameter value of +inf (t,s = 1,0).
207
208 d[] is filled with the cubic inflection function coefficients. See "Resolution Independent
209 Curve Rendering using Programmable Graphics Hardware", 4.2 Curve Categorization:
210
211 If the input points contain infinities or NaN, the return values are undefined.
212
213 https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
214 */
215 SkCubicType SkClassifyCubic(const SkPoint p[4], double t[2] = nullptr, double s[2] = nullptr,
216 double d[4] = nullptr);
217
218 ///////////////////////////////////////////////////////////////////////////////
219
220 enum SkRotationDirection {
221 kCW_SkRotationDirection,
222 kCCW_SkRotationDirection
223 };
224
225 struct SkConic {
SkConicSkConic226 SkConic() {}
SkConicSkConic227 SkConic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar w) {
228 fPts[0] = p0;
229 fPts[1] = p1;
230 fPts[2] = p2;
231 fW = w;
232 }
SkConicSkConic233 SkConic(const SkPoint pts[3], SkScalar w) {
234 memcpy(fPts, pts, sizeof(fPts));
235 fW = w;
236 }
237
238 SkPoint fPts[3];
239 SkScalar fW;
240
setSkConic241 void set(const SkPoint pts[3], SkScalar w) {
242 memcpy(fPts, pts, 3 * sizeof(SkPoint));
243 fW = w;
244 }
245
setSkConic246 void set(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar w) {
247 fPts[0] = p0;
248 fPts[1] = p1;
249 fPts[2] = p2;
250 fW = w;
251 }
252
253 /**
254 * Given a t-value [0...1] return its position and/or tangent.
255 * If pos is not null, return its position at the t-value.
256 * If tangent is not null, return its tangent at the t-value. NOTE the
257 * tangent value's length is arbitrary, and only its direction should
258 * be used.
259 */
260 void evalAt(SkScalar t, SkPoint* pos, SkVector* tangent = nullptr) const;
261 bool SK_WARN_UNUSED_RESULT chopAt(SkScalar t, SkConic dst[2]) const;
262 void chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const;
263 void chop(SkConic dst[2]) const;
264
265 SkPoint evalAt(SkScalar t) const;
266 SkVector evalTangentAt(SkScalar t) const;
267
268 void computeAsQuadError(SkVector* err) const;
269 bool asQuadTol(SkScalar tol) const;
270
271 /**
272 * return the power-of-2 number of quads needed to approximate this conic
273 * with a sequence of quads. Will be >= 0.
274 */
275 int SK_API computeQuadPOW2(SkScalar tol) const;
276
277 /**
278 * Chop this conic into N quads, stored continguously in pts[], where
279 * N = 1 << pow2. The amount of storage needed is (1 + 2 * N)
280 */
281 int SK_API SK_WARN_UNUSED_RESULT chopIntoQuadsPOW2(SkPoint pts[], int pow2) const;
282
283 bool findXExtrema(SkScalar* t) const;
284 bool findYExtrema(SkScalar* t) const;
285 bool chopAtXExtrema(SkConic dst[2]) const;
286 bool chopAtYExtrema(SkConic dst[2]) const;
287
288 void computeTightBounds(SkRect* bounds) const;
289 void computeFastBounds(SkRect* bounds) const;
290
291 /** Find the parameter value where the conic takes on its maximum curvature.
292 *
293 * @param t output scalar for max curvature. Will be unchanged if
294 * max curvature outside 0..1 range.
295 *
296 * @return true if max curvature found inside 0..1 range, false otherwise
297 */
298 // bool findMaxCurvature(SkScalar* t) const; // unimplemented
299
300 static SkScalar TransformW(const SkPoint[3], SkScalar w, const SkMatrix&);
301
302 enum {
303 kMaxConicsForArc = 5
304 };
305 static int BuildUnitArc(const SkVector& start, const SkVector& stop, SkRotationDirection,
306 const SkMatrix*, SkConic conics[kMaxConicsForArc]);
307 };
308
309 // inline helpers are contained in a namespace to avoid external leakage to fragile SkNx members
310 namespace {
311
312 /**
313 * use for : eval(t) == A * t^2 + B * t + C
314 */
315 struct SkQuadCoeff {
SkQuadCoeffSkQuadCoeff316 SkQuadCoeff() {}
317
SkQuadCoeffSkQuadCoeff318 SkQuadCoeff(const Sk2s& A, const Sk2s& B, const Sk2s& C)
319 : fA(A)
320 , fB(B)
321 , fC(C)
322 {
323 }
324
SkQuadCoeffSkQuadCoeff325 SkQuadCoeff(const SkPoint src[3]) {
326 fC = from_point(src[0]);
327 Sk2s P1 = from_point(src[1]);
328 Sk2s P2 = from_point(src[2]);
329 fB = times_2(P1 - fC);
330 fA = P2 - times_2(P1) + fC;
331 }
332
evalSkQuadCoeff333 Sk2s eval(SkScalar t) {
334 Sk2s tt(t);
335 return eval(tt);
336 }
337
evalSkQuadCoeff338 Sk2s eval(const Sk2s& tt) {
339 return (fA * tt + fB) * tt + fC;
340 }
341
342 Sk2s fA;
343 Sk2s fB;
344 Sk2s fC;
345 };
346
347 struct SkConicCoeff {
SkConicCoeffSkConicCoeff348 SkConicCoeff(const SkConic& conic) {
349 Sk2s p0 = from_point(conic.fPts[0]);
350 Sk2s p1 = from_point(conic.fPts[1]);
351 Sk2s p2 = from_point(conic.fPts[2]);
352 Sk2s ww(conic.fW);
353
354 Sk2s p1w = p1 * ww;
355 fNumer.fC = p0;
356 fNumer.fA = p2 - times_2(p1w) + p0;
357 fNumer.fB = times_2(p1w - p0);
358
359 fDenom.fC = Sk2s(1);
360 fDenom.fB = times_2(ww - fDenom.fC);
361 fDenom.fA = Sk2s(0) - fDenom.fB;
362 }
363
evalSkConicCoeff364 Sk2s eval(SkScalar t) {
365 Sk2s tt(t);
366 Sk2s numer = fNumer.eval(tt);
367 Sk2s denom = fDenom.eval(tt);
368 return numer / denom;
369 }
370
371 SkQuadCoeff fNumer;
372 SkQuadCoeff fDenom;
373 };
374
375 struct SkCubicCoeff {
SkCubicCoeffSkCubicCoeff376 SkCubicCoeff(const SkPoint src[4]) {
377 Sk2s P0 = from_point(src[0]);
378 Sk2s P1 = from_point(src[1]);
379 Sk2s P2 = from_point(src[2]);
380 Sk2s P3 = from_point(src[3]);
381 Sk2s three(3);
382 fA = P3 + three * (P1 - P2) - P0;
383 fB = three * (P2 - times_2(P1) + P0);
384 fC = three * (P1 - P0);
385 fD = P0;
386 }
387
evalSkCubicCoeff388 Sk2s eval(SkScalar t) {
389 Sk2s tt(t);
390 return eval(tt);
391 }
392
evalSkCubicCoeff393 Sk2s eval(const Sk2s& t) {
394 return ((fA * t + fB) * t + fC) * t + fD;
395 }
396
397 Sk2s fA;
398 Sk2s fB;
399 Sk2s fC;
400 Sk2s fD;
401 };
402
403 }
404
405 #include "SkTemplates.h"
406
407 /**
408 * Help class to allocate storage for approximating a conic with N quads.
409 */
410 class SkAutoConicToQuads {
411 public:
SkAutoConicToQuads()412 SkAutoConicToQuads() : fQuadCount(0) {}
413
414 /**
415 * Given a conic and a tolerance, return the array of points for the
416 * approximating quad(s). Call countQuads() to know the number of quads
417 * represented in these points.
418 *
419 * The quads are allocated to share end-points. e.g. if there are 4 quads,
420 * there will be 9 points allocated as follows
421 * quad[0] == pts[0..2]
422 * quad[1] == pts[2..4]
423 * quad[2] == pts[4..6]
424 * quad[3] == pts[6..8]
425 */
computeQuads(const SkConic & conic,SkScalar tol)426 const SkPoint* computeQuads(const SkConic& conic, SkScalar tol) {
427 int pow2 = conic.computeQuadPOW2(tol);
428 fQuadCount = 1 << pow2;
429 SkPoint* pts = fStorage.reset(1 + 2 * fQuadCount);
430 fQuadCount = conic.chopIntoQuadsPOW2(pts, pow2);
431 return pts;
432 }
433
computeQuads(const SkPoint pts[3],SkScalar weight,SkScalar tol)434 const SkPoint* computeQuads(const SkPoint pts[3], SkScalar weight,
435 SkScalar tol) {
436 SkConic conic;
437 conic.set(pts, weight);
438 return computeQuads(conic, tol);
439 }
440
countQuads()441 int countQuads() const { return fQuadCount; }
442
443 private:
444 enum {
445 kQuadCount = 8, // should handle most conics
446 kPointCount = 1 + 2 * kQuadCount,
447 };
448 SkAutoSTMalloc<kPointCount, SkPoint> fStorage;
449 int fQuadCount; // #quads for current usage
450 };
451
452 #endif
453