freetype/raster: Implement round joins.
R=r, rsc CC=golang-dev, rog http://codereview.appspot.com/1746043
This commit is contained in:
Nigel Tao
parent
c95fb230fe
commit
4d90648d2c
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@ -0,0 +1,98 @@
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// Copyright 2010 The Freetype-Go Authors. All rights reserved.
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// Use of this source code is governed by your choice of either the
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// FreeType License or the GNU General Public License version 2,
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// both of which can be found in the LICENSE file.
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// This program visualizes the quadratic approximation to the circle, used to
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// implement round joins when stroking paths. The approximation is used in the
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// stroking code for arcs between 0 and 45 degrees, but is visualized here
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// between 0 and 90 degrees. The discrepancy between the approximation and the
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// true circle is clearly visible at angles above 65 degrees.
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package main
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import (
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"bufio"
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"fmt"
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"image"
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"image/png"
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"log"
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"math"
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"os"
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"freetype-go.googlecode.com/hg/freetype/raster"
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)
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func main() {
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const (
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n = 17
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r = 256 * 80
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)
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s := raster.Fixed(r * math.Sqrt(2) / 2)
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t := raster.Fixed(r * math.Tan(math.Pi/8))
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m := image.NewRGBA(800, 600)
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for y := 0; y < m.Height(); y++ {
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for x := 0; x < m.Width(); x++ {
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m.Pixel[y][x] = image.RGBAColor{63, 63, 63, 255}
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}
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}
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mp := raster.NewRGBAPainter(m)
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mp.SetColor(image.Black)
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z := raster.NewRasterizer(800, 600)
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for i := 0; i < n; i++ {
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cx := raster.Fixed(25600 + 51200*(i%4))
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cy := raster.Fixed(2560 + 32000*(i/4))
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c := raster.Point{cx, cy}
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theta := math.Pi * (0.5 + 0.5*float64(i)/(n-1))
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dx := raster.Fixed(r * math.Cos(theta))
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dy := raster.Fixed(r * math.Sin(theta))
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d := raster.Point{dx, dy}
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// Draw a quarter-circle approximated by two quadratic segments,
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// with each segment spanning 45 degrees.
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z.Start(c)
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z.Add1(c.Add(raster.Point{r, 0}))
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z.Add2(c.Add(raster.Point{r, t}), c.Add(raster.Point{s, s}))
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z.Add2(c.Add(raster.Point{t, r}), c.Add(raster.Point{0, r}))
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// Add another quadratic segment whose angle ranges between 0 and 90 degrees.
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// For an explanation of the magic constants 22, 150, 181 and 256, read the
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// comments in the freetype/raster package.
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dot := 256 * d.Dot(raster.Point{0, r}) / (r * r)
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multiple := raster.Fixed(150 - 22*(dot-181)/(256-181))
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z.Add2(c.Add(raster.Point{dx, r + dy}.Mul(multiple)), c.Add(d))
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// Close the curve.
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z.Add1(c)
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}
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z.Rasterize(mp)
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for i := 0; i < n; i++ {
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cx := raster.Fixed(25600 + 51200*(i%4))
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cy := raster.Fixed(2560 + 32000*(i/4))
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for j := 0; j < n; j++ {
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theta := math.Pi * float64(j) / (n - 1)
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dx := raster.Fixed(r * math.Cos(theta))
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dy := raster.Fixed(r * math.Sin(theta))
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m.Set(int((cx+dx)/256), int((cy+dy)/256), image.Yellow)
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}
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}
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// Save that RGBA image to disk.
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f, err := os.Open("out.png", os.O_CREAT|os.O_WRONLY, 0600)
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if err != nil {
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log.Stderr(err)
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os.Exit(1)
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}
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defer f.Close()
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b := bufio.NewWriter(f)
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err = png.Encode(b, m)
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if err != nil {
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log.Stderr(err)
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os.Exit(1)
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}
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err = b.Flush()
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if err != nil {
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log.Stderr(err)
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os.Exit(1)
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}
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fmt.Println("Wrote out.png OK.")
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}
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@ -13,6 +13,9 @@ import (
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// A Fixed is a 24.8 fixed point number.
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type Fixed int32
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// A Fixed64 is a 48.16 fixed point number.
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type Fixed64 int64
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// String returns a human-readable representation of a 24.8 fixed point number.
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// For example, the number one-and-a-quarter becomes "1:064".
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func (x Fixed) String() string {
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@ -23,6 +26,16 @@ func (x Fixed) String() string {
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return fmt.Sprintf("%d:%03d", int32(i), int32(f))
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}
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// String returns a human-readable representation of a 48.16 fixed point number.
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// For example, the number one-and-a-quarter becomes "1:00064".
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func (x Fixed64) String() string {
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i, f := x/65536, x%65536
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if f < 0 {
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f = -f
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}
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return fmt.Sprintf("%d:%05d", int64(i), int64(f))
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}
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// maxAbs returns the maximum of abs(a) and abs(b).
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func maxAbs(a, b Fixed) Fixed {
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if a < 0 {
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@ -58,6 +71,18 @@ func (p Point) Mul(k Fixed) Point {
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return Point{p.X * k / 256, p.Y * k / 256}
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}
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// Neg returns the vector -p, or equivalently p rotated by 180 degrees.
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func (p Point) Neg() Point {
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return Point{-p.X, -p.Y}
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}
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// Dot returns the dot product p·q.
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func (p Point) Dot(q Point) Fixed64 {
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px, py := int64(p.X), int64(p.Y)
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qx, qy := int64(q.X), int64(q.Y)
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return Fixed64(px*qx + py*qy)
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}
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// Len returns the length of the vector p.
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func (p Point) Len() Fixed {
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// TODO(nigeltao): use fixed point math.
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@ -73,22 +98,64 @@ func (p Point) Norm(length Fixed) Point {
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if d == 0 {
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return Point{0, 0}
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}
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// TODO(nigeltao): should we check for overflow?
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return Point{p.X * length / d, p.Y * length / d}
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s, t := int64(length), int64(d)
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x := int64(p.X) * s / t
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y := int64(p.Y) * s / t
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return Point{Fixed(x), Fixed(y)}
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}
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// RotateCW returns the vector p rotated clockwise by 90 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.RotateCW is {0, 1}.
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func (p Point) RotateCW() Point {
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// Rot45CW returns the vector p rotated clockwise by 45 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot45CW is {1/√2, 1/√2}.
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func (p Point) Rot45CW() Point {
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// 181/256 is approximately 1/√2, or sin(π/4).
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px, py := int64(p.X), int64(p.Y)
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qx := (+px - py) * 181 / 256
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qy := (+px + py) * 181 / 256
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return Point{Fixed(qx), Fixed(qy)}
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}
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// Rot90CW returns the vector p rotated clockwise by 90 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot90CW is {0, 1}.
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func (p Point) Rot90CW() Point {
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return Point{-p.Y, p.X}
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}
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// RotateCCW returns the vector p rotated counter-clockwise by 90 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.RotateCCW is {0, -1}.
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func (p Point) RotateCCW() Point {
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// Rot135CW returns the vector p rotated clockwise by 135 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot135CW is {-1/√2, 1/√2}.
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func (p Point) Rot135CW() Point {
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// 181/256 is approximately 1/√2, or sin(π/4).
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px, py := int64(p.X), int64(p.Y)
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qx := (-px - py) * 181 / 256
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qy := (+px - py) * 181 / 256
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return Point{Fixed(qx), Fixed(qy)}
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}
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// Rot45CCW returns the vector p rotated counter-clockwise by 45 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot45CCW is {1/√2, -1/√2}.
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func (p Point) Rot45CCW() Point {
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// 181/256 is approximately 1/√2, or sin(π/4).
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px, py := int64(p.X), int64(p.Y)
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qx := (+px + py) * 181 / 256
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qy := (-px + py) * 181 / 256
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return Point{Fixed(qx), Fixed(qy)}
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}
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// Rot90CCW returns the vector p rotated counter-clockwise by 90 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot90CCW is {0, -1}.
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func (p Point) Rot90CCW() Point {
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return Point{p.Y, -p.X}
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}
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// Rot135CCW returns the vector p rotated counter-clockwise by 135 degrees.
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// Note that the Y-axis grows downwards, so {1, 0}.Rot135CCW is {-1/√2, -1/√2}.
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func (p Point) Rot135CCW() Point {
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// 181/256 is approximately 1/√2, or sin(π/4).
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px, py := int64(p.X), int64(p.Y)
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qx := (-px + py) * 181 / 256
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qy := (-px - py) * 181 / 256
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return Point{Fixed(qx), Fixed(qy)}
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}
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// An Adder accumulates points on a curve.
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type Adder interface {
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// Start starts a new curve at the given point.
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@ -258,10 +325,10 @@ func addCap(p Adder, cap Cap, center, end Point) {
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switch cap {
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case RoundCap:
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// The cubic Bézier approximation to a circle involves the magic number
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// (sqrt(2) - 1) * 4/3, which is approximately 141 / 256.
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// (√2 - 1) * 4/3, which is approximately 141/256.
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const k = 141
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d := end.Sub(center)
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e := d.RotateCCW()
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e := d.Rot90CCW()
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side := center.Add(e)
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start := center.Sub(d)
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d, e = d.Mul(k), e.Mul(k)
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@ -271,7 +338,7 @@ func addCap(p Adder, cap Cap, center, end Point) {
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p.Add1(end)
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case SquareCap:
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d := end.Sub(center)
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e := d.RotateCCW()
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e := d.Rot90CCW()
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side := center.Add(e)
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p.Add1(side.Sub(d))
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p.Add1(side.Add(d))
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@ -279,6 +346,110 @@ func addCap(p Adder, cap Cap, center, end Point) {
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}
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}
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func addJoin(lhs, rhs Adder, join Join, a, anorm, bnorm Point) {
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switch join {
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case RoundJoin:
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dot := anorm.Rot90CW().Dot(bnorm)
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if dot >= 0 {
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addArc(lhs, a, anorm, bnorm)
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rhs.Add1(a.Sub(bnorm))
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} else {
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lhs.Add1(a.Add(bnorm))
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addArc(rhs, a, anorm.Neg(), bnorm.Neg())
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}
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case BevelJoin:
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lhs.Add1(a.Add(bnorm))
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rhs.Add1(a.Sub(bnorm))
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case MiterJoin:
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panic("freetype/raster: miter join unimplemented")
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}
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}
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// addArc adds a circular arc from pivot+n0 to pivot+n1 to p. The shorter of
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// the two possible arcs is taken, i.e. the one spanning <= 180 degrees.
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// The two vectors n0 and n1 must be of equal length.
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func addArc(p Adder, pivot, n0, n1 Point) {
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// r2 is the square of the length of n0.
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r2 := n0.Dot(n0)
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if r2 < 4096 {
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// The arc radius is so small that we collapse to a straight line.
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p.Add1(pivot.Add(n1))
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return
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}
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// We approximate the arc by 0, 1, 2 or 3 45-degree quadratic segments plus
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// a final quadratic segment from s to n1. Each 45-degree segment has control
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// points {1, 0}, {1, tan(π/8)} and {1/√2, 1/√2} suitably scaled, rotated and
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// translated. tan(π/8) is approximately 106/256.
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const t = 106
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var s Point
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// We determine which octant the angle between n0 and n1 is in via three dot products.
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// m0, m1 and m2 are n0 rotated clockwise by 45, 90 and 135 degrees.
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m0 := n0.Rot45CW()
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m1 := n0.Rot90CW()
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m2 := m0.Rot90CW()
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if m1.Dot(n1) >= 0 {
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if n0.Dot(n1) >= 0 {
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if m2.Dot(n1) <= 0 {
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// n1 is between 0 and 45 degrees clockwise of n0.
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s = n0
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} else {
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// n1 is between 45 and 90 degrees clockwise of n0.
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p.Add2(pivot.Add(n0).Add(m1.Mul(t)), pivot.Add(m0))
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s = m0
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}
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} else {
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pm1, n0t := pivot.Add(m1), n0.Mul(t)
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p.Add2(pivot.Add(n0).Add(m1.Mul(t)), pivot.Add(m0))
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p.Add2(pm1.Add(n0t), pm1)
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if m0.Dot(n1) >= 0 {
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// n1 is between 90 and 135 degrees clockwise of n0.
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s = m1
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} else {
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// n1 is between 135 and 180 degrees clockwise of n0.
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p.Add2(pm1.Sub(n0t), pivot.Add(m2))
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s = m2
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}
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}
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} else {
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if n0.Dot(n1) >= 0 {
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if m0.Dot(n1) >= 0 {
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// n1 is between 0 and 45 degrees counter-clockwise of n0.
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s = n0
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} else {
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// n1 is between 45 and 90 degrees counter-clockwise of n0.
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p.Add2(pivot.Add(n0).Sub(m1.Mul(t)), pivot.Sub(m2))
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s = m2.Neg()
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}
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} else {
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pm1, n0t := pivot.Sub(m1), n0.Mul(t)
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p.Add2(pivot.Add(n0).Sub(m1.Mul(t)), pivot.Sub(m2))
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p.Add2(pm1.Add(n0t), pm1)
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if m2.Dot(n1) <= 0 {
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// n1 is between 90 and 135 degrees counter-clockwise of n0.
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s = m1.Neg()
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} else {
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// n1 is between 135 and 180 degrees counter-clockwise of n0.
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p.Add2(pm1.Sub(n0t), pivot.Sub(m0))
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s = m0.Neg()
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}
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}
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}
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// The final quadratic segment has two endpoints s and n1 and the middle
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// control point is a multiple of s.Add(n1), i.e. it is on the angle bisector
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// of those two points. The multiple ranges between 128/256 and 150/256 as
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// the angle between s and n1 ranges between 0 and 45 degrees.
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// When the angle is 0 degrees (i.e. s and n1 are coincident) then s.Add(n1)
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// is twice s and so the middle control point of the degenerate quadratic
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// segment should be half s.Add(n1), and half = 128/256.
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// When the angle is 45 degrees then 150/256 is the ratio of the lengths of
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// the two vectors {1, tan(π/8)} and {1 + 1/√2, 1/√2}.
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// d is the normalized dot product between s and n1. Since the angle ranges
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// between 0 and 45 degrees then d ranges between 256/256 and 181/256.
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d := 256 * s.Dot(n1) / r2
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multiple := Fixed(150 - 22*(d-181)/(256-181))
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p.Add2(pivot.Add(s.Add(n1).Mul(multiple)), pivot.Add(n1))
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}
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// stroke adds the stroked Path q to p, where q consists of exactly one curve.
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func stroke(p Adder, q Path, width Fixed, cap Cap, join Join) {
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// Stroking is implemented by deriving two paths each width/2 apart from q.
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@ -286,27 +457,24 @@ func stroke(p Adder, q Path, width Fixed, cap Cap, join Join) {
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// path is accumulated in r, and once we've finished adding the LHS to p
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// we add the RHS in reverse order.
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r := Path(make([]Fixed, 0, len(q)))
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var start Point
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var start, anorm Point
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a := Point{q[1], q[2]}
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i := 4
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for i < len(q) {
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switch q[i] {
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case 1:
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bx, by := q[i+1], q[i+2]
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delta := Point{bx - a.X, by - a.Y}
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normal := delta.Norm(width / 2).RotateCCW()
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b := Point{q[i+1], q[i+2]}
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bnorm := b.Sub(a).Norm(width / 2).Rot90CCW()
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if i == 4 {
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start = Point{a.X + normal.X, a.Y + normal.Y}
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start = a.Add(bnorm)
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p.Start(start)
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r.Start(Point{a.X - normal.X, a.Y - normal.Y})
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r.Start(a.Sub(bnorm))
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} else {
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// TODO(nigeltao): handle joins.
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p.Add1(Point{a.X + normal.X, a.Y + normal.Y})
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r.Add1(Point{a.X - normal.X, a.Y - normal.Y})
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addJoin(p, &r, join, a, anorm, bnorm)
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}
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p.Add1(Point{bx + normal.X, by + normal.Y})
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r.Add1(Point{bx - normal.X, by - normal.Y})
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a = Point{q[i+1], q[i+2]}
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p.Add1(b.Add(bnorm))
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r.Add1(b.Sub(bnorm))
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a, anorm = b, bnorm
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i += 4
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case 2:
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panic("freetype/raster: stroke unimplemented for quadratic segments")
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