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op/clip: separate function for approximating arcs with quadratic curves

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Both Path.Arc and the internal stroke package needs to support arcs;
this change isolates the approximation computation into a function we
can move to internal/stroke in a follow-up.

Signed-off-by: Elias Naur <mail@eliasnaur.com>
This commit is contained in:
Elias Naur
2021-03-23 15:01:00 +01:00
parent 8c8d1dc16f
commit f9cf6ff20a
1 changed files with 23 additions and 35 deletions
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op/clip/clip.go
+23 -35
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@@ -173,8 +173,16 @@ func (p *Path) QuadTo(ctrl, to f32.Point) {
func (p *Path) Arc(f1, f2 f32.Point, angle float32) {
f1 = f1.Add(p.pen)
f2 = f2.Add(p.pen)
c, rx, ry, beg, alpha := arcFrom(f1, f2, p.pen)
p.arc(alpha, c, rx, ry, beg, float64(angle))
const segments = 16
m := arcTransform(p.pen, f1, f2, angle, segments)
for i := 0; i < segments; i++ {
p0 := p.pen
p1 := m.Transform(p0)
p2 := m.Transform(p1)
ctl := p1.Mul(2).Sub(p0.Add(p2).Mul(.5))
p.QuadTo(ctl, p2)
}
}
func dist(p1, p2 f32.Point) float64 {
@@ -189,8 +197,16 @@ func dist(p1, p2 f32.Point) float64 {
return math.Hypot(dx, dy)
}
func arcFrom(f1, f2, p f32.Point) (c f32.Point, rx, ry, start, alpha float64) {
c = f32.Point{
// 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, segments int) f32.Affine2D {
c := f32.Point{
X: 0.5 * (f1.X + f2.X),
Y: 0.5 * (f1.Y + f2.Y),
}
@@ -202,6 +218,7 @@ func arcFrom(f1, f2, p f32.Point) (c f32.Point, rx, ry, start, alpha float64) {
f := dist(f1, c)
b := math.Sqrt(a*a - f*f)
var rx, ry, alpha, start float64
switch {
case a > b:
rx = a
@@ -239,21 +256,8 @@ func arcFrom(f1, f2, p f32.Point) (c f32.Point, rx, ry, start, alpha float64) {
}
start -= alpha
return c, rx, ry, start, alpha
}
// arc records an elliptical arc centered at c, with radii rx and ry,
// starting at angle beg and stopping at end, in radians.
//
// 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 (p *Path) arc(alpha float64, c f32.Point, rx, ry, beg, delta float64) {
const n = 16
var (
θ = delta / n
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
@@ -272,23 +276,7 @@ func (p *Path) arc(alpha float64, c f32.Point, rx, ry, beg, delta float64) {
// 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.
step := func(p f32.Point) f32.Point {
q := ref.Transform(p)
q = rot.Transform(q)
q = inv.Transform(q)
return q
}
for i := 0; i < n; i++ {
p0 := p.pen
p1 := step(p0)
p2 := step(p1)
ctl := f32.Pt(
2*p1.X-0.5*(p0.X+p2.X),
2*p1.Y-0.5*(p0.Y+p2.Y),
)
p.QuadTo(ctl, p2)
}
return inv.Mul(rot).Mul(ref)
}
// Cube records a cubic Bézier from the pen through