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734 lines (671 loc) · 20.6 KB
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package curve
import (
"iter"
"math"
)
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)
}
// 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
}
// / 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(math.Pow(est, 3)*2.5e-6, 3e-2) * lplc
if estGauss8Error < accuracy {
return arclenQuadratureCore(gaussLegendreCoeffs8Half[:], dm, dm1, dm2)
}
estGauss16Error := min(math.Pow(est, 6)*1.5e-11, 9e-3) * lplc
if estGauss16Error < accuracy {
return arclenQuadratureCore(gaussLegendreCoeffs16Half[:], dm, dm1, dm2)
}
estGauss24Error := min(math.Pow(est, 9)*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 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,
}
}
// / Subdivide 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
}
// / Convert 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 `QuadBez`.
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
}
// regularize preprocesses a cubic Bézier to ease numerical robustness.
//
// If the cubic Bézier segment has zero or near-zero derivatives, perturb
// the control points to make it easier to process (especially offset and
// stroke), avoiding numerical robustness problems.
func (c CubicBez) regularize(dimension float64) CubicBez {
out := c
// First step: if control point is too near the endpoint, nudge it away
// along the tangent.
dim2 := dimension * dimension
if out.P0.DistanceSquared(out.P1) < dim2 {
d02 := out.P0.DistanceSquared(out.P2)
if d02 >= dim2 {
// TODO: moderate if this would move closer to p3
out.P1 = out.P0.Lerp(out.P2, math.Sqrt(dim2/d02))
} else {
out.P1 = out.P0.Lerp(out.P3, 1.0/3.0)
out.P2 = out.P3.Lerp(out.P0, 1.0/3.0)
return out
}
}
if out.P3.DistanceSquared(out.P2) < dim2 {
d13 := out.P1.DistanceSquared(out.P2)
if d13 >= dim2 {
// TODO: moderate if this would move closer to p0
out.P2 = out.P3.Lerp(out.P1, math.Sqrt(dim2/d13))
} else {
out.P1 = out.P0.Lerp(out.P3, 1.0/3.0)
out.P2 = out.P3.Lerp(out.P0, 1.0/3.0)
return out
}
}
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
}
// / 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),
}
}
// / Find the nearest point, using subdivision.
func (c CubicBez) Nearest(pt Point, accuracy float64) (distSq, t float64) {
var bestR option[float64]
bestT := 0.0
for qq := range c.Quadratics(accuracy) {
t0, t1, q := qq.Start, qq.End, qq.Segment
qDistSq, qT := q.Nearest(pt, accuracy)
if !bestR.isSet || qDistSq < bestR.value {
bestT = t0 + qT*(t1-t0)
bestR.set(qDistSq)
}
}
return bestR.value, bestT
}
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
}
// / Determine the inflection points.
// /
// / Return value is t parameter for the inflection points of the curve segment.
// / There are a maximum of two for a cubic Bézier.
// /
// / See <https://www.caffeineowl.com/graphics/2d/vectorial/cubic-inflexion.html>
// / for the theory.
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
}
// / Return a [`QuadSpline`] approximating this cubic Bézier.
// /
// / Returns `None` 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 `None` 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
}
// / Convert multiple cubic Bézier curves to quadratic splines.
// /
// / Ensures that the resulting splines have the same number of control points.
// /
// / Rust port of cu2qu [cubic_approx_quadratic](https://github.com/fonttools/fonttools/blob/3b9a73ff8379ab49d3ce35aaaaf04b3a7d9d1655/Lib/fontTools/cu2qu/cu2qu.py#L322)
//
// CubicsToQuadraticSplines converts multiple cubic Béziers to quadratic
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
}