From 4d90648d2cdb6347a9c11f708a585575b7a90a0c Mon Sep 17 00:00:00 2001 From: Nigel Tao Date: Wed, 30 Jun 2010 15:10:17 +1000 Subject: [PATCH] freetype/raster: Implement round joins. R=r, rsc CC=golang-dev, rog http://codereview.appspot.com/1746043 --- example/round/main.go | 98 ++++++++++++++++++ freetype/raster/geom.go | 214 +++++++++++++++++++++++++++++++++++----- 2 files changed, 289 insertions(+), 23 deletions(-) create mode 100644 example/round/main.go diff --git a/example/round/main.go b/example/round/main.go new file mode 100644 index 0000000..c048b3a --- /dev/null +++ b/example/round/main.go @@ -0,0 +1,98 @@ +// Copyright 2010 The Freetype-Go Authors. All rights reserved. +// Use of this source code is governed by your choice of either the +// FreeType License or the GNU General Public License version 2, +// both of which can be found in the LICENSE file. + +// This program visualizes the quadratic approximation to the circle, used to +// implement round joins when stroking paths. The approximation is used in the +// stroking code for arcs between 0 and 45 degrees, but is visualized here +// between 0 and 90 degrees. The discrepancy between the approximation and the +// true circle is clearly visible at angles above 65 degrees. +package main + +import ( + "bufio" + "fmt" + "image" + "image/png" + "log" + "math" + "os" + + "freetype-go.googlecode.com/hg/freetype/raster" +) + +func main() { + const ( + n = 17 + r = 256 * 80 + ) + s := raster.Fixed(r * math.Sqrt(2) / 2) + t := raster.Fixed(r * math.Tan(math.Pi/8)) + + m := image.NewRGBA(800, 600) + for y := 0; y < m.Height(); y++ { + for x := 0; x < m.Width(); x++ { + m.Pixel[y][x] = image.RGBAColor{63, 63, 63, 255} + } + } + mp := raster.NewRGBAPainter(m) + mp.SetColor(image.Black) + z := raster.NewRasterizer(800, 600) + + for i := 0; i < n; i++ { + cx := raster.Fixed(25600 + 51200*(i%4)) + cy := raster.Fixed(2560 + 32000*(i/4)) + c := raster.Point{cx, cy} + theta := math.Pi * (0.5 + 0.5*float64(i)/(n-1)) + dx := raster.Fixed(r * math.Cos(theta)) + dy := raster.Fixed(r * math.Sin(theta)) + d := raster.Point{dx, dy} + // Draw a quarter-circle approximated by two quadratic segments, + // with each segment spanning 45 degrees. + z.Start(c) + z.Add1(c.Add(raster.Point{r, 0})) + z.Add2(c.Add(raster.Point{r, t}), c.Add(raster.Point{s, s})) + z.Add2(c.Add(raster.Point{t, r}), c.Add(raster.Point{0, r})) + // Add another quadratic segment whose angle ranges between 0 and 90 degrees. + // For an explanation of the magic constants 22, 150, 181 and 256, read the + // comments in the freetype/raster package. + dot := 256 * d.Dot(raster.Point{0, r}) / (r * r) + multiple := raster.Fixed(150 - 22*(dot-181)/(256-181)) + z.Add2(c.Add(raster.Point{dx, r + dy}.Mul(multiple)), c.Add(d)) + // Close the curve. + z.Add1(c) + } + z.Rasterize(mp) + + for i := 0; i < n; i++ { + cx := raster.Fixed(25600 + 51200*(i%4)) + cy := raster.Fixed(2560 + 32000*(i/4)) + for j := 0; j < n; j++ { + theta := math.Pi * float64(j) / (n - 1) + dx := raster.Fixed(r * math.Cos(theta)) + dy := raster.Fixed(r * math.Sin(theta)) + m.Set(int((cx+dx)/256), int((cy+dy)/256), image.Yellow) + } + } + + // Save that RGBA image to disk. + f, err := os.Open("out.png", os.O_CREAT|os.O_WRONLY, 0600) + if err != nil { + log.Stderr(err) + os.Exit(1) + } + defer f.Close() + b := bufio.NewWriter(f) + err = png.Encode(b, m) + if err != nil { + log.Stderr(err) + os.Exit(1) + } + err = b.Flush() + if err != nil { + log.Stderr(err) + os.Exit(1) + } + fmt.Println("Wrote out.png OK.") +} diff --git a/freetype/raster/geom.go b/freetype/raster/geom.go index 5de33eb..409cc9d 100644 --- a/freetype/raster/geom.go +++ b/freetype/raster/geom.go @@ -13,6 +13,9 @@ import ( // A Fixed is a 24.8 fixed point number. type Fixed int32 +// A Fixed64 is a 48.16 fixed point number. +type Fixed64 int64 + // String returns a human-readable representation of a 24.8 fixed point number. // For example, the number one-and-a-quarter becomes "1:064". func (x Fixed) String() string { @@ -23,6 +26,16 @@ func (x Fixed) String() string { return fmt.Sprintf("%d:%03d", int32(i), int32(f)) } +// String returns a human-readable representation of a 48.16 fixed point number. +// For example, the number one-and-a-quarter becomes "1:00064". +func (x Fixed64) String() string { + i, f := x/65536, x%65536 + if f < 0 { + f = -f + } + return fmt.Sprintf("%d:%05d", int64(i), int64(f)) +} + // maxAbs returns the maximum of abs(a) and abs(b). func maxAbs(a, b Fixed) Fixed { if a < 0 { @@ -58,6 +71,18 @@ func (p Point) Mul(k Fixed) Point { return Point{p.X * k / 256, p.Y * k / 256} } +// Neg returns the vector -p, or equivalently p rotated by 180 degrees. +func (p Point) Neg() Point { + return Point{-p.X, -p.Y} +} + +// Dot returns the dot product p·q. +func (p Point) Dot(q Point) Fixed64 { + px, py := int64(p.X), int64(p.Y) + qx, qy := int64(q.X), int64(q.Y) + return Fixed64(px*qx + py*qy) +} + // Len returns the length of the vector p. func (p Point) Len() Fixed { // TODO(nigeltao): use fixed point math. @@ -73,22 +98,64 @@ func (p Point) Norm(length Fixed) Point { if d == 0 { return Point{0, 0} } - // TODO(nigeltao): should we check for overflow? - return Point{p.X * length / d, p.Y * length / d} + s, t := int64(length), int64(d) + x := int64(p.X) * s / t + y := int64(p.Y) * s / t + return Point{Fixed(x), Fixed(y)} } -// RotateCW returns the vector p rotated clockwise by 90 degrees. -// Note that the Y-axis grows downwards, so {1, 0}.RotateCW is {0, 1}. -func (p Point) RotateCW() Point { +// Rot45CW returns the vector p rotated clockwise by 45 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot45CW is {1/√2, 1/√2}. +func (p Point) Rot45CW() Point { + // 181/256 is approximately 1/√2, or sin(π/4). + px, py := int64(p.X), int64(p.Y) + qx := (+px - py) * 181 / 256 + qy := (+px + py) * 181 / 256 + return Point{Fixed(qx), Fixed(qy)} +} + +// Rot90CW returns the vector p rotated clockwise by 90 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot90CW is {0, 1}. +func (p Point) Rot90CW() Point { return Point{-p.Y, p.X} } -// RotateCCW returns the vector p rotated counter-clockwise by 90 degrees. -// Note that the Y-axis grows downwards, so {1, 0}.RotateCCW is {0, -1}. -func (p Point) RotateCCW() Point { +// Rot135CW returns the vector p rotated clockwise by 135 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot135CW is {-1/√2, 1/√2}. +func (p Point) Rot135CW() Point { + // 181/256 is approximately 1/√2, or sin(π/4). + px, py := int64(p.X), int64(p.Y) + qx := (-px - py) * 181 / 256 + qy := (+px - py) * 181 / 256 + return Point{Fixed(qx), Fixed(qy)} +} + +// Rot45CCW returns the vector p rotated counter-clockwise by 45 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot45CCW is {1/√2, -1/√2}. +func (p Point) Rot45CCW() Point { + // 181/256 is approximately 1/√2, or sin(π/4). + px, py := int64(p.X), int64(p.Y) + qx := (+px + py) * 181 / 256 + qy := (-px + py) * 181 / 256 + return Point{Fixed(qx), Fixed(qy)} +} + +// Rot90CCW returns the vector p rotated counter-clockwise by 90 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot90CCW is {0, -1}. +func (p Point) Rot90CCW() Point { return Point{p.Y, -p.X} } +// Rot135CCW returns the vector p rotated counter-clockwise by 135 degrees. +// Note that the Y-axis grows downwards, so {1, 0}.Rot135CCW is {-1/√2, -1/√2}. +func (p Point) Rot135CCW() Point { + // 181/256 is approximately 1/√2, or sin(π/4). + px, py := int64(p.X), int64(p.Y) + qx := (-px + py) * 181 / 256 + qy := (-px - py) * 181 / 256 + return Point{Fixed(qx), Fixed(qy)} +} + // An Adder accumulates points on a curve. type Adder interface { // Start starts a new curve at the given point. @@ -258,10 +325,10 @@ func addCap(p Adder, cap Cap, center, end Point) { switch cap { case RoundCap: // The cubic Bézier approximation to a circle involves the magic number - // (sqrt(2) - 1) * 4/3, which is approximately 141 / 256. + // (√2 - 1) * 4/3, which is approximately 141/256. const k = 141 d := end.Sub(center) - e := d.RotateCCW() + e := d.Rot90CCW() side := center.Add(e) start := center.Sub(d) d, e = d.Mul(k), e.Mul(k) @@ -271,7 +338,7 @@ func addCap(p Adder, cap Cap, center, end Point) { p.Add1(end) case SquareCap: d := end.Sub(center) - e := d.RotateCCW() + e := d.Rot90CCW() side := center.Add(e) p.Add1(side.Sub(d)) p.Add1(side.Add(d)) @@ -279,6 +346,110 @@ func addCap(p Adder, cap Cap, center, end Point) { } } +func addJoin(lhs, rhs Adder, join Join, a, anorm, bnorm Point) { + switch join { + case RoundJoin: + dot := anorm.Rot90CW().Dot(bnorm) + if dot >= 0 { + addArc(lhs, a, anorm, bnorm) + rhs.Add1(a.Sub(bnorm)) + } else { + lhs.Add1(a.Add(bnorm)) + addArc(rhs, a, anorm.Neg(), bnorm.Neg()) + } + case BevelJoin: + lhs.Add1(a.Add(bnorm)) + rhs.Add1(a.Sub(bnorm)) + case MiterJoin: + panic("freetype/raster: miter join unimplemented") + } +} + +// addArc adds a circular arc from pivot+n0 to pivot+n1 to p. The shorter of +// the two possible arcs is taken, i.e. the one spanning <= 180 degrees. +// The two vectors n0 and n1 must be of equal length. +func addArc(p Adder, pivot, n0, n1 Point) { + // r2 is the square of the length of n0. + r2 := n0.Dot(n0) + if r2 < 4096 { + // The arc radius is so small that we collapse to a straight line. + p.Add1(pivot.Add(n1)) + return + } + // We approximate the arc by 0, 1, 2 or 3 45-degree quadratic segments plus + // a final quadratic segment from s to n1. Each 45-degree segment has control + // points {1, 0}, {1, tan(π/8)} and {1/√2, 1/√2} suitably scaled, rotated and + // translated. tan(π/8) is approximately 106/256. + const t = 106 + var s Point + // We determine which octant the angle between n0 and n1 is in via three dot products. + // m0, m1 and m2 are n0 rotated clockwise by 45, 90 and 135 degrees. + m0 := n0.Rot45CW() + m1 := n0.Rot90CW() + m2 := m0.Rot90CW() + if m1.Dot(n1) >= 0 { + if n0.Dot(n1) >= 0 { + if m2.Dot(n1) <= 0 { + // n1 is between 0 and 45 degrees clockwise of n0. + s = n0 + } else { + // n1 is between 45 and 90 degrees clockwise of n0. + p.Add2(pivot.Add(n0).Add(m1.Mul(t)), pivot.Add(m0)) + s = m0 + } + } else { + pm1, n0t := pivot.Add(m1), n0.Mul(t) + p.Add2(pivot.Add(n0).Add(m1.Mul(t)), pivot.Add(m0)) + p.Add2(pm1.Add(n0t), pm1) + if m0.Dot(n1) >= 0 { + // n1 is between 90 and 135 degrees clockwise of n0. + s = m1 + } else { + // n1 is between 135 and 180 degrees clockwise of n0. + p.Add2(pm1.Sub(n0t), pivot.Add(m2)) + s = m2 + } + } + } else { + if n0.Dot(n1) >= 0 { + if m0.Dot(n1) >= 0 { + // n1 is between 0 and 45 degrees counter-clockwise of n0. + s = n0 + } else { + // n1 is between 45 and 90 degrees counter-clockwise of n0. + p.Add2(pivot.Add(n0).Sub(m1.Mul(t)), pivot.Sub(m2)) + s = m2.Neg() + } + } else { + pm1, n0t := pivot.Sub(m1), n0.Mul(t) + p.Add2(pivot.Add(n0).Sub(m1.Mul(t)), pivot.Sub(m2)) + p.Add2(pm1.Add(n0t), pm1) + if m2.Dot(n1) <= 0 { + // n1 is between 90 and 135 degrees counter-clockwise of n0. + s = m1.Neg() + } else { + // n1 is between 135 and 180 degrees counter-clockwise of n0. + p.Add2(pm1.Sub(n0t), pivot.Sub(m0)) + s = m0.Neg() + } + } + } + // The final quadratic segment has two endpoints s and n1 and the middle + // control point is a multiple of s.Add(n1), i.e. it is on the angle bisector + // of those two points. The multiple ranges between 128/256 and 150/256 as + // the angle between s and n1 ranges between 0 and 45 degrees. + // When the angle is 0 degrees (i.e. s and n1 are coincident) then s.Add(n1) + // is twice s and so the middle control point of the degenerate quadratic + // segment should be half s.Add(n1), and half = 128/256. + // When the angle is 45 degrees then 150/256 is the ratio of the lengths of + // the two vectors {1, tan(π/8)} and {1 + 1/√2, 1/√2}. + // d is the normalized dot product between s and n1. Since the angle ranges + // between 0 and 45 degrees then d ranges between 256/256 and 181/256. + d := 256 * s.Dot(n1) / r2 + multiple := Fixed(150 - 22*(d-181)/(256-181)) + p.Add2(pivot.Add(s.Add(n1).Mul(multiple)), pivot.Add(n1)) +} + // stroke adds the stroked Path q to p, where q consists of exactly one curve. func stroke(p Adder, q Path, width Fixed, cap Cap, join Join) { // Stroking is implemented by deriving two paths each width/2 apart from q. @@ -286,27 +457,24 @@ func stroke(p Adder, q Path, width Fixed, cap Cap, join Join) { // path is accumulated in r, and once we've finished adding the LHS to p // we add the RHS in reverse order. r := Path(make([]Fixed, 0, len(q))) - var start Point + var start, anorm Point a := Point{q[1], q[2]} i := 4 for i < len(q) { switch q[i] { case 1: - bx, by := q[i+1], q[i+2] - delta := Point{bx - a.X, by - a.Y} - normal := delta.Norm(width / 2).RotateCCW() + b := Point{q[i+1], q[i+2]} + bnorm := b.Sub(a).Norm(width / 2).Rot90CCW() if i == 4 { - start = Point{a.X + normal.X, a.Y + normal.Y} + start = a.Add(bnorm) p.Start(start) - r.Start(Point{a.X - normal.X, a.Y - normal.Y}) + r.Start(a.Sub(bnorm)) } else { - // TODO(nigeltao): handle joins. - p.Add1(Point{a.X + normal.X, a.Y + normal.Y}) - r.Add1(Point{a.X - normal.X, a.Y - normal.Y}) + addJoin(p, &r, join, a, anorm, bnorm) } - p.Add1(Point{bx + normal.X, by + normal.Y}) - r.Add1(Point{bx - normal.X, by - normal.Y}) - a = Point{q[i+1], q[i+2]} + p.Add1(b.Add(bnorm)) + r.Add1(b.Sub(bnorm)) + a, anorm = b, bnorm i += 4 case 2: panic("freetype/raster: stroke unimplemented for quadratic segments")