This repository was archived by the owner on Apr 2, 2026. It is now read-only.
Repository navigation
Expand file tree
/
Copy pathcubic.go
More file actions
739 lines (668 loc) · 20.5 KB
/
Copy pathcubic.go
File metadata and controls
739 lines (668 loc) · 20.5 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
package curve
import (
10BC8
"iter"
"math"
"slices"
"sort"
"honnef.co/go/stuff/math/polyroot"
)
const maxSplineSplit = 100
var _ Shape = CubicBez{}
var _ ParametricCurve = CubicBez{}
type CubicBez struct {
P0 Point
P1 Point
P2 Point
P3 Point
}
func (q CubicBez) IsInf() bool {
return q.P0.IsInf() || q.P1.IsInf() || q.P2.IsInf() || q.P3.IsInf()
}
func (q CubicBez) IsNaN() bool {
return q.P0.IsNaN() || q.P1.IsNaN() || q.P2.IsNaN() || q.P3.IsNaN()
}
// BoundingBox implements [Shape].
func (c CubicBez) BoundingBox() Rect {
return BoundingBox(c)
}
func (c CubicBez) Path(tolerance float64) BezPath { return slices.Collect(c.PathElements(tolerance)) }
// PathElements implements [Shape].
func (c CubicBez) PathElements(tolerance float64) iter.Seq[PathElement] {
return func(yield func(PathElement) bool) {
_ = yield(MoveTo(c.P0)) &&
yield(CubicTo(c.P1, c.P2, c.P3))
}
}
// Perimeter implements [Shape].
func (c CubicBez) Perimeter(accuracy float64) float64 {
return c.Arclen(accuracy)
}
// Winding implements [Shape].
func (c CubicBez) Winding(pt Point) int {
return 0
}
// Arclen returns the arclength of a cubic Bézier segment.
//
// This is an adaptive subdivision approach using Legendre-Gauss quadrature
func (c CubicBez) Arclen(accuracy float64) float64 {
return c.arclen(accuracy, 0)
}
func (c CubicBez) arclen(accuracy float64, depth int) float64 {
d03 := c.P3.Sub(c.P0)
d01 := c.P1.Sub(c.P0)
d12 := c.P2.Sub(c.P1)
d23 := c.P3.Sub(c.P2)
lplc := d01.Hypot() + d12.Hypot() + d23.Hypot() - d03.Hypot()
dd1 := d12.Sub(d01)
dd2 := d23.Sub(d12)
// It might be faster to do direct multiplies, the data dependencies would be shorter.
// The following values don't have the factor of 3 for first deriv
dm := d01.Add(d23).Mul(0.25).Add(d12.Mul(0.5)) // first derivative at midpoint
dm1 := dd2.Add(dd1).Mul(0.5) // second derivative at midpoint
dm2 := dd2.Sub(dd1).Mul(0.25) // 0.5 * (third derivative at midpoint)
var est float64
for _, coeff := range gaussLegendreCoeffs8 {
wi, xi := coeff[0], coeff[1]
dNorm2 := dm.Add(dm1.Mul(xi)).Add(dm2.Mul(xi * xi)).Hypot2()
ddNorm2 := dm1.Add(dm2.Mul(2.0 * xi)).Hypot2()
f := ddNorm2 / dNorm2
est += wi * f
}
if math.IsNaN(est) {
// dNorm2 will be 0 as c approaches a singularity
est = 0
}
estGauss8Error := min(pow3(est)*2.5e-6, 3e-2) * lplc
if estGauss8Error < accuracy {
return arclenQuadratureCore(gaussLegendreCoeffs8Half[:], dm, dm1, dm2)
}
estGauss16Error := min(pow6(est)*1.5e-11, 9e-3) * lplc
if estGauss16Error < accuracy {
return arclenQuadratureCore(gaussLegendreCoeffs16Half[:], dm, dm1, dm2)
}
estGauss24Error := min(pow9(est)*3.5e-16, 3.5e-3) * lplc
if estGauss24Error < accuracy || depth >= 20 {
return arclenQuadratureCore(gaussLegendreCoeffs24Half[:], dm, dm1, dm2)
}
c0, c1 := c.Subdivide()
return c0.arclen(accuracy*0.5, depth+1) + c1.arclen(accuracy*0.5, depth+1)
}
func (cb CubicBez) Eval(t float64) Point {
mt := 1.0 - t
a := Vec2(cb.P0).Mul(mt * mt * mt)
b := Vec2(cb.P1).Mul(mt * mt * 3.0)
c := Vec2(cb.P2).Mul(mt * 3.0)
d := Vec2(cb.P3)
v := a.Add(b.Add(c.Add(d.Mul(t)).Mul(t)).Mul(t))
return Point(v)
}
// Subdivide subdivides the cubic into halves, using de Casteljau.
func (c CubicBez) Subdivide() (CubicBez, CubicBez) {
pm := c.Eval(0.5)
return CubicBez{
c.P0,
c.P0.Midpoint(c.P1),
Point(Vec2(c.P0).Add(Vec2(c.P1).Mul(2.0)).Add(Vec2(c.P2)).Mul(0.25)),
pm,
},
CubicBez{
pm,
Point(Vec2(c.P1).Add(Vec2(c.P2).Mul(2.0)).Add(Vec2(c.P3)).Mul(0.25)),
c.P2.Midpoint(c.P3),
c.P3,
}
}
// SubdivideCurve subdivides the cubic into halves, using de Casteljau.
func (c CubicBez) SubdivideCurve() (ParametricCurve, ParametricCurve) {
return c.Subdivide()
}
func (c CubicBez) Start() Point {
return c.P0
}
func (c CubicBez) End() Point {
return c.P3
}
type CubicToQuadraticSegment struct {
Start, End float64
Segment QuadBez
}
// Quadratics converts the cubic Béziers to quadratic Béziers.
//
// The iterator returns the start and end parameter in the cubic of each quadratic
// segment, along with the quadratic.
//
// Note that the resulting quadratic Béziers are not in general G1 continuous;
// they are optimized for minimizing distance error.
//
// This iterator will always produce at least one value.
func (c CubicBez) Quadratics(accuracy float64) iter.Seq[CubicToQuadraticSegment] {
// The maximum error, as a vector from the cubic to the best approximating
// quadratic, is proportional to the third derivative, which is constant
// across the segment. Thus, the error scales down as the third power of
// the number of subdivisions. Our strategy then is to subdivide t evenly.
//
// This is an overestimate of the error because only the component
// perpendicular to the first derivative is important. But the simplicity is
// appealing.
return func(yield func(CubicToQuadraticSegment) bool) {
// This magic number is the square of 36 / sqrt(3).
// See: https://web.archive.org/web/20210108052742/http://caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
maxHypot2 := 432.0 * accuracy * accuracy
p1x2 := Vec2(c.P1).Mul(3).Sub(Vec2(c.P0))
p2x2 := Vec2(c.P2).Mul(3).Sub(Vec2(c.P3))
err := p2x2.Sub(p1x2).Hypot2()
n := max(int(math.Ceil(math.Sqrt(math.Cbrt(err/maxHypot2)))), 1)
for i := range n {
t0 := float64(i) / float64(n)
t1 := float64(i+1) / float64(n)
seg := c.Subsegment(t0, t1)
p1x2 := Vec2(seg.P1).Mul(3).Sub(Vec2(seg.P0))
p2x2 := Vec2(seg.P2).Mul(3).Sub(Vec2(seg.P3))
result := QuadBez{seg.P0, Point(p1x2.Add(p2x2).Mul(1.0 / 4.0)), seg.P3}
if !yield(CubicToQuadraticSegment{t0, t1, result}) {
return
}
}
}
}
func (c CubicBez) Subsegment(t0, t1 float64) CubicBez {
p0 := c.Eval(t0)
p3 := c.Eval(t1)
d := c.Differentiate()
scale := (t1 - t0) * (1.0 / 3.0)
p1 := p0.Translate(Vec2(d.Eval(t0)).Mul(scale))
p2 := p3.Translate(Vec2(d.Eval(t1)).Mul(scale).Negate())
return CubicBez{p0, p1, p2, p3}
}
func (c CubicBez) SubsegmentCurve(t0, t1 float64) ParametricCurve {
return c.Subsegment(t0, t1)
}
func (c CubicBez) Differentiate() QuadBez {
return QuadBez{
Point(c.P1.Sub(c.P0).Mul(3)),
Point(c.P2.Sub(c.P1).Mul(3)),
Point(c.P3.Sub(c.P2).Mul(3)),
}
}
func arclenQuadratureCore(coeffs [][2]float64, dm Vec2, dm1 Vec2, dm2 Vec2) float64 {
var sum float64
for _, coeff := range coeffs {
wi, xi := coeff[0], coeff[1]
d := dm.Add(dm2.Mul(xi * xi))
dpx := d.Add(dm1.Mul(xi)).Hypot()
dmx := d.Sub(dm1.Mul(xi)).Hypot()
sum += math.Sqrt(2.25) * wi * (dpx + dmx)
}
return sum
}
func (c CubicBez) Extrema() ([MaxExtrema]float64, int) {
// two calls to oneCoord, up to 2 roots per call, for a total of 4 possible values.
var out [MaxExtrema]float64
var outN int
oneCoord := func(d0, d1, d2 float64) {
a := d0 - 2*d1 + d2
b := 2 * (d1 - d0)
c := d0
roots, n := SolveQuadratic(c, b, a)
for _, t := range roots[:n] {
if t > 0.0 && t < 1.0 {
out[outN] = t
outN++
}
}
}
d0 := c.P1.Sub(c.P0)
d1 := c.P2.Sub(c.P1)
d2 := c.P3.Sub(c.P2)
oneCoord(d0.X, d1.X, d2.X)
oneCoord(d0.Y, d1.Y, d2.Y)
sort.Float64s(out[:outN])
return out, outN
}
// regularizeCusp preprocesses a cubic Bézier to ease numerical robustness.
//
// If the cubic Bézier segment has zero or near-zero derivatives as an interior
// cusp, perturb the control points to make curvature finite, avoiding
// numerical robustness problems in offset and stroke.
func (c CubicBez) regularizeCusp(dimension float64) CubicBez {
out := c
// If control point is too near the endpoint, nudge it away along the
// tangent.
if cuspType, ok := c.detectCusp(dimension); ok {
d01 := out.P1.Sub(out.P0)
d01h := d01.Hypot()
d23 := out.P3.Sub(out.P2)
d23h := d23.Hypot()
switch cuspType {
case CuspTypeLoop:
out.P1 = out.P1.Translate(d01.Mul(dimension / d01h))
out.P2 = out.P2.Translate(d23.Mul(dimension / d23h).Negate())
case CuspTypeDoubleInflection:
// Avoid making control distance smaller than dimension
if d01h > 2.0*dimension {
out.P1 = out.P1.Translate(d01.Mul(dimension / d01h).Negate())
}
if d23h > 2.0*dimension {
out.P2 = out.P2.Translate(d23.Mul(dimension / d23h))
}
}
}
return out
}
// CustType defines the classification result for cusp detection.
type CuspType int
const (
// Cusp is a loop.
CuspTypeLoop CuspType = iota + 1
// Cusp has two closely spaced inflection points.
CuspTypeDoubleInflection
)
// detectCusp detects whether there is a cusp.
//
// It returns a cusp classification if there is a cusp with curvature greater than
// the reciprocal of the given dimension.
func (c CubicBez) detectCusp(dimension float64) (CuspType, bool) {
d01 := c.P1.Sub(c.P0)
d02 := c.P2.Sub(c.P0)
d03 := c.P3.Sub(c.P0)
d12 := c.P2.Sub(c.P1)
d23 := c.P3.Sub(c.P2)
det012 := d01.Cross(d02)
det123 := d12.Cross(d23)
det013 := d01.Cross(d03)
det023 := d02.Cross(d03)
if det012*det123 > 0.0 && det012*det013 < 0.0 && det012*det023 < 0.0 {
q := c.Differentiate()
// accuracy isn't used for quadratic nearest
nearestDist, nearestT := q.Nearest(Point{}, 1e-9)
// detect whether curvature at minimum derivative exceeds 1/dimension,
// without division.
d := q.Eval(nearestT)
d2 := q.Differentiate().Eval(nearestT)
cross := Vec2(d).Cross(Vec2(d2))
if nearestDist*nearestDist*nearestDist <= cross*dimension*cross*dimension {
a := 3.0*det012 + det023 - 2.0*det013
b := -3.0*det012 + det013
c := det012
d := b*b - 4.0*a*c
if d > 0.0 {
return CuspTypeDoubleInflection, true
} else {
return CuspTypeLoop, true
}
}
}
return 0, false
}
func (c CubicBez) Transform(aff Affine) CubicBez {
return CubicBez{
P0: c.P0.Transform(aff),
P1: c.P1.Transform(aff),
P2: c.P2.Transform(aff),
P3: c.P3.Transform(aff),
}
}
// Nearest finds the nearest point, using a quintic solver.
func (c CubicBez) Nearest(pt Point, accuracy float64) (distSq, t float64) {
var (
tBest float64
rBest = math.Inf(1)
)
evalT := func(pt Point, t float64, p0 Point) {
r := p0.Sub(pt).Hypot2()
if r < rBest {
rBest = r
tBest = t
}
}
// Reparameterize 'c - p' as 'q0 + q1 t + q2 t² + q3 t³'.
q0 := c.P0.Sub(pt)
q1 := (c.P1.Sub(c.P0)).Mul(3.0)
q2 := (Vec2(c.P0).Sub(Vec2(c.P1).Mul(2)).Add(Vec2(c.P2))).Mul(3)
q3 := Vec2(c.P0).Negate().Add(Vec2(c.P1).Mul(3)).Sub(Vec2(c.P2).Mul(3)).Add(Vec2(c.P3))
// Coefficients of the degree-5 polynomial (c - p) · tangent.
c0 := q0.Dot(q1)
c1 := q1.Hypot2() + 2.0*q2.Dot(q0)
c2 := 3.0 * (q2.Dot(q1) + q3.Dot(q0))
c3 := 4.0*q3.Dot(q1) + 2.0*q2.Hypot2()
c4 := 5.0 * q3.Dot(q2)
c5 := 3.0 * q3.Hypot2()
roots := polyroot.NewPolynomial(c0, c1, c2, c3, c4, c5).Roots(0, 1, accuracy, nil)
for _, t := range roots {
evalT(pt, t, c.Eval(t))
}
// If we found all 5 critical points, we can skip evaluating the endpoints.
if len(roots) != 5 {
evalT(pt, 0.0, c.P0)
evalT(pt, 1.0, c.P3)
}
return rBest, tBest
}
func (c CubicBez) SignedArea() float64 {
v := c.P0.X*(6.0*c.P1.Y+3.0*c.P2.Y+c.P3.Y) +
3.0*(c.P1.X*(-2.0*c.P0.Y+c.P2.Y+c.P3.Y)-c.P2.X*(c.P0.Y+c.P1.Y-2.0*c.P3.Y)) -
c.P3.X*(c.P0.Y+3.0*c.P1.Y+6.0*c.P2.Y)
return v * (1.0 / 20.0)
}
func (c CubicBez) Tangents() (Vec2, Vec2) {
const epsilon = 1e-12
d01 := c.P1.Sub(c.P0)
var d0, d1 Vec2
if d01.Hypot2() > epsilon {
d0 = d01
} else {
d02 := c.P2.Sub(c.P0)
if d02.Hypot2() > epsilon {
d0 = d02
} else {
d0 = c.P3.Sub(c.P0)
}
}
d23 := c.P3.Sub(c.P2)
if d23.Hypot2() > epsilon {
d1 = d23
} else {
d13 := c.P3.Sub(c.P1)
if d13.Hypot2() > epsilon {
d1 = d13
} else {
d1 = c.P3.Sub(c.P0)
}
}
return d0, d1
}
func (c CubicBez) Seg() PathSegment {
return PathSegment{Kind: CubicKind, P0: c.P0, P1: c.P1, P2: c.P2, P3: c.P3}
}
func (c CubicBez) IntersectLine(line Line) ([3]LineIntersection, int) {
const epsilon = 1e-9
p0 := line.P0
p1 := line.P1
dx := p1.X - p0.X
dy := p1.Y - p0.Y
// The basic technique here is to determine x and y as a cubic polynomial
// as a function of t. Then plug those values into the line equation for the
// probe line (giving a sort of signed distance from the probe line) and solve
// that for t.
px0, px1, px2, px3 := cubicBezCoefficients(c.P0.X, c.P1.X, c.P2.X, c.P3.X)
py0, py1, py2, py3 := cubicBezCoefficients(c.P0.Y, c.P1.Y, c.P2.Y, c.P3.Y)
c0 := dy*(px0-p0.X) - dx*(py0-p0.Y)
c1 := dy*px1 - dx*py1
c2 := dy*px2 - dx*py2
c3 := dy*px3 - dx*py3
invlen2 := 1.0 / (dx*dx + dy*dy)
ts, n := SolveCubic(c0, c1, c2, c3)
var ret [3]LineIntersection
var retN int
for _, t := range ts[:n] {
if t >= -epsilon && t <= 1+epsilon {
x := px0 + t*px1 + t*t*px2 + t*t*t*px3
y := py0 + t*py1 + t*t*py2 + t*t*t*py3
u := ((x-p0.X)*dx + (y-p0.Y)*dy) * invlen2
if u >= 0.0 && u <= 1.0 {
ret[retN] = LineIntersection{u, t}
retN++
}
}
}
return ret, retN
}
// Return polynomial coefficients given cubic bezier coordinates.
func cubicBezCoefficients(x0, x1, x2, x3 float64) (_, _, _, _ float64) {
p0 := x0
p1 := 3.0*x1 - 3.0*x0
p2 := 3.0*x2 - 6.0*x1 + 3.0*x0
p3 := x3 - 3.0*x2 + 3.0*x1 - x0
return p0, p1, p2, p3
}
// Inflections returns the inflection points.
//
// The function returns up to two inflection points in the first return
// parameter, with the second parameter specifying the number of points
// returned.
func (cb CubicBez) Inflections() ([2]float64, int) {
a := cb.P1.Sub(cb.P0)
b := cb.P2.Sub(cb.P1).Sub(a)
c := cb.P3.Sub(cb.P0).Sub(cb.P2.Sub(cb.P1).Mul(3))
nums, n := SolveQuadratic(a.Cross(b), a.Cross(c), b.Cross(c))
var out [2]float64
var outN int
for _, num := range nums[:n] {
if num >= 0 && num <= 1 {
out[outN] = num
outN++
}
}
return out, outN
}
// ApproxQuadSpline returns a quadratic B-spline approximating this cubic
// Bézier.
//
// Returns false if no suitable approximation is found within the given
// tolerance.
func (cb CubicBez) ApproxQuadSpline(accuracy float64) (QuadBSpline, bool) {
for i := range maxSplineSplit {
if spline, ok := cb.approxQuadSplineN(i+1, accuracy); ok {
return spline, true
}
}
return nil, false
}
// Approximate a cubic curve with a quadratic spline of n curves
func (c CubicBez) approxQuadSplineN(n int, accuracy float64) (QuadBSpline, bool) {
if n == 1 {
qb, ok := c.tryApproxQuadratic(accuracy)
if !ok {
return nil, false
}
return QuadBSpline{qb.P0, qb.P1, qb.P2}, true
}
cubicsNext_, cubicsDone := iter.Pull(c.splitIntoN(n))
defer cubicsDone()
cubicsNext := func() CubicBez {
v, ok := cubicsNext_()
if !ok {
panic("unreachable")
}
return v
}
nextCubic := cubicsNext()
nextQ1 := nextCubic.approxQuadControl(0.0)
q2 := c.P0
var d1 Vec2
spline := []Point{c.P0, nextQ1}
for i := 1; i <= n; i++ {
currentCubic := nextCubic
q0 := q2
q1 := nextQ1
if i < n {
nextCubic = cubicsNext()
nextQ1 = nextCubic.approxQuadControl(float64(i) / float64(n-1))
spline = append(spline, nextQ1)
q2 = q1.Midpoint(nextQ1)
} else {
q2 = currentCubic.P3
}
d0 := d1
d1 = Vec2(q2).Sub(Vec2(currentCubic.P3))
if d1.Hypot() > accuracy ||
!(CubicBez{
Point(d0),
// XXX is negate fine
q0.Lerp(q1, 2.0/3.0).Translate(Vec2(currentCubic.P1).Negate()),
q2.Lerp(q1, 2.0/3.0).Translate(Vec2(currentCubic.P2).Negate()),
Point(d1),
}.fitsInside(accuracy)) {
return nil, false
}
}
spline = append(spline, c.P3)
return QuadBSpline(spline), true
}
func (c CubicBez) splitIntoN(n int) iter.Seq[CubicBez] {
// We special case some values of n so that we produce the same results as Kurbo,
// which in turn produces the same results as cu2qu.
switch n {
case 1:
return func(yield func(CubicBez) bool) { yield(c) }
case 2:
return func(yield func(CubicBez) bool) {
l, r := c.Subdivide()
_ = yield(l) && yield(r)
}
case 3:
return func(yield func(CubicBez) bool) {
l, m, r := c.subdivide3()
_ = yield(l) && yield(m) && yield(r)
}
case 4:
return func(yield func(CubicBez) bool) {
l, r := c.Subdivide()
ll, lr := l.Subdivide()
rl, rr := r.Subdivide()
_ = yield(ll) &&
yield(lr) &&
yield(rl) &&
yield(rr)
}
case 6:
return func(yield func(CubicBez) bool) {
l, r := c.Subdivide()
l1, l2, l3 := l.subdivide3()
r1, r2, r3 := r.subdivide3()
_ = yield(l1) &&
yield(l2) &&
yield(l3) &&
yield(r1) &&
yield(r2) &&
yield(r3)
}
}
return func(yield func(CubicBez) bool) {
a, b, c, d := c.parameters()
dt := 1.0 / float64(n)
delta2 := dt * dt
delta3 := dt * delta2
for i := range n {
t1 := float64(i) * dt
t1_2 := t1 * t1
a1 := a.Mul(delta3)
// Note that we don't simplify a * 3.0 * t1 to a * (3.0 * t1) because
// multiplication isn't associative in floating point math, and we want our
// results to match cu2qu's.
b1 := a.Mul(3.0).Mul(t1).Add(b).Mul(delta2)
c1 := b.Mul(2.0).Mul(t1).Add(c).Add(a.Mul(3.0).Mul(t1_2)).Mul(dt)
d1 := a.Mul(t1).Mul(t1_2).Add(b.Mul(t1_2)).Add(c.Mul(t1)).Add(d)
// Rust port of cu2qu [calc_cubic_points](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L63-L68)
p0 := Point(d1)
p1 := Point(c1.Div(3)).Translate(d1)
p2 := Point(b1.Add(c1).Div(3)).Translate(Vec2(p1))
p3 := Point(a1.Add(d1).Add(c1).Add(b1))
result := CubicBez{p0, p1, p2, p3}
if !yield(result) {
break
}
}
}
}
func (cb CubicBez) parameters() (Vec2, Vec2, Vec2, Vec2) {
c := cb.P1.Sub(cb.P0).Mul(3.0)
b := cb.P2.Sub(cb.P1).Mul(3.0).Sub(c)
d := Vec2(cb.P0)
a := Vec2(cb.P3).Sub(d).Sub(c).Sub(b)
return a, b, c, d
}
func (c CubicBez) subdivide3() (CubicBez, CubicBez, CubicBez) {
p0, p1, p2, p3 :=
Vec2(c.P0),
Vec2(c.P1),
Vec2(c.P2),
Vec2(c.P3)
// The original Python cu2qu code does not use the division operator to divide by
// 27 but instead uses multiplication by the reciprocal 1 / 27. We want to match
// it exactly to avoid any floating point differences.
//
// Source: https://github.com/fonttools/fonttools/blob/85c80be/Lib/fontTools/cu2qu/cu2qu.py#L215-L218
// See also: https://github.com/linebender/kurbo/issues/272
mid1 := Point(p0.Mul(8).Add(p1.Mul(12)).Add(p2.Mul(6)).Add(p3).Mul(1.0 / 27.0))
deriv1 := p3.Add(p2.Mul(3)).Sub(p0.Mul(4)).Mul(1.0 / 27.0)
mid2 := Point(p0.Add(p1.Mul(6)).Add(p2.Mul(12)).Add(p3.Mul(8)).Mul(1.0 / 27.0))
deriv2 := p3.Mul(4).Sub(p1.Mul(3)).Sub(p0).Mul(1.0 / 27.0)
left := CubicBez{
c.P0,
Point(p0.Mul(2).Add(p1).Div(3)),
// XXX is mid2 + (-deriv2) the same as mid2 - deriv2, under float math?
mid1.Translate(deriv1.Negate()),
mid1,
}
// XXX is mid2 + (-deriv2) the same as mid2 - deriv2, under float math?
mid := CubicBez{mid1, mid1.Translate(deriv1), mid2.Translate(deriv2.Negate()), mid2}
right := CubicBez{
mid2,
mid2.Translate(deriv2),
Point(p2.Add(p3.Mul(2)).Div(3)),
c.P3,
}
return left, mid, right
}
// Does this curve fit inside the given distance from the origin?
//
// Rust port of cu2qu [cubic_farthest_fit_inside](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L281)
func (c CubicBez) fitsInside(distance float64) bool {
if Vec2(c.P2).Hypot() <= distance && Vec2(c.P1).Hypot() <= distance {
return true
}
mid := Vec2(c.P0).Add(Vec2(c.P1).Add(Vec2(c.P2)).Mul(3)).Add(Vec2(c.P3)).Mul(0.125)
if mid.Hypot() > distance {
return false
}
// Split in two. Note that cu2qu here uses a 3/8 subdivision. I don't know why.
left, right := c.Subdivide()
return left.fitsInside(distance) && right.fitsInside(distance)
}
func (c CubicBez) approxQuadControl(t float64) Point {
p1 := c.P0.Translate(c.P1.Sub(c.P0).Mul(1.5))
p2 := c.P3.Translate(c.P2.Sub(c.P3).Mul(1.5))
return p1.Lerp(p2, t)
}
// Approximate a cubic with a single quadratic
//
// Returns a quadratic approximating the given cubic that maintains
// endpoint tangents if that is within tolerance, or false otherwise.
func (c CubicBez) tryApproxQuadratic(accuracy float64) (QuadBez, bool) {
q1, ok := Line{c.P0, c.P1}.CrossingPoint(Line{c.P2, c.P3})
if !ok {
return QuadBez{}, false
}
c1 := c.P0.Lerp(q1, 2.0/3.0)
c2 := c.P3.Lerp(q1, 2.0/3.0)
if !(CubicBez{
Point{},
// XXX negate ok?
c1.Translate(Vec2(c.P1).Negate()),
c2.Translate(Vec2(c.P2).Negate()),
Point{},
}.fitsInside(accuracy)) {
return QuadBez{}, false
}
return QuadBez{c.P0, q1, c.P3}, true
}
// CubicsToQuadraticSplines converts multiple cubic Bézier curves to quadratic
// splines. It ensures that the resulting splines have the same number of
// control points.
func CubicsToQuadraticSplines(curves []CubicBez, accuracy float64) ([]QuadBSpline, bool) {
result := make([]QuadBSpline, 0, len(curves))
outer:
for i := 1; i <= maxSplineSplit; i++ {
result = result[:0]
for _, curve := range curves {
spline, ok := curve.approxQuadSplineN(i, accuracy)
if !ok {
continue outer
}
result = append(result, spline)
}
return result, true
}
return nil, false
}