vector: Better interpolation of Bézier curves

Updates #844

vector/path.go

@@ -18,8 +18,6 @@
 package vector
 
 import (
-	"math"
-
 	"github.com/hajimehoshi/ebiten/v2"
 )
 
@@ -51,46 +49,76 @@ func (p *Path) LineTo(x, y float32) {
 	p.cur = point{x: x, y: y}
 }
 
-// nseg returns a number of segments based on the given two points (x0, y0) and (x1, y1).
-func nseg(x0, y0, x1, y1 float32) int {
-	distx := x1 - x0
-	if distx < 0 {
-		distx = -distx
-	}
-	disty := y1 - y0
-	if disty < 0 {
-		disty = -disty
-	}
-	dist := distx
-	if dist < disty {
-		dist = disty
-	}
-
-	return int(math.Ceil(float64(dist)))
-}
-
 // QuadTo adds a quadratic Bézier curve to the path.
 func (p *Path) QuadTo(cpx, cpy, x, y float32) {
-	// TODO: Split more appropriate number of segments.
-	c := p.cur
-	num := nseg(c.x, c.y, x, y)
-	for t := float32(0.0); t <= 1; t += 1.0 / float32(num) {
-		xf := (1-t)*(1-t)*c.x + 2*t*(1-t)*cpx + t*t*x
-		yf := (1-t)*(1-t)*c.y + 2*t*(1-t)*cpy + t*t*y
-		p.LineTo(xf, yf)
+	p.quadTo(cpx, cpy, x, y, 0)
+}
+
+// isPointCloseToSegment detects the distance between a segment (x0, y0)-(x1, y1) and a point (x, y) is less than allow.
+func isPointCloseToSegment(x, y, x0, y0, x1, y1 float32, allow float32) bool {
+	// Line passing through (x0, y0) and (x1, y1) in the form of ax + by + c = 0
+	a := y1 - y0
+	b := -(x1 - x0)
+	c := (x1-x0)*y0 - (y1-y0)*x0
+
+	// The distance between a line ax+by+c=0 and (x0, y0) is
+	//     |ax0 + by0 + c| / √(a² + b²)
+	return allow*allow*(a*a+b*b) > (a*x+b*y+c)*(a*x+b*y+c)
+}
+
+func (p *Path) quadTo(x1, y1, x2, y2 float32, level int) {
+	if level > 10 {
+		return
 	}
+
+	x0 := p.cur.x
+	y0 := p.cur.y
+	if isPointCloseToSegment(x1, y1, x0, y0, x2, y2, 1) {
+		p.LineTo(x2, y2)
+		return
+	}
+
+	x01 := (x0 + x1) / 2
+	y01 := (y0 + y1) / 2
+	x12 := (x1 + x2) / 2
+	y12 := (y1 + y2) / 2
+	x012 := (x01 + x12) / 2
+	y012 := (y01 + y12) / 2
+	p.quadTo(x01, y01, x012, y012, level+1)
+	p.quadTo(x12, y12, x2, y2, level+1)
 }
 
 // CubicTo adds a cubic Bézier curve to the path.
 func (p *Path) CubicTo(cp0x, cp0y, cp1x, cp1y, x, y float32) {
-	// TODO: Split more appropriate number of segments.
-	c := p.cur
-	num := nseg(c.x, c.y, x, y)
-	for t := float32(0.0); t <= 1; t += 1.0 / float32(num) {
-		xf := (1-t)*(1-t)*(1-t)*c.x + 3*(1-t)*(1-t)*t*cp0x + 3*(1-t)*t*t*cp1x + t*t*t*x
-		yf := (1-t)*(1-t)*(1-t)*c.y + 3*(1-t)*(1-t)*t*cp0y + 3*(1-t)*t*t*cp1y + t*t*t*y
-		p.LineTo(xf, yf)
+	p.cubicTo(cp0x, cp0y, cp1x, cp1y, x, y, 0)
+}
+
+func (p *Path) cubicTo(x1, y1, x2, y2, x3, y3 float32, level int) {
+	if level > 10 {
+		return
 	}
+
+	x0 := p.cur.x
+	y0 := p.cur.y
+	if isPointCloseToSegment(x1, y1, x0, y0, x3, y3, 1) && isPointCloseToSegment(x2, y2, x0, y0, x3, y3, 1) {
+		p.LineTo(x3, y3)
+		return
+	}
+
+	x01 := (x0 + x1) / 2
+	y01 := (y0 + y1) / 2
+	x12 := (x1 + x2) / 2
+	y12 := (y1 + y2) / 2
+	x23 := (x2 + x3) / 2
+	y23 := (y2 + y3) / 2
+	x012 := (x01 + x12) / 2
+	y012 := (y01 + y12) / 2
+	x123 := (x12 + x23) / 2
+	y123 := (y12 + y23) / 2
+	x0123 := (x012 + x123) / 2
+	y0123 := (y012 + y123) / 2
+	p.cubicTo(x01, y01, x012, y012, x0123, y0123, level+1)
+	p.cubicTo(x123, y123, x23, y23, x3, y3, level+1)
 }
 
 // AppendVerticesAndIndices appends vertices and indices for this path and returns them.