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Remove Point methods other than Add, Sub, Mul.

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This is in anticipation of removing the Point type entirely. Instead, we will
use the fixed.Point26_6 type from golang.org/x/image/math/fixed.
This commit is contained in:
Nigel Tao 2015-08-13 21:17:25 +10:00
parent 84ba7fbd7a
commit d91c1d6844
3 changed files with 95 additions and 78 deletions
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9
example/round/main.go
View File

@ -29,6 +29,13 @@ import (
"github.com/golang/freetype/raster"
)
// pDot returns the dot product p·q.
func pDot(p, q raster.Point) raster.Fix64 {
px, py := int64(p.X), int64(p.Y)
qx, qy := int64(q.X), int64(q.Y)
return raster.Fix64(px*qx + py*qy)
}
func main() {
const (
n = 17
@ -60,7 +67,7 @@ func main() {
// 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{X: 0, Y: r}) / (r * r)
dot := 256 * pDot(d, raster.Point{X: 0, Y: r}) / (r * r)
multiple := raster.Fix32(150 - 22*(dot-181)/(256-181))
z.Add2(c.Add(raster.Point{X: dx, Y: r + dy}.Mul(multiple)), c.Add(d))
// Close the curve.

50
raster/geom.go
View File

@ -76,30 +76,30 @@ func (p Point) Mul(k Fix32) 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 {
// pNeg returns the vector -p, or equivalently p rotated by 180 degrees.
func pNeg(p Point) Point {
return Point{-p.X, -p.Y}
}
// Dot returns the dot product p·q.
func (p Point) Dot(q Point) Fix64 {
// pDot returns the dot product p·q.
func pDot(p, q Point) Fix64 {
px, py := int64(p.X), int64(p.Y)
qx, qy := int64(q.X), int64(q.Y)
return Fix64(px*qx + py*qy)
}
// Len returns the length of the vector p.
func (p Point) Len() Fix32 {
// pLen returns the length of the vector p.
func pLen(p Point) Fix32 {
// TODO(nigeltao): use fixed point math.
x := float64(p.X)
y := float64(p.Y)
return Fix32(math.Sqrt(x*x + y*y))
}
// Norm returns the vector p normalized to the given length, or the zero Point
// if p is degenerate.
func (p Point) Norm(length Fix32) Point {
d := p.Len()
// pNorm returns the vector p normalized to the given length, or zero if p is
// degenerate.
func pNorm(p Point, length Fix32) Point {
d := pLen(p)
if d == 0 {
return Point{}
}
@ -109,9 +109,10 @@ func (p Point) Norm(length Fix32) Point {
return Point{Fix32(x), Fix32(y)}
}
// Rot45CW returns the vector p rotated clockwise by 45 degrees.
// pRot45CW 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 {
func pRot45CW(p Point) Point {
// 181/256 is approximately 1/√2, or sin(π/4).
px, py := int64(p.X), int64(p.Y)
qx := (+px - py) * 181 / 256
@ -119,15 +120,17 @@ func (p Point) Rot45CW() Point {
return Point{Fix32(qx), Fix32(qy)}
}
// Rot90CW returns the vector p rotated clockwise by 90 degrees.
// pRot90CW 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 {
func pRot90CW(p Point) Point {
return Point{-p.Y, p.X}
}
// Rot135CW returns the vector p rotated clockwise by 135 degrees.
// pRot135CW 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 {
func pRot135CW(p Point) Point {
// 181/256 is approximately 1/√2, or sin(π/4).
px, py := int64(p.X), int64(p.Y)
qx := (-px - py) * 181 / 256
@ -135,9 +138,10 @@ func (p Point) Rot135CW() Point {
return Point{Fix32(qx), Fix32(qy)}
}
// Rot45CCW returns the vector p rotated counter-clockwise by 45 degrees.
// pRot45CCW 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 {
func pRot45CCW(p Point) Point {
// 181/256 is approximately 1/√2, or sin(π/4).
px, py := int64(p.X), int64(p.Y)
qx := (+px + py) * 181 / 256
@ -145,15 +149,17 @@ func (p Point) Rot45CCW() Point {
return Point{Fix32(qx), Fix32(qy)}
}
// Rot90CCW returns the vector p rotated counter-clockwise by 90 degrees.
// pRot90CCW 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 {
func pRot90CCW(p Point) Point {
return Point{p.Y, -p.X}
}
// Rot135CCW returns the vector p rotated counter-clockwise by 135 degrees.
// pRot135CCW 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 {
func pRot135CCW(p Point) Point {
// 181/256 is approximately 1/√2, or sin(π/4).
px, py := int64(p.X), int64(p.Y)
qx := (-px + py) * 181 / 256

114
raster/stroke.go
View File

@ -45,7 +45,7 @@ func roundCapper(p Adder, halfWidth Fix32, pivot, n1 Point) {
// The cubic Bézier approximation to a circle involves the magic number
// (√2 - 1) * 4/3, which is approximately 141/256.
const k = 141
e := n1.Rot90CCW()
e := pRot90CCW(n1)
side := pivot.Add(e)
start, end := pivot.Sub(n1), pivot.Add(n1)
d, e := n1.Mul(k), e.Mul(k)
@ -64,7 +64,7 @@ func buttCapper(p Adder, halfWidth Fix32, pivot, n1 Point) {
var SquareCapper Capper = CapperFunc(squareCapper)
func squareCapper(p Adder, halfWidth Fix32, pivot, n1 Point) {
e := n1.Rot90CCW()
e := pRot90CCW(n1)
side := pivot.Add(e)
p.Add1(side.Sub(n1))
p.Add1(side.Add(n1))
@ -75,13 +75,13 @@ func squareCapper(p Adder, halfWidth Fix32, pivot, n1 Point) {
var RoundJoiner Joiner = JoinerFunc(roundJoiner)
func roundJoiner(lhs, rhs Adder, haflWidth Fix32, pivot, n0, n1 Point) {
dot := n0.Rot90CW().Dot(n1)
dot := pDot(pRot90CW(n0), n1)
if dot >= 0 {
addArc(lhs, pivot, n0, n1)
rhs.Add1(pivot.Sub(n1))
} else {
lhs.Add1(pivot.Add(n1))
addArc(rhs, pivot, n0.Neg(), n1.Neg())
addArc(rhs, pivot, pNeg(n0), pNeg(n1))
}
}
@ -94,30 +94,31 @@ func bevelJoiner(lhs, rhs Adder, haflWidth Fix32, pivot, n0, n1 Point) {
}
// 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.
// 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)
r2 := pDot(n0, n0)
if r2 < epsilon {
// 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.
// 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 tpo8 = 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 {
// 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 := pRot45CW(n0)
m1 := pRot90CW(n0)
m2 := pRot90CW(m0)
if pDot(m1, n1) >= 0 {
if pDot(n0, n1) >= 0 {
if pDot(m2, n1) <= 0 {
// n1 is between 0 and 45 degrees clockwise of n0.
s = n0
} else {
@ -129,7 +130,7 @@ func addArc(p Adder, pivot, n0, n1 Point) {
pm1, n0t := pivot.Add(m1), n0.Mul(tpo8)
p.Add2(pivot.Add(n0).Add(m1.Mul(tpo8)), pivot.Add(m0))
p.Add2(pm1.Add(n0t), pm1)
if m0.Dot(n1) >= 0 {
if pDot(m0, n1) >= 0 {
// n1 is between 90 and 135 degrees clockwise of n0.
s = m1
} else {
@ -139,41 +140,44 @@ func addArc(p Adder, pivot, n0, n1 Point) {
}
}
} else {
if n0.Dot(n1) >= 0 {
if m0.Dot(n1) >= 0 {
if pDot(n0, n1) >= 0 {
if pDot(m0, 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(tpo8)), pivot.Sub(m2))
s = m2.Neg()
s = pNeg(m2)
}
} else {
pm1, n0t := pivot.Sub(m1), n0.Mul(tpo8)
p.Add2(pivot.Add(n0).Sub(m1.Mul(tpo8)), pivot.Sub(m2))
p.Add2(pm1.Add(n0t), pm1)
if m2.Dot(n1) <= 0 {
if pDot(m2, n1) <= 0 {
// n1 is between 90 and 135 degrees counter-clockwise of n0.
s = m1.Neg()
s = pNeg(m1)
} else {
// n1 is between 135 and 180 degrees counter-clockwise of n0.
p.Add2(pm1.Sub(n0t), pivot.Sub(m0))
s = m0.Neg()
s = pNeg(m0)
}
}
}
// 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.
// 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
d := 256 * pDot(s, n1) / r2
multiple := Fix32(150 - 22*(d-181)/(256-181))
p.Add2(pivot.Add(s.Add(n1).Mul(multiple)), pivot.Add(n1))
}
@ -186,13 +190,13 @@ func midpoint(a, b Point) Point {
// angleGreaterThan45 returns whether the angle between two vectors is more
// than 45 degrees.
func angleGreaterThan45(v0, v1 Point) bool {
v := v0.Rot45CCW()
return v.Dot(v1) < 0 || v.Rot90CW().Dot(v1) < 0
v := pRot45CCW(v0)
return pDot(v, v1) < 0 || pDot(pRot90CW(v), v1) < 0
}
// interpolate returns the point (1-t)*a + t*b.
func interpolate(a, b Point, t Fix64) Point {
s := 65536 - t
s := 1<<16 - t
x := s*Fix64(a.X) + t*Fix64(b.X)
y := s*Fix64(a.Y) + t*Fix64(b.Y)
return Point{Fix32(x >> 16), Fix32(y >> 16)}
@ -268,14 +272,14 @@ func (k *stroker) addNonCurvy2(b, c Point) {
c := ps[2*t+0]
ab := b.Sub(a)
bc := c.Sub(b)
abIsSmall := ab.Dot(ab) < Fix64(1<<16)
bcIsSmall := bc.Dot(bc) < Fix64(1<<16)
abIsSmall := pDot(ab, ab) < Fix64(1<<16)
bcIsSmall := pDot(bc, bc) < Fix64(1<<16)
if abIsSmall && bcIsSmall {
// Approximate the segment by a circular arc.
cnorm = bc.Norm(k.u).Rot90CCW()
cnorm = pRot90CCW(pNorm(bc, k.u))
mac := midpoint(a, c)
addArc(k.p, mac, anorm, cnorm)
addArc(&k.r, mac, anorm.Neg(), cnorm.Neg())
addArc(&k.r, mac, pNeg(anorm), pNeg(cnorm))
} else if depth < maxDepth && angleGreaterThan45(ab, bc) {
// Divide the segment in two and push both halves on the stack.
mab := midpoint(a, b)
@ -290,8 +294,8 @@ func (k *stroker) addNonCurvy2(b, c Point) {
continue
} else {
// Translate the control points.
bnorm := c.Sub(a).Norm(k.u).Rot90CCW()
cnorm = bc.Norm(k.u).Rot90CCW()
bnorm := pRot90CCW(pNorm(c.Sub(a), k.u))
cnorm = pRot90CCW(pNorm(bc, k.u))
k.p.Add2(b.Add(bnorm), c.Add(cnorm))
k.r.Add2(b.Sub(bnorm), c.Sub(cnorm))
}
@ -307,7 +311,7 @@ func (k *stroker) addNonCurvy2(b, c Point) {
// Add1 adds a linear segment to the stroker.
func (k *stroker) Add1(b Point) {
bnorm := b.Sub(k.a).Norm(k.u).Rot90CCW()
bnorm := pRot90CCW(pNorm(b.Sub(k.a), k.u))
if len(k.r) == 0 {
k.p.Start(k.a.Add(bnorm))
k.r.Start(k.a.Sub(bnorm))
@ -323,7 +327,7 @@ func (k *stroker) Add1(b Point) {
func (k *stroker) Add2(b, c Point) {
ab := b.Sub(k.a)
bc := c.Sub(b)
abnorm := ab.Norm(k.u).Rot90CCW()
abnorm := pRot90CCW(pNorm(ab, k.u))
if len(k.r) == 0 {
k.p.Start(k.a.Add(abnorm))
k.r.Start(k.a.Sub(abnorm))
@ -332,10 +336,10 @@ func (k *stroker) Add2(b, c Point) {
}
// Approximate nearly-degenerate quadratics by linear segments.
abIsSmall := ab.Dot(ab) < epsilon
bcIsSmall := bc.Dot(bc) < epsilon
abIsSmall := pDot(ab, ab) < epsilon
bcIsSmall := pDot(bc, bc) < epsilon
if abIsSmall || bcIsSmall {
acnorm := c.Sub(k.a).Norm(k.u).Rot90CCW()
acnorm := pRot90CCW(pNorm(c.Sub(k.a), k.u))
k.p.Add1(c.Add(acnorm))
k.r.Add1(c.Sub(acnorm))
k.a, k.anorm = c, acnorm
@ -345,7 +349,7 @@ func (k *stroker) Add2(b, c Point) {
// The quadratic segment (k.a, b, c) has a point of maximum curvature.
// If this occurs at an end point, we process the segment as a whole.
t := curviest2(k.a, b, c)
if t <= 0 || t >= 65536 {
if t <= 0 || 65536 <= t {
k.addNonCurvy2(b, c)
return
}
@ -359,13 +363,13 @@ func (k *stroker) Add2(b, c Point) {
// If the vectors ab and bc are close to being in opposite directions,
// then the decomposition can become unstable, so we approximate the
// quadratic segment by two linear segments joined by an arc.
bcnorm := bc.Norm(k.u).Rot90CCW()
if abnorm.Dot(bcnorm) < -Fix64(k.u)*Fix64(k.u)*2047/2048 {
pArc := abnorm.Dot(bc) < 0
bcnorm := pRot90CCW(pNorm(bc, k.u))
if pDot(abnorm, bcnorm) < -Fix64(k.u)*Fix64(k.u)*2047/2048 {
pArc := pDot(abnorm, bc) < 0
k.p.Add1(mabc.Add(abnorm))
if pArc {
z := abnorm.Rot90CW()
z := pRot90CW(abnorm)
addArc(k.p, mabc, abnorm, z)
addArc(k.p, mabc, z, bcnorm)
}
@ -374,9 +378,9 @@ func (k *stroker) Add2(b, c Point) {
k.r.Add1(mabc.Sub(abnorm))
if !pArc {
z := abnorm.Rot90CW()
addArc(&k.r, mabc, abnorm.Neg(), z)
addArc(&k.r, mabc, z, bcnorm.Neg())
z := pRot90CW(abnorm)
addArc(&k.r, mabc, pNeg(abnorm), z)
addArc(&k.r, mabc, z, pNeg(bcnorm))
}
k.r.Add1(mabc.Sub(bcnorm))
k.r.Add1(c.Sub(bcnorm))
@ -423,7 +427,7 @@ func (k *stroker) stroke(q Path) {
}
// TODO(nigeltao): if q is a closed curve then we should join the first and
// last segments instead of capping them.
k.cr.Cap(k.p, k.u, q.lastPoint(), k.anorm.Neg())
k.cr.Cap(k.p, k.u, q.lastPoint(), pNeg(k.anorm))
addPathReversed(k.p, k.r)
pivot := q.firstPoint()
k.cr.Cap(k.p, k.u, pivot, pivot.Sub(Point{k.r[1], k.r[2]}))