-
Notifications
You must be signed in to change notification settings - Fork 0
/
stroke.go
748 lines (670 loc) · 19 KB
/
stroke.go
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
740
741
742
743
744
745
746
747
748
// SPDX-License-Identifier: Unlicense OR MIT
// Most of the algorithms to compute strokes and their offsets have been
// extracted, adapted from (and used as a reference implementation):
// - github.com/tdewolff/canvas (Licensed under MIT)
//
// These algorithms have been implemented from:
// Fast, precise flattening of cubic Bézier path and offset curves
// Thomas F. Hain, et al.
//
// An electronic version is available at:
// https://seant23.files.wordpress.com/2010/11/fastpreciseflatteningofbeziercurve.pdf
//
// Possible improvements (in term of speed and/or accuracy) on these
// algorithms are:
//
// - Polar Stroking: New Theory and Methods for Stroking Paths,
// M. Kilgard
// https://arxiv.org/pdf/2007.00308.pdf
//
// - https://raphlinus.github.io/graphics/curves/2019/12/23/flatten-quadbez.html
// R. Levien
// Package stroke implements conversion of strokes to filled outlines. It is used as a
// fallback for stroke configurations not natively supported by the renderer.
package stroke
import (
"encoding/binary"
"math"
"github.com/Seikaijyu/gio/internal/f32"
"github.com/Seikaijyu/gio/internal/ops"
"github.com/Seikaijyu/gio/internal/scene"
)
// The following are copies of types from op/clip to avoid a circular import of
// that package.
// TODO: when the old renderer is gone, this package can be merged with
// op/clip, eliminating the duplicate types.
type StrokeStyle struct {
Width float32
}
// strokeTolerance is used to reconcile rounding errors arising
// when splitting quads into smaller and smaller segments to approximate
// them into straight lines, and when joining back segments.
//
// The magic value of 0.01 was found by striking a compromise between
// aesthetic looking (curves did look like curves, even after linearization)
// and speed.
const strokeTolerance = 0.01
type QuadSegment struct {
From, Ctrl, To f32.Point
}
type StrokeQuad struct {
Contour uint32
Quad QuadSegment
}
type strokeState struct {
p0, p1 f32.Point // p0 is the start point, p1 the end point.
n0, n1 f32.Point // n0 is the normal vector at the start point, n1 at the end point.
r0, r1 float32 // r0 is the curvature at the start point, r1 at the end point.
ctl f32.Point // ctl is the control point of the quadratic Bézier segment.
}
type StrokeQuads []StrokeQuad
func (qs *StrokeQuads) pen() f32.Point {
return (*qs)[len(*qs)-1].Quad.To
}
func (qs *StrokeQuads) lineTo(pt f32.Point) {
end := qs.pen()
*qs = append(*qs, StrokeQuad{
Quad: QuadSegment{
From: end,
Ctrl: end.Add(pt).Mul(0.5),
To: pt,
},
})
}
func (qs *StrokeQuads) arc(f1, f2 f32.Point, angle float32) {
pen := qs.pen()
m, segments := ArcTransform(pen, f1.Add(pen), f2.Add(pen), angle)
for i := 0; i < segments; i++ {
p0 := qs.pen()
p1 := m.Transform(p0)
p2 := m.Transform(p1)
ctl := p1.Mul(2).Sub(p0.Add(p2).Mul(.5))
*qs = append(*qs, StrokeQuad{
Quad: QuadSegment{
From: p0, Ctrl: ctl, To: p2,
},
})
}
}
// split splits a slice of quads into slices of quads grouped
// by contours (ie: splitted at move-to boundaries).
func (qs StrokeQuads) split() []StrokeQuads {
if len(qs) == 0 {
return nil
}
var (
c uint32
o []StrokeQuads
i = len(o)
)
for _, q := range qs {
if q.Contour != c {
c = q.Contour
i = len(o)
o = append(o, StrokeQuads{})
}
o[i] = append(o[i], q)
}
return o
}
func (qs StrokeQuads) stroke(stroke StrokeStyle) StrokeQuads {
var (
o StrokeQuads
hw = 0.5 * stroke.Width
)
for _, ps := range qs.split() {
rhs, lhs := ps.offset(hw, stroke)
switch lhs {
case nil:
o = o.append(rhs)
default:
// Closed path.
// Inner path should go opposite direction to cancel outer path.
switch {
case ps.ccw():
lhs = lhs.reverse()
o = o.append(rhs)
o = o.append(lhs)
default:
rhs = rhs.reverse()
o = o.append(lhs)
o = o.append(rhs)
}
}
}
return o
}
// offset returns the right-hand and left-hand sides of the path, offset by
// the half-width hw.
// The stroke handles how segments are joined and ends are capped.
func (qs StrokeQuads) offset(hw float32, stroke StrokeStyle) (rhs, lhs StrokeQuads) {
var (
states []strokeState
beg = qs[0].Quad.From
end = qs[len(qs)-1].Quad.To
closed = beg == end
)
for i := range qs {
q := qs[i].Quad
var (
n0 = strokePathNorm(q.From, q.Ctrl, q.To, 0, hw)
n1 = strokePathNorm(q.From, q.Ctrl, q.To, 1, hw)
r0 = strokePathCurv(q.From, q.Ctrl, q.To, 0)
r1 = strokePathCurv(q.From, q.Ctrl, q.To, 1)
)
states = append(states, strokeState{
p0: q.From,
p1: q.To,
n0: n0,
n1: n1,
r0: r0,
r1: r1,
ctl: q.Ctrl,
})
}
for i, state := range states {
rhs = rhs.append(strokeQuadBezier(state, +hw, strokeTolerance))
lhs = lhs.append(strokeQuadBezier(state, -hw, strokeTolerance))
// join the current and next segments
if hasNext := i+1 < len(states); hasNext || closed {
var next strokeState
switch {
case hasNext:
next = states[i+1]
case closed:
next = states[0]
}
if state.n1 != next.n0 {
strokePathRoundJoin(&rhs, &lhs, hw, state.p1, state.n1, next.n0, state.r1, next.r0)
}
}
}
if closed {
rhs.close()
lhs.close()
return rhs, lhs
}
qbeg := &states[0]
qend := &states[len(states)-1]
// Default to counter-clockwise direction.
lhs = lhs.reverse()
strokePathCap(stroke, &rhs, hw, qend.p1, qend.n1)
rhs = rhs.append(lhs)
strokePathCap(stroke, &rhs, hw, qbeg.p0, qbeg.n0.Mul(-1))
rhs.close()
return rhs, nil
}
func (qs *StrokeQuads) close() {
p0 := (*qs)[len(*qs)-1].Quad.To
p1 := (*qs)[0].Quad.From
if p1 == p0 {
return
}
*qs = append(*qs, StrokeQuad{
Quad: QuadSegment{
From: p0,
Ctrl: p0.Add(p1).Mul(0.5),
To: p1,
},
})
}
// ccw returns whether the path is counter-clockwise.
func (qs StrokeQuads) ccw() bool {
// Use the Shoelace formula:
// https://en.wikipedia.org/wiki/Shoelace_formula
var area float32
for _, ps := range qs.split() {
for i := 1; i < len(ps); i++ {
pi := ps[i].Quad.To
pj := ps[i-1].Quad.To
area += (pi.X - pj.X) * (pi.Y + pj.Y)
}
}
return area <= 0.0
}
func (qs StrokeQuads) reverse() StrokeQuads {
if len(qs) == 0 {
return nil
}
ps := make(StrokeQuads, 0, len(qs))
for i := range qs {
q := qs[len(qs)-1-i]
q.Quad.To, q.Quad.From = q.Quad.From, q.Quad.To
ps = append(ps, q)
}
return ps
}
func (qs StrokeQuads) append(ps StrokeQuads) StrokeQuads {
switch {
case len(ps) == 0:
return qs
case len(qs) == 0:
return ps
}
// Consolidate quads and smooth out rounding errors.
// We need to also check for the strokeTolerance to correctly handle
// join/cap points or on-purpose disjoint quads.
p0 := qs[len(qs)-1].Quad.To
p1 := ps[0].Quad.From
if p0 != p1 && lenPt(p0.Sub(p1)) < strokeTolerance {
qs = append(qs, StrokeQuad{
Quad: QuadSegment{
From: p0,
Ctrl: p0.Add(p1).Mul(0.5),
To: p1,
},
})
}
return append(qs, ps...)
}
func (q QuadSegment) Transform(t f32.Affine2D) QuadSegment {
q.From = t.Transform(q.From)
q.Ctrl = t.Transform(q.Ctrl)
q.To = t.Transform(q.To)
return q
}
// strokePathNorm returns the normal vector at t.
func strokePathNorm(p0, p1, p2 f32.Point, t, d float32) f32.Point {
switch t {
case 0:
n := p1.Sub(p0)
if n.X == 0 && n.Y == 0 {
return f32.Point{}
}
n = rot90CW(n)
return normPt(n, d)
case 1:
n := p2.Sub(p1)
if n.X == 0 && n.Y == 0 {
return f32.Point{}
}
n = rot90CW(n)
return normPt(n, d)
}
panic("impossible")
}
func rot90CW(p f32.Point) f32.Point { return f32.Pt(+p.Y, -p.X) }
func normPt(p f32.Point, l float32) f32.Point {
d := math.Hypot(float64(p.X), float64(p.Y))
l64 := float64(l)
if math.Abs(d-l64) < 1e-10 {
return f32.Point{}
}
n := float32(l64 / d)
return f32.Point{X: p.X * n, Y: p.Y * n}
}
func lenPt(p f32.Point) float32 {
return float32(math.Hypot(float64(p.X), float64(p.Y)))
}
func perpDot(p, q f32.Point) float32 {
return p.X*q.Y - p.Y*q.X
}
func angleBetween(n0, n1 f32.Point) float64 {
return math.Atan2(float64(n1.Y), float64(n1.X)) -
math.Atan2(float64(n0.Y), float64(n0.X))
}
// strokePathCurv returns the curvature at t, along the quadratic Bézier
// curve defined by the triplet (beg, ctl, end).
func strokePathCurv(beg, ctl, end f32.Point, t float32) float32 {
var (
d1p = quadBezierD1(beg, ctl, end, t)
d2p = quadBezierD2(beg, ctl, end, t)
// Negative when bending right, ie: the curve is CW at this point.
a = float64(perpDot(d1p, d2p))
)
// We check early that the segment isn't too line-like and
// save a costly call to math.Pow that will be discarded by dividing
// with a too small 'a'.
if math.Abs(a) < 1e-10 {
return float32(math.NaN())
}
return float32(math.Pow(float64(d1p.X*d1p.X+d1p.Y*d1p.Y), 1.5) / a)
}
// quadBezierSample returns the point on the Bézier curve at t.
//
// B(t) = (1-t)^2 P0 + 2(1-t)t P1 + t^2 P2
func quadBezierSample(p0, p1, p2 f32.Point, t float32) f32.Point {
t1 := 1 - t
c0 := t1 * t1
c1 := 2 * t1 * t
c2 := t * t
o := p0.Mul(c0)
o = o.Add(p1.Mul(c1))
o = o.Add(p2.Mul(c2))
return o
}
// quadBezierD1 returns the first derivative of the Bézier curve with respect to t.
//
// B'(t) = 2(1-t)(P1 - P0) + 2t(P2 - P1)
func quadBezierD1(p0, p1, p2 f32.Point, t float32) f32.Point {
p10 := p1.Sub(p0).Mul(2 * (1 - t))
p21 := p2.Sub(p1).Mul(2 * t)
return p10.Add(p21)
}
// quadBezierD2 returns the second derivative of the Bézier curve with respect to t:
//
// B''(t) = 2(P2 - 2P1 + P0)
func quadBezierD2(p0, p1, p2 f32.Point, t float32) f32.Point {
p := p2.Sub(p1.Mul(2)).Add(p0)
return p.Mul(2)
}
func strokeQuadBezier(state strokeState, d, flatness float32) StrokeQuads {
// Gio strokes are only quadratic Bézier curves, w/o any inflection point.
// So we just have to flatten them.
var qs StrokeQuads
return flattenQuadBezier(qs, state.p0, state.ctl, state.p1, d, flatness)
}
// flattenQuadBezier splits a Bézier quadratic curve into linear sub-segments,
// themselves also encoded as Bézier (degenerate, flat) quadratic curves.
func flattenQuadBezier(qs StrokeQuads, p0, p1, p2 f32.Point, d, flatness float32) StrokeQuads {
var (
t float32
flat64 = float64(flatness)
)
for t < 1 {
s2 := float64((p2.X-p0.X)*(p1.Y-p0.Y) - (p2.Y-p0.Y)*(p1.X-p0.X))
den := math.Hypot(float64(p1.X-p0.X), float64(p1.Y-p0.Y))
if s2*den == 0.0 {
break
}
s2 /= den
t = 2.0 * float32(math.Sqrt(flat64/3.0/math.Abs(s2)))
if t >= 1.0 {
break
}
var q0, q1, q2 f32.Point
q0, q1, q2, p0, p1, p2 = quadBezierSplit(p0, p1, p2, t)
qs.addLine(q0, q1, q2, 0, d)
}
qs.addLine(p0, p1, p2, 1, d)
return qs
}
func (qs *StrokeQuads) addLine(p0, ctrl, p1 f32.Point, t, d float32) {
switch i := len(*qs); i {
case 0:
p0 = p0.Add(strokePathNorm(p0, ctrl, p1, 0, d))
default:
// Address possible rounding errors and use previous point.
p0 = (*qs)[i-1].Quad.To
}
p1 = p1.Add(strokePathNorm(p0, ctrl, p1, 1, d))
*qs = append(*qs,
StrokeQuad{
Quad: QuadSegment{
From: p0,
Ctrl: p0.Add(p1).Mul(0.5),
To: p1,
},
},
)
}
// quadInterp returns the interpolated point at t.
func quadInterp(p, q f32.Point, t float32) f32.Point {
return f32.Pt(
(1-t)*p.X+t*q.X,
(1-t)*p.Y+t*q.Y,
)
}
// quadBezierSplit returns the pair of triplets (from,ctrl,to) Bézier curve,
// split before (resp. after) the provided parametric t value.
func quadBezierSplit(p0, p1, p2 f32.Point, t float32) (f32.Point, f32.Point, f32.Point, f32.Point, f32.Point, f32.Point) {
var (
b0 = p0
b1 = quadInterp(p0, p1, t)
b2 = quadBezierSample(p0, p1, p2, t)
a0 = b2
a1 = quadInterp(p1, p2, t)
a2 = p2
)
return b0, b1, b2, a0, a1, a2
}
// strokePathRoundJoin joins the two paths rhs and lhs, creating an arc.
func strokePathRoundJoin(rhs, lhs *StrokeQuads, hw float32, pivot, n0, n1 f32.Point, r0, r1 float32) {
rp := pivot.Add(n1)
lp := pivot.Sub(n1)
angle := angleBetween(n0, n1)
switch {
case angle <= 0:
// Path bends to the right, ie. CW (or 180 degree turn).
c := pivot.Sub(lhs.pen())
lhs.arc(c, c, float32(angle))
lhs.lineTo(lp) // Add a line to accommodate for rounding errors.
rhs.lineTo(rp)
default:
// Path bends to the left, ie. CCW.
c := pivot.Sub(rhs.pen())
rhs.arc(c, c, float32(angle))
rhs.lineTo(rp) // Add a line to accommodate for rounding errors.
lhs.lineTo(lp)
}
}
// strokePathCap caps the provided path qs, according to the provided stroke operation.
func strokePathCap(stroke StrokeStyle, qs *StrokeQuads, hw float32, pivot, n0 f32.Point) {
strokePathRoundCap(qs, hw, pivot, n0)
}
// strokePathRoundCap caps the start or end of a path with a round cap.
func strokePathRoundCap(qs *StrokeQuads, hw float32, pivot, n0 f32.Point) {
c := pivot.Sub(qs.pen())
qs.arc(c, c, math.Pi)
}
// ArcTransform computes a transformation that can be used for generating quadratic bézier
// curve approximations for an arc.
//
// The math is extracted from the following paper:
//
// "Drawing an elliptical arc using polylines, quadratic or
// cubic Bezier curves", L. Maisonobe
//
// An electronic version may be found at:
//
// http://spaceroots.org/documents/ellipse/elliptical-arc.pdf
func ArcTransform(p, f1, f2 f32.Point, angle float32) (transform f32.Affine2D, segments int) {
const segmentsPerCircle = 16
const anglePerSegment = 2 * math.Pi / segmentsPerCircle
s := angle / anglePerSegment
if s < 0 {
s = -s
}
segments = int(math.Ceil(float64(s)))
if segments <= 0 {
segments = 1
}
var rx, ry, alpha float64
if f1 == f2 {
// degenerate case of a circle.
rx = dist(f1, p)
ry = rx
} else {
// semi-major axis: 2a = |PF1| + |PF2|
a := 0.5 * (dist(f1, p) + dist(f2, p))
// semi-minor axis: c^2 = a^2 - b^2 (c: focal distance)
c := dist(f1, f2) * 0.5
b := math.Sqrt(a*a - c*c)
switch {
case a > b:
rx = a
ry = b
default:
rx = b
ry = a
}
if f1.X == f2.X {
// special case of a "vertical" ellipse.
alpha = math.Pi / 2
if f1.Y < f2.Y {
alpha = -alpha
}
} else {
x := float64(f1.X-f2.X) * 0.5
if x < 0 {
x = -x
}
alpha = math.Acos(x / c)
}
}
var (
θ = angle / float32(segments)
ref f32.Affine2D // transform from absolute frame to ellipse-based one
rot f32.Affine2D // rotation matrix for each segment
inv f32.Affine2D // transform from ellipse-based frame to absolute one
)
center := f32.Point{
X: 0.5 * (f1.X + f2.X),
Y: 0.5 * (f1.Y + f2.Y),
}
ref = ref.Offset(f32.Point{}.Sub(center))
ref = ref.Rotate(f32.Point{}, float32(-alpha))
ref = ref.Scale(f32.Point{}, f32.Point{
X: float32(1 / rx),
Y: float32(1 / ry),
})
inv = ref.Invert()
rot = rot.Rotate(f32.Point{}, 0.5*θ)
// Instead of invoking math.Sincos for every segment, compute a rotation
// matrix once and apply for each segment.
// Before applying the rotation matrix rot, transform the coordinates
// to a frame centered to the ellipse (and warped into a unit circle), then rotate.
// Finally, transform back into the original frame.
return inv.Mul(rot).Mul(ref), segments
}
func dist(p1, p2 f32.Point) float64 {
var (
x1 = float64(p1.X)
y1 = float64(p1.Y)
x2 = float64(p2.X)
y2 = float64(p2.Y)
dx = x2 - x1
dy = y2 - y1
)
return math.Hypot(dx, dy)
}
func StrokePathCommands(style StrokeStyle, scene []byte) StrokeQuads {
quads := decodeToStrokeQuads(scene)
return quads.stroke(style)
}
// decodeToStrokeQuads decodes scene commands to quads ready to stroke.
func decodeToStrokeQuads(pathData []byte) StrokeQuads {
quads := make(StrokeQuads, 0, 2*len(pathData)/(scene.CommandSize+4))
scratch := make([]QuadSegment, 0, 10)
for len(pathData) >= scene.CommandSize+4 {
contour := binary.LittleEndian.Uint32(pathData)
cmd := ops.DecodeCommand(pathData[4:])
switch cmd.Op() {
case scene.OpLine:
var q QuadSegment
q.From, q.To = scene.DecodeLine(cmd)
q.Ctrl = q.From.Add(q.To).Mul(.5)
quad := StrokeQuad{
Contour: contour,
Quad: q,
}
quads = append(quads, quad)
case scene.OpGap:
// Ignore gaps for strokes.
case scene.OpQuad:
var q QuadSegment
q.From, q.Ctrl, q.To = scene.DecodeQuad(cmd)
quad := StrokeQuad{
Contour: contour,
Quad: q,
}
quads = append(quads, quad)
case scene.OpCubic:
from, ctrl0, ctrl1, to := scene.DecodeCubic(cmd)
scratch = SplitCubic(from, ctrl0, ctrl1, to, scratch[:0])
for _, q := range scratch {
quad := StrokeQuad{
Contour: contour,
Quad: q,
}
quads = append(quads, quad)
}
default:
panic("unsupported scene command")
}
pathData = pathData[scene.CommandSize+4:]
}
return quads
}
func SplitCubic(from, ctrl0, ctrl1, to f32.Point, quads []QuadSegment) []QuadSegment {
// Set the maximum distance proportionally to the longest side
// of the bounding rectangle.
hull := f32.Rectangle{
Min: from,
Max: ctrl0,
}.Canon().Union(f32.Rectangle{
Min: ctrl1,
Max: to,
}.Canon())
l := hull.Dx()
if h := hull.Dy(); h > l {
l = h
}
maxDist := l * 0.001
approxCubeTo(&quads, 0, maxDist*maxDist, from, ctrl0, ctrl1, to)
return quads
}
// approxCubeTo approximates a cubic Bézier by a series of quadratic
// curves.
func approxCubeTo(quads *[]QuadSegment, splits int, maxDistSq float32, from, ctrl0, ctrl1, to f32.Point) int {
// The idea is from
// https://caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
// where a quadratic approximates a cubic by eliminating its t³ term
// from its polynomial expression anchored at the starting point:
//
// P(t) = pen + 3t(ctrl0 - pen) + 3t²(ctrl1 - 2ctrl0 + pen) + t³(to - 3ctrl1 + 3ctrl0 - pen)
//
// The control point for the new quadratic Q1 that shares starting point, pen, with P is
//
// C1 = (3ctrl0 - pen)/2
//
// The reverse cubic anchored at the end point has the polynomial
//
// P'(t) = to + 3t(ctrl1 - to) + 3t²(ctrl0 - 2ctrl1 + to) + t³(pen - 3ctrl0 + 3ctrl1 - to)
//
// The corresponding quadratic Q2 that shares the end point, to, with P has control
// point
//
// C2 = (3ctrl1 - to)/2
//
// The combined quadratic Bézier, Q, shares both start and end points with its cubic
// and use the midpoint between the two curves Q1 and Q2 as control point:
//
// C = (3ctrl0 - pen + 3ctrl1 - to)/4
// using, q0 := 3ctrl0 - pen, q1 := 3ctrl1 - to
// C = (q0 + q1)/4
q0 := ctrl0.Mul(3).Sub(from)
q1 := ctrl1.Mul(3).Sub(to)
c := q0.Add(q1).Mul(1.0 / 4.0)
const maxSplits = 32
if splits >= maxSplits {
*quads = append(*quads, QuadSegment{From: from, Ctrl: c, To: to})
return splits
}
// The maximum distance between the cubic P and its approximation Q given t
// can be shown to be
//
// d = sqrt(3)/36 * |to - 3ctrl1 + 3ctrl0 - pen|
// reusing, q0 := 3ctrl0 - pen, q1 := 3ctrl1 - to
// d = sqrt(3)/36 * |-q1 + q0|
//
// To save a square root, compare d² with the squared tolerance.
v := q0.Sub(q1)
d2 := (v.X*v.X + v.Y*v.Y) * 3 / (36 * 36)
if d2 <= maxDistSq {
*quads = append(*quads, QuadSegment{From: from, Ctrl: c, To: to})
return splits
}
// De Casteljau split the curve and approximate the halves.
t := float32(0.5)
c0 := from.Add(ctrl0.Sub(from).Mul(t))
c1 := ctrl0.Add(ctrl1.Sub(ctrl0).Mul(t))
c2 := ctrl1.Add(to.Sub(ctrl1).Mul(t))
c01 := c0.Add(c1.Sub(c0).Mul(t))
c12 := c1.Add(c2.Sub(c1).Mul(t))
c0112 := c01.Add(c12.Sub(c01).Mul(t))
splits++
splits = approxCubeTo(quads, splits, maxDistSq, from, c0, c01, c0112)
splits = approxCubeTo(quads, splits, maxDistSq, c0112, c12, c2, to)
return splits
}