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
170 /** Returns the cubic classification.
171
172 t[],s[] are set to the two homogeneous parameter values at which points the lines L & M
173 intersect with K, sorted from smallest to largest and oriented so positive values of the
174 implicit are on the "left" side. For a serpentine curve they are the inflection points. For a
175 loop they are the double point. For a local cusp, they are both equal and denote the cusp point.
176 For a cusp at an infinite parameter value, one will be the local inflection point and the other
177 +inf (t,s = 1,0). If the curve is degenerate (i.e. quadratic or linear) they are both set to a
178 parameter value of +inf (t,s = 1,0).
179
180 d[] is filled with the cubic inflection function coefficients. See "Resolution Independent
181 Curve Rendering using Programmable Graphics Hardware", 4.2 Curve Categorization:
182
183 https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
184 */
185 SkCubicType SkClassifyCubic(const SkPoint p[4], double t[2] = nullptr, double s[2] = nullptr,
186 double d[4] = nullptr);
187
188 ///////////////////////////////////////////////////////////////////////////////
189
190 enum SkRotationDirection {
191 kCW_SkRotationDirection,
192 kCCW_SkRotationDirection
193 };
194
195 struct SkConic {
SkConicSkConic196 SkConic() {}
SkConicSkConic197 SkConic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar w) {
198 fPts[0] = p0;
199 fPts[1] = p1;
200 fPts[2] = p2;
201 fW = w;
202 }
SkConicSkConic203 SkConic(const SkPoint pts[3], SkScalar w) {
204 memcpy(fPts, pts, sizeof(fPts));
205 fW = w;
206 }
207
208 SkPoint fPts[3];
209 SkScalar fW;
210
setSkConic211 void set(const SkPoint pts[3], SkScalar w) {
212 memcpy(fPts, pts, 3 * sizeof(SkPoint));
213 fW = w;
214 }
215
setSkConic216 void set(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar w) {
217 fPts[0] = p0;
218 fPts[1] = p1;
219 fPts[2] = p2;
220 fW = w;
221 }
222
223 /**
224 * Given a t-value [0...1] return its position and/or tangent.
225 * If pos is not null, return its position at the t-value.
226 * If tangent is not null, return its tangent at the t-value. NOTE the
227 * tangent value's length is arbitrary, and only its direction should
228 * be used.
229 */
230 void evalAt(SkScalar t, SkPoint* pos, SkVector* tangent = nullptr) const;
231 bool SK_WARN_UNUSED_RESULT chopAt(SkScalar t, SkConic dst[2]) const;
232 void chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const;
233 void chop(SkConic dst[2]) const;
234
235 SkPoint evalAt(SkScalar t) const;
236 SkVector evalTangentAt(SkScalar t) const;
237
238 void computeAsQuadError(SkVector* err) const;
239 bool asQuadTol(SkScalar tol) const;
240
241 /**
242 * return the power-of-2 number of quads needed to approximate this conic
243 * with a sequence of quads. Will be >= 0.
244 */
245 int computeQuadPOW2(SkScalar tol) const;
246
247 /**
248 * Chop this conic into N quads, stored continguously in pts[], where
249 * N = 1 << pow2. The amount of storage needed is (1 + 2 * N)
250 */
251 int SK_WARN_UNUSED_RESULT chopIntoQuadsPOW2(SkPoint pts[], int pow2) const;
252
253 bool findXExtrema(SkScalar* t) const;
254 bool findYExtrema(SkScalar* t) const;
255 bool chopAtXExtrema(SkConic dst[2]) const;
256 bool chopAtYExtrema(SkConic dst[2]) const;
257
258 void computeTightBounds(SkRect* bounds) const;
259 void computeFastBounds(SkRect* bounds) const;
260
261 /** Find the parameter value where the conic takes on its maximum curvature.
262 *
263 * @param t output scalar for max curvature. Will be unchanged if
264 * max curvature outside 0..1 range.
265 *
266 * @return true if max curvature found inside 0..1 range, false otherwise
267 */
268 // bool findMaxCurvature(SkScalar* t) const; // unimplemented
269
270 static SkScalar TransformW(const SkPoint[3], SkScalar w, const SkMatrix&);
271
272 enum {
273 kMaxConicsForArc = 5
274 };
275 static int BuildUnitArc(const SkVector& start, const SkVector& stop, SkRotationDirection,
276 const SkMatrix*, SkConic conics[kMaxConicsForArc]);
277 };
278
279 // inline helpers are contained in a namespace to avoid external leakage to fragile SkNx members
280 namespace {
281
282 /**
283 * use for : eval(t) == A * t^2 + B * t + C
284 */
285 struct SkQuadCoeff {
SkQuadCoeffSkQuadCoeff286 SkQuadCoeff() {}
287
SkQuadCoeffSkQuadCoeff288 SkQuadCoeff(const Sk2s& A, const Sk2s& B, const Sk2s& C)
289 : fA(A)
290 , fB(B)
291 , fC(C)
292 {
293 }
294
SkQuadCoeffSkQuadCoeff295 SkQuadCoeff(const SkPoint src[3]) {
296 fC = from_point(src[0]);
297 Sk2s P1 = from_point(src[1]);
298 Sk2s P2 = from_point(src[2]);
299 fB = times_2(P1 - fC);
300 fA = P2 - times_2(P1) + fC;
301 }
302
evalSkQuadCoeff303 Sk2s eval(SkScalar t) {
304 Sk2s tt(t);
305 return eval(tt);
306 }
307
evalSkQuadCoeff308 Sk2s eval(const Sk2s& tt) {
309 return (fA * tt + fB) * tt + fC;
310 }
311
312 Sk2s fA;
313 Sk2s fB;
314 Sk2s fC;
315 };
316
317 struct SkConicCoeff {
SkConicCoeffSkConicCoeff318 SkConicCoeff(const SkConic& conic) {
319 Sk2s p0 = from_point(conic.fPts[0]);
320 Sk2s p1 = from_point(conic.fPts[1]);
321 Sk2s p2 = from_point(conic.fPts[2]);
322 Sk2s ww(conic.fW);
323
324 Sk2s p1w = p1 * ww;
325 fNumer.fC = p0;
326 fNumer.fA = p2 - times_2(p1w) + p0;
327 fNumer.fB = times_2(p1w - p0);
328
329 fDenom.fC = Sk2s(1);
330 fDenom.fB = times_2(ww - fDenom.fC);
331 fDenom.fA = Sk2s(0) - fDenom.fB;
332 }
333
evalSkConicCoeff334 Sk2s eval(SkScalar t) {
335 Sk2s tt(t);
336 Sk2s numer = fNumer.eval(tt);
337 Sk2s denom = fDenom.eval(tt);
338 return numer / denom;
339 }
340
341 SkQuadCoeff fNumer;
342 SkQuadCoeff fDenom;
343 };
344
345 struct SkCubicCoeff {
SkCubicCoeffSkCubicCoeff346 SkCubicCoeff(const SkPoint src[4]) {
347 Sk2s P0 = from_point(src[0]);
348 Sk2s P1 = from_point(src[1]);
349 Sk2s P2 = from_point(src[2]);
350 Sk2s P3 = from_point(src[3]);
351 Sk2s three(3);
352 fA = P3 + three * (P1 - P2) - P0;
353 fB = three * (P2 - times_2(P1) + P0);
354 fC = three * (P1 - P0);
355 fD = P0;
356 }
357
evalSkCubicCoeff358 Sk2s eval(SkScalar t) {
359 Sk2s tt(t);
360 return eval(tt);
361 }
362
evalSkCubicCoeff363 Sk2s eval(const Sk2s& t) {
364 return ((fA * t + fB) * t + fC) * t + fD;
365 }
366
367 Sk2s fA;
368 Sk2s fB;
369 Sk2s fC;
370 Sk2s fD;
371 };
372
373 }
374
375 #include "SkTemplates.h"
376
377 /**
378 * Help class to allocate storage for approximating a conic with N quads.
379 */
380 class SkAutoConicToQuads {
381 public:
SkAutoConicToQuads()382 SkAutoConicToQuads() : fQuadCount(0) {}
383
384 /**
385 * Given a conic and a tolerance, return the array of points for the
386 * approximating quad(s). Call countQuads() to know the number of quads
387 * represented in these points.
388 *
389 * The quads are allocated to share end-points. e.g. if there are 4 quads,
390 * there will be 9 points allocated as follows
391 * quad[0] == pts[0..2]
392 * quad[1] == pts[2..4]
393 * quad[2] == pts[4..6]
394 * quad[3] == pts[6..8]
395 */
computeQuads(const SkConic & conic,SkScalar tol)396 const SkPoint* computeQuads(const SkConic& conic, SkScalar tol) {
397 int pow2 = conic.computeQuadPOW2(tol);
398 fQuadCount = 1 << pow2;
399 SkPoint* pts = fStorage.reset(1 + 2 * fQuadCount);
400 fQuadCount = conic.chopIntoQuadsPOW2(pts, pow2);
401 return pts;
402 }
403
computeQuads(const SkPoint pts[3],SkScalar weight,SkScalar tol)404 const SkPoint* computeQuads(const SkPoint pts[3], SkScalar weight,
405 SkScalar tol) {
406 SkConic conic;
407 conic.set(pts, weight);
408 return computeQuads(conic, tol);
409 }
410
countQuads()411 int countQuads() const { return fQuadCount; }
412
413 private:
414 enum {
415 kQuadCount = 8, // should handle most conics
416 kPointCount = 1 + 2 * kQuadCount,
417 };
418 SkAutoSTMalloc<kPointCount, SkPoint> fStorage;
419 int fQuadCount; // #quads for current usage
420 };
421
422 #endif
423