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Copy pathquadbez.go
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362 lines (323 loc) · 8.81 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"
"honnef.co/go/stuff/container/maybe"
"honnef.co/go/stuff/math/polyroot"
)
var _ Shape = QuadBez{}
var _ ParametricCurve = QuadBez{}
type QuadBez struct {
P0 Point
P1 Point
P2 Point
}
func (q QuadBez) BoundingBox() Rect {
return BoundingBox(q)
}
func (q QuadBez) Path(tolerance float64, out BezPath) BezPath {
out.MoveTo(q.P0)
out.QuadTo(q.P1, q.P2)
return out
}
// Raise raises the order by one, returning a cubic Bézier segment that exactly
// represents this quadratic.
func (q QuadBez) Raise() CubicBez {
return CubicBez{
q.P0,
q.P0.Translate(q.P1.Sub(q.P0).Mul(2.0 / 3.0)),
q.P2.Translate(q.P1.Sub(q.P2).Mul(2.0 / 3.0)),
q.P2,
}
}
func (q QuadBez) IsInf() bool {
return q.P0.IsInf() || q.P1.IsInf() || q.P2.IsInf()
}
func (q QuadBez) IsNaN() bool {
return q.P0.IsNaN() || q.P1.IsNaN() || q.P2.IsNaN()
}
// PathLength returns the arc length of the quadratic Bézier segment.
//
// This computation is based on an analytical formula. Since that formula suffers
// from numerical instability when the curve is very close to a straight line, we
// detect that case and fall back to Legendre-Gauss quadrature.
//
// Overall accuracy should be better than 1e-13 over the entire range.
func (q QuadBez) PathLength(accuracy float64) float64 {
d2 := Vec2(q.P0).Sub(Vec2(q.P1).Mul(2)).Add(Vec2(q.P2))
a := d2.Hypot2()
d1 := q.P1.Sub(q.P0)
c := d1.Hypot2()
if a < 5e-4*c {
// This case happens for nearly straight Béziers.
//
// Calculate arc length using Legendre-Gauss quadrature using formula from Behdad
// in https://github.com/Pomax/BezierInfo-2/issues/77
v0 := Vec2(q.P0).Mul(-0.492943519233745).
Add(Vec2(q.P1).Mul(0.430331482911935)).
Add(Vec2(q.P2).Mul(0.0626120363218102)).
Hypot()
v1 := q.P2.Sub(q.P0).Mul(0.4444444444444444).Hypot()
v2 := Vec2(q.P0).Mul(-0.0626120363218102).
Sub(Vec2(q.P1).Mul(0.430331482911935)).
Add(Vec2(q.P2).Mul(0.492943519233745)).
Hypot()
return v0 + v1 + v2
}
b := 2.0 * d2.Dot(d1)
sabc := math.Sqrt(a + b + c)
a2 := math.Pow(a, -0.5)
a32 := a2 * a2 * a2
c2 := 2.0 * math.Sqrt(c)
baC2 := b*a2 + c2
v0 := 0.25*a2*a2*b*(2.0*sabc-c2) + sabc
// TODO: justify and fine-tune this exact constant.
if baC2 < 1e-13 {
// This case happens for Béziers with a sharp kink.
return v0
} else {
return v0 + 0.25*a32*(4.0*c*a-b*b)*math.Log(((2.0*a+b)*a2+2.0*sabc)/baC2)
}
}
func (q QuadBez) Eval(t float64) Point {
mt := 1.0 - t
a := Vec2(q.P0).Mul(mt * mt)
b := Vec2(q.P1).Mul(mt * 2.0)
c := Vec2(q.P2).Mul(t)
d := b.Add(c)
return Point(a.Add(d.Mul(t)))
}
func (q QuadBez) Subdivide() (QuadBez, QuadBez) {
pm := q.Eval(0.5)
return QuadBez{q.P0, q.P0.Midpoint(q.P1), pm},
QuadBez{pm, q.P1.Midpoint(q.P2), q.P2}
}
func (q QuadBez) SubdivideCurve() (ParametricCurve, ParametricCurve) {
return q.Subdivide()
}
func (q QuadBez) Subsegment(t0 float64, t1 float64) QuadBez {
p0 := q.Eval(t0)
p2 := q.Eval(t1)
p1 := p0.Translate(q.P1.Sub(q.P0).Lerp(q.P2.Sub(q.P1), t0).Mul(t1 - t0))
return QuadBez{p0, p1, p2}
}
func (q QuadBez) SubsegmentCurve(t0 float64, t1 float64) ParametricCurve {
return q.Subsegment(t0, t1)
}
func (q QuadBez) Differentiate() Line {
return Line{
Point(q.P1.Sub(q.P0).Mul(2)),
Point(q.P2.Sub(q.P1).Mul(2)),
}
}
func (q QuadBez) Start() Point {
return q.P0
}
func (q QuadBez) End() Point {
return q.P2
}
func (q QuadBez) Extrema() ([MaxExtrema]float64, int) {
// Finding the extrema of a quadratic bezier means finding the roots in the
// quadratic's first derivative, which is a line.
var out [MaxExtrema]float64
var outN int
d0 := q.P1.Sub(q.P0)
d1 := q.P2.Sub(q.P1)
dd := d1.Sub(d0)
if dd.X != 0.0 {
t := -d0.X / dd.X
if t > 0.0 && t < 1.0 {
out[outN] = t
outN++
}
}
if dd.Y != 0 {
t := -d0.Y / dd.Y
if t > 0.0 && t < 1.0 {
out[outN] = t
outN++
if outN == 2 && out[0] > t {
out[0], out[1] = out[1], out[0]
}
}
}
return out, outN
}
func (q QuadBez) Nearest(pt Point, accuracy float64) (distSq, outT float64) {
// Find the nearest point, using analytical algorithm based on cubic root finding.
evalT := func(pt Point, tBest *float64, rBest *maybe.Option[float64], t float64, p0 Point) {
r := p0.Sub(pt).Hypot2()
if v, ok := rBest.Get(); !ok || r < v {
*rBest = maybe.Some(r)
*tBest = t
}
}
tryT := func(
q *QuadBez,
pt Point,
tBest *float64,
rBest *maybe.Option[float64],
t float64,
) bool {
evalT(pt, tBest, rBest, t, q.Eval(t))
return false
}
d0 := q.P1.Sub(q.P0)
d1 := Vec2(q.P0).Add(Vec2(q.P2)).Sub(Vec2(q.P1).Mul(2.0))
d := q.P0.Sub(pt)
c0 := d.Dot(d0)
c1 := 2.0*d0.Hypot2() + d.Dot(d1)
c2 := 3.0 * d1.Dot(d0)
c3 := d1.Hypot2()
roots := polyroot.NewPolynomial(c0, c1, c2, c3).Roots(0, 1, 0, nil)
var rBest maybe.Option[float64]
tBest := 0.0
needEnds := len(roots) == 0
for _, t := range roots {
b := tryT(&q, pt, &tBest, &rBest, t)
if b {
needEnds = true
}
}
if needEnds {
evalT(pt, &tBest, &rBest, 0.0, q.P0)
evalT(pt, &tBest, &rBest, 1.0, q.P2)
}
return rBest.UnwrapOr(0), tBest
}
func (q QuadBez) Transform(aff Affine) QuadBez {
return QuadBez{
P0: q.P0.Transform(aff),
P1: q.P1.Transform(aff),
P2: q.P2.Transform(aff),
}
}
func (q QuadBez) Area() float64 {
v := q.P0.X*(2.0*q.P1.Y+q.P2.Y) +
2.0*(q.P1.X*(q.P2.Y-q.P0.Y)) -
q.P2.X*(q.P0.Y+2.0*q.P1.Y)
return v * (1.0 / 6.0)
}
func (q QuadBez) Tangents() (Vec2, Vec2) {
const epsilon = 1e-12
d01 := q.P1.Sub(q.P0)
var d0, d1 Vec2
if d01.Hypot2() > epsilon {
d0 = d01
} else {
d0 = q.P2.Sub(q.P0)
}
d12 := q.P2.Sub(q.P1)
if d12.Hypot2() > epsilon {
d1 = d12
} else {
d1 = q.P2.Sub(q.P0)
}
return d0, d1
}
// An approximation to $\int (1 + 4x^2) ^ -0.25 dx$
//
// This is used for flattening curves.
func approxParabolaIntegral(x float64) float64 {
const d = 0.67
return x / (1.0 - d + math.Sqrt(math.Sqrt(pow4(d)+0.25*x*x)))
}
// An approximation to the inverse parabola integral.
func approxParabolaInvIntegral(x float64) float64 {
const b = 0.39
return x * (1.0 - b + math.Sqrt(b*b+0.25*x*x))
}
// Maps a value from 0..1 to 0..1.
func (q QuadBez) determineSubdivT(params *flattenParams, x float64) float64 {
a := params.a0 + (params.a2-params.a0)*x
u := approxParabolaInvIntegral(a)
return (u - params.u0) * params.uscale
}
// Estimate the number of subdivisions for flattening.
func (q QuadBez) estimateSubdiv(sqrtTol float64) flattenParams {
// Determine transformation to $y = x^2$ parabola.
d01 := q.P1.Sub(q.P0)
d12 := q.P2.Sub(q.P1)
dd := d01.Sub(d12)
cross := q.P2.Sub(q.P0).Cross(dd)
x0 := d01.Dot(dd) * (1.0 / cross)
x2 := d12.Dot(dd) * (1.0 / cross)
scale := math.Abs(cross / (dd.Hypot() * (x2 - x0)))
// Compute number of subdivisions needed.
a0 := approxParabolaIntegral(x0)
a2 := approxParabolaIntegral(x2)
var val float64
if !math.IsInf(scale, 0) {
da := math.Abs(a2 - a0)
sqrtScale := math.Sqrt(scale)
if math.Signbit(x0) == math.Signbit(x2) {
val = da * sqrtScale
} else {
// Handle cusp case (segment contains curvature maximum)
xmin := sqrtTol / sqrtScale
val = sqrtTol * da / approxParabolaIntegral(xmin)
}
}
u0 := approxParabolaInvIntegral(a0)
u2 := approxParabolaInvIntegral(a2)
uscale := 1.0 / (u2 - u0)
return flattenParams{
a0,
a2,
u0,
uscale,
val,
}
}
type flattenParams struct {
a0 float64
a2 float64
u0 float64
uscale float64
// The number of subdivisions * 2 * sqrtTol.
val float64
}
func (q QuadBez) Seg() PathSegment {
return PathSegment{Kind: QuadKind, P0: q.P0, P1: q.P1, P2: q.P2}
}
func (q QuadBez) 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 quadratic 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 := quadBezCoefficients(q.P0.X, q.P1.X, q.P2.X)
py0, py1, py2 := quadBezCoefficients(q.P0.Y, q.P1.Y, q.P2.Y)
c0 := dy*(px0-p0.X) - dx*(py0-p0.Y)
c1 := dy*px1 - dx*py1
c2 := dy*px2 - dx*py2
invlen2 := 1.0 / (dx*dx + dy*dy)
ts := polyroot.NewPolynomial(c0, c1, c2).Roots(-epsilon, 1+epsilon, 0, nil)
var ret [3]LineIntersection
var retN int
for _, t := range ts {
x := px0 + t*px1 + t*t*px2
y := py0 + t*py1 + t*t*py2
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 quadBezCoefficients(x0, x1, x2 float64) (_, _, _ float64) {
p0 := x0
p1 := 2.0*x1 - 2.0*x0
p2 := x2 - 2.0*x1 + x0
return p0, p1, p2
}