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Copy patharc.go
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220 lines (187 loc) · 6.14 KB
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// SPDX-FileCopyrightText: 2018 Raph Levien
// SPDX-FileCopyrightText: 2024 Dominik Honnef and contributors
//
// SPDX-License-Identifier: MIT
// SPDX-FileAttributionText: https://github.com/linebender/kurbo
package curve
import (
"math"
)
type Arc struct {
Center Point
Radii Vec2
StartAngle Angle
SweepAngle Angle
XRotation Angle
}
var _ Shape = Arc{}
var _ ParametricCurve = Arc{}
func (a Arc) Path(tolerance float64, out BezPath) BezPath {
out.MoveTo(a.Start())
sweepAngle := clampAngle(a.SweepAngle)
scaledError := max(a.Radii.X, a.Radii.Y) / tolerance
// Number of subdivisions per ellipse based on error tolerance.
// Note: this may slightly underestimate the error for quadrants.
nError := max(math.Pow(1.1163*scaledError, 1.0/6.0), 3.999_999)
n := math.Ceil(nError * math.Abs(sweepAngle) * (1.0 / (2.0 * math.Pi)))
angleStep := sweepAngle / n
armLen := math.Copysign((4.0/3.0)*math.Tan(math.Abs(0.25*angleStep)), sweepAngle)
angle0 := normalizeAngle(a.StartAngle)
p0 := sampleEllipse(a.Radii, a.XRotation, angle0)
for range int(n) {
angle1 := angle0 + angleStep
p1 := p0.Add(sampleEllipse(a.Radii, a.XRotation, angle0+math.Pi/2).Mul(armLen))
p3 := sampleEllipse(a.Radii, a.XRotation, angle1)
p2 := p3.Sub(sampleEllipse(a.Radii, a.XRotation, angle1+math.Pi/2).Mul(armLen))
angle0 = angle1
p0 = p3
out.CubicTo(
a.Center.Translate(p1),
a.Center.Translate(p2),
a.Center.Translate(p3),
)
}
return out
}
// Take the ellipse radii, how the radii are rotated, and the sweep angle, and return a
// point on the ellipse.
func sampleEllipse(radii Vec2, xRotation, angle Angle) Vec2 {
sin, cos := math.Sincos(angle)
u := radii.X * cos
v := radii.Y * sin
return rotatePt(Vec2{u, v}, xRotation)
}
// Rotate pt about the origin by angle radians.
func rotatePt(pt Vec2, angle Angle) Vec2 {
sin, cos := math.Sincos(angle)
return Vec2{
X: pt.X*cos - pt.Y*sin,
Y: pt.X*sin + pt.Y*cos,
}
}
func (a Arc) BoundingBox() Rect {
bbox := NewRectFromPoints(a.Start(), a.End())
startAngle := normalizeAngle(a.StartAngle)
sweepAngle := clampAngle(a.SweepAngle)
if math.Abs(sweepAngle) == 2*math.Pi {
return NewEllipse(a.Center, a.Radii, a.XRotation).BoundingBox()
}
containsAngle := func(angle Angle) bool {
if sweepAngle >= 0 {
end := startAngle + sweepAngle
if end <= 2*math.Pi {
return angle >= startAngle && angle <= end
}
return angle >= startAngle || angle <= normalizeAngle(end)
}
end := startAngle + sweepAngle
if end >= 0 {
return angle <= startAngle && angle >= end
}
return angle <= startAngle || angle >= normalizeAngle(end)
}
// These are the angles where the rotated ellipse reaches an extremum
// in x or y. A tight bounding box can only change at the arc endpoints or at
// one of these stationary points, so we test each extremum and keep only those
// that lie on the swept portion of the ellipse.
for _, angle := range [...]Angle{
math.Atan2(-a.Radii.Y*math.Sin(a.XRotation), a.Radii.X*math.Cos(a.XRotation)),
math.Atan2(-a.Radii.Y*math.Sin(a.XRotation), a.Radii.X*math.Cos(a.XRotation)) + math.Pi,
math.Atan2(a.Radii.Y*math.Cos(a.XRotation), a.Radii.X*math.Sin(a.XRotation)),
math.Atan2(a.Radii.Y*math.Cos(a.XRotation), a.Radii.X*math.Sin(a.XRotation)) + math.Pi,
} {
if containsAngle(angle) {
bbox = bbox.UnionPoint(a.Center.Translate(sampleEllipse(a.Radii, a.XRotation, angle)))
}
}
return bbox
}
func (a Arc) PathLength(accuracy float64) float64 {
if a.SweepAngle == 0 {
return 0
}
if a.Radii.IsInf() {
return math.Inf(1)
}
radii := Vec(math.Abs(a.Radii.X), math.Abs(a.Radii.Y))
sweepAngle := math.Abs(clampAngle(a.SweepAngle))
if radii.X == radii.Y {
return radii.X * sweepAngle
}
if sweepAngle == 2*math.Pi {
return NewEllipse(a.Center, radii, a.XRotation).PathLength(accuracy)
}
integrand := func(theta Angle) float64 {
sin, cos := math.Sincos(theta)
return math.Hypot(radii.X*sin, radii.Y*cos)
}
startAngle := normalizeAngle(a.StartAngle)
dir := math.Copysign(1, a.SweepAngle)
var integrate func(start, end Angle, accuracy float64, depth int) float64
integrateInterval := func(sweepStart, sweepEnd Angle, coeffs [][2]float64) float64 {
mid := 0.5 * (sweepStart + sweepEnd)
halfRange := 0.5 * (sweepEnd - sweepStart)
sum := 0.0
for _, coeff := range coeffs {
sum += coeff[0] * integrand(startAngle+dir*(mid+halfRange*coeff[1]))
}
return sum * halfRange
}
integrate = func(sweepStart, sweepEnd Angle, accuracy float64, depth int) float64 {
i8 := integrateInterval(sweepStart, sweepEnd, gaussLegendreCoeffs8[:])
i16 := integrateInterval(sweepStart, sweepEnd, gaussLegendreCoeffs16[:])
if math.Abs(i16-i8) <= accuracy || depth >= 20 {
return i16
}
mid := 0.5 * (sweepStart + sweepEnd)
return integrate(sweepStart, mid, accuracy*0.5, depth+1) + integrate(mid, sweepEnd, accuracy*0.5, depth+1)
}
return integrate(0, sweepAngle, accuracy, 0)
}
func (a Arc) Translate(v Vec2) Arc {
a.Center = a.Center.Translate(v)
return a
}
// Reverse returns a copy of this arc in the opposite direction.
//
// The new arc will sweep towards the original arc's start angle.
func (a Arc) Reverse() Arc {
sweepAngle := clampAngle(a.SweepAngle)
return Arc{
Center: a.Center,
Radii: a.Radii,
StartAngle: normalizeAngle(a.StartAngle + sweepAngle),
SweepAngle: -sweepAngle,
XRotation: a.XRotation,
}
}
func (a Arc) AngleAt(t float64) float64 {
return normalizeAngle(a.StartAngle + clampAngle(a.SweepAngle)*t)
}
func (a Arc) Eval(t float64) Point {
return a.Center.Translate(sampleEllipse(a.Radii, a.XRotation, a.AngleAt(t)))
}
func (a Arc) Subsegment(start, end float64) Arc {
return Arc{
Center: a.Center,
Radii: a.Radii,
StartAngle: a.AngleAt(start),
SweepAngle: clampAngle(clampAngle(a.SweepAngle) * (end - start)),
XRotation: a.XRotation,
}
}
func (a Arc) SubsegmentCurve(start, end float64) ParametricCurve {
return a.Subsegment(start, end)
}
func (a Arc) Subdivide() (Arc, Arc) {
return a.Subsegment(0.0, 0.5), a.Subsegment(0.5, 1.0)
}
func (a Arc) SubdivideCurve() (ParametricCurve, ParametricCurve) {
return a.Subdivide()
}
func (a Arc) Start() Point {
return a.Eval(0)
}
func (a Arc) End() Point {
return a.Eval(1)
}