484 lines
15 KiB
Go
484 lines
15 KiB
Go
// 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 (or
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// any later version), both of which can be found in the LICENSE file.
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package raster
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import (
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"golang.org/x/image/math/fixed"
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)
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// Two points are considered practically equal if the square of the distance
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// between them is less than one quarter (i.e. 1024 / 4096).
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const epsilon = fixed.Int52_12(1024)
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// A Capper signifies how to begin or end a stroked path.
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type Capper interface {
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// Cap adds a cap to p given a pivot point and the normal vector of a
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// terminal segment. The normal's length is half of the stroke width.
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Cap(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6)
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}
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// The CapperFunc type adapts an ordinary function to be a Capper.
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type CapperFunc func(Adder, fixed.Int26_6, fixed.Point26_6, fixed.Point26_6)
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func (f CapperFunc) Cap(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6) {
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f(p, halfWidth, pivot, n1)
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}
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// A Joiner signifies how to join interior nodes of a stroked path.
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type Joiner interface {
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// Join adds a join to the two sides of a stroked path given a pivot
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// point and the normal vectors of the trailing and leading segments.
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// Both normals have length equal to half of the stroke width.
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Join(lhs, rhs Adder, halfWidth fixed.Int26_6, pivot, n0, n1 fixed.Point26_6)
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}
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// The JoinerFunc type adapts an ordinary function to be a Joiner.
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type JoinerFunc func(lhs, rhs Adder, halfWidth fixed.Int26_6, pivot, n0, n1 fixed.Point26_6)
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func (f JoinerFunc) Join(lhs, rhs Adder, halfWidth fixed.Int26_6, pivot, n0, n1 fixed.Point26_6) {
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f(lhs, rhs, halfWidth, pivot, n0, n1)
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}
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// RoundCapper adds round caps to a stroked path.
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var RoundCapper Capper = CapperFunc(roundCapper)
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func roundCapper(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6) {
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// The cubic Bézier approximation to a circle involves the magic number
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// (√2 - 1) * 4/3, which is approximately 35/64.
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const k = 35
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e := pRot90CCW(n1)
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side := pivot.Add(e)
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start, end := pivot.Sub(n1), pivot.Add(n1)
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d, e := n1.Mul(k), e.Mul(k)
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p.Add3(start.Add(e), side.Sub(d), side)
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p.Add3(side.Add(d), end.Add(e), end)
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}
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// ButtCapper adds butt caps to a stroked path.
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var ButtCapper Capper = CapperFunc(buttCapper)
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func buttCapper(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6) {
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p.Add1(pivot.Add(n1))
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}
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// SquareCapper adds square caps to a stroked path.
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var SquareCapper Capper = CapperFunc(squareCapper)
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func squareCapper(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6) {
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e := pRot90CCW(n1)
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side := pivot.Add(e)
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p.Add1(side.Sub(n1))
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p.Add1(side.Add(n1))
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p.Add1(pivot.Add(n1))
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}
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// RoundJoiner adds round joins to a stroked path.
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var RoundJoiner Joiner = JoinerFunc(roundJoiner)
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func roundJoiner(lhs, rhs Adder, haflWidth fixed.Int26_6, pivot, n0, n1 fixed.Point26_6) {
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dot := pDot(pRot90CW(n0), n1)
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if dot >= 0 {
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addArc(lhs, pivot, n0, n1)
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rhs.Add1(pivot.Sub(n1))
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} else {
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lhs.Add1(pivot.Add(n1))
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addArc(rhs, pivot, pNeg(n0), pNeg(n1))
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}
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}
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// BevelJoiner adds bevel joins to a stroked path.
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var BevelJoiner Joiner = JoinerFunc(bevelJoiner)
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func bevelJoiner(lhs, rhs Adder, haflWidth fixed.Int26_6, pivot, n0, n1 fixed.Point26_6) {
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lhs.Add1(pivot.Add(n1))
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rhs.Add1(pivot.Sub(n1))
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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. The
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// two vectors n0 and n1 must be of equal length.
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func addArc(p Adder, pivot, n0, n1 fixed.Point26_6) {
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// r2 is the square of the length of n0.
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r2 := pDot(n0, n0)
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if r2 < epsilon {
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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
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// control points {1, 0}, {1, tan(π/8)} and {1/√2, 1/√2} suitably scaled,
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// rotated and translated. tan(π/8) is approximately 27/64.
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const tpo8 = 27
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var s fixed.Point26_6
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// We determine which octant the angle between n0 and n1 is in via three
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// dot products. m0, m1 and m2 are n0 rotated clockwise by 45, 90 and 135
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// degrees.
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m0 := pRot45CW(n0)
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m1 := pRot90CW(n0)
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m2 := pRot90CW(m0)
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if pDot(m1, n1) >= 0 {
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if pDot(n0, n1) >= 0 {
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if pDot(m2, 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(tpo8)), 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(tpo8)
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p.Add2(pivot.Add(n0).Add(m1.Mul(tpo8)), pivot.Add(m0))
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p.Add2(pm1.Add(n0t), pm1)
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if pDot(m0, 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 pDot(n0, n1) >= 0 {
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if pDot(m0, 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(tpo8)), pivot.Sub(m2))
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s = pNeg(m2)
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}
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} else {
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pm1, n0t := pivot.Sub(m1), n0.Mul(tpo8)
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p.Add2(pivot.Add(n0).Sub(m1.Mul(tpo8)), pivot.Sub(m2))
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p.Add2(pm1.Add(n0t), pm1)
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if pDot(m2, n1) <= 0 {
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// n1 is between 90 and 135 degrees counter-clockwise of n0.
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s = pNeg(m1)
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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 = pNeg(m0)
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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
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// bisector of those two points. The multiple ranges between 128/256 and
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// 150/256 as the angle between s and n1 ranges between 0 and 45 degrees.
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//
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// When the angle is 0 degrees (i.e. s and n1 are coincident) then
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// s.Add(n1) is twice s and so the middle control point of the degenerate
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// quadratic segment should be half s.Add(n1), and half = 128/256.
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//
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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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//
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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 * pDot(s, n1) / r2
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multiple := fixed.Int26_6(150-(150-128)*(d-181)/(256-181)) >> 2
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p.Add2(pivot.Add(s.Add(n1).Mul(multiple)), pivot.Add(n1))
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}
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// midpoint returns the midpoint of two Points.
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func midpoint(a, b fixed.Point26_6) fixed.Point26_6 {
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return fixed.Point26_6{(a.X + b.X) / 2, (a.Y + b.Y) / 2}
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}
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// angleGreaterThan45 returns whether the angle between two vectors is more
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// than 45 degrees.
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func angleGreaterThan45(v0, v1 fixed.Point26_6) bool {
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v := pRot45CCW(v0)
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return pDot(v, v1) < 0 || pDot(pRot90CW(v), v1) < 0
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}
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// interpolate returns the point (1-t)*a + t*b.
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func interpolate(a, b fixed.Point26_6, t fixed.Int52_12) fixed.Point26_6 {
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s := 1<<12 - t
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x := s*fixed.Int52_12(a.X) + t*fixed.Int52_12(b.X)
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y := s*fixed.Int52_12(a.Y) + t*fixed.Int52_12(b.Y)
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return fixed.Point26_6{fixed.Int26_6(x >> 12), fixed.Int26_6(y >> 12)}
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}
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// curviest2 returns the value of t for which the quadratic parametric curve
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// (1-t)²*a + 2*t*(1-t).b + t²*c has maximum curvature.
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//
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// The curvature of the parametric curve f(t) = (x(t), y(t)) is
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// |x′y″-y′x″| / (x′²+y′²)^(3/2).
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//
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// Let d = b-a and e = c-2*b+a, so that f′(t) = 2*d+2*e*t and f″(t) = 2*e.
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// The curvature's numerator is (2*dx+2*ex*t)*(2*ey)-(2*dy+2*ey*t)*(2*ex),
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// which simplifies to 4*dx*ey-4*dy*ex, which is constant with respect to t.
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//
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// Thus, curvature is extreme where the denominator is extreme, i.e. where
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// (x′²+y′²) is extreme. The first order condition is that
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// 2*x′*x″+2*y′*y″ = 0, or (dx+ex*t)*ex + (dy+ey*t)*ey = 0.
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// Solving for t gives t = -(dx*ex+dy*ey) / (ex*ex+ey*ey).
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func curviest2(a, b, c fixed.Point26_6) fixed.Int52_12 {
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dx := int64(b.X - a.X)
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dy := int64(b.Y - a.Y)
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ex := int64(c.X - 2*b.X + a.X)
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ey := int64(c.Y - 2*b.Y + a.Y)
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if ex == 0 && ey == 0 {
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return 2048
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}
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return fixed.Int52_12(-4096 * (dx*ex + dy*ey) / (ex*ex + ey*ey))
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}
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// A stroker holds state for stroking a path.
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type stroker struct {
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// p is the destination that records the stroked path.
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p Adder
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// u is the half-width of the stroke.
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u fixed.Int26_6
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// cr and jr specify how to end and connect path segments.
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cr Capper
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jr Joiner
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// r is the reverse path. Stroking a path involves constructing two
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// parallel paths 2*u apart. The first path is added immediately to p,
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// the second path is accumulated in r and eventually added in reverse.
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r Path
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// a is the most recent segment point. anorm is the segment normal of
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// length u at that point.
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a, anorm fixed.Point26_6
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}
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// addNonCurvy2 adds a quadratic segment to the stroker, where the segment
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// defined by (k.a, b, c) achieves maximum curvature at either k.a or c.
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func (k *stroker) addNonCurvy2(b, c fixed.Point26_6) {
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// We repeatedly divide the segment at its middle until it is straight
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// enough to approximate the stroke by just translating the control points.
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// ds and ps are stacks of depths and points. t is the top of the stack.
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const maxDepth = 5
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var (
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ds [maxDepth + 1]int
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ps [2*maxDepth + 3]fixed.Point26_6
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t int
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)
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// Initially the ps stack has one quadratic segment of depth zero.
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ds[0] = 0
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ps[2] = k.a
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ps[1] = b
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ps[0] = c
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anorm := k.anorm
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var cnorm fixed.Point26_6
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for {
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depth := ds[t]
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a := ps[2*t+2]
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b := ps[2*t+1]
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c := ps[2*t+0]
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ab := b.Sub(a)
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bc := c.Sub(b)
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abIsSmall := pDot(ab, ab) < fixed.Int52_12(1<<12)
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bcIsSmall := pDot(bc, bc) < fixed.Int52_12(1<<12)
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if abIsSmall && bcIsSmall {
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// Approximate the segment by a circular arc.
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cnorm = pRot90CCW(pNorm(bc, k.u))
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mac := midpoint(a, c)
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addArc(k.p, mac, anorm, cnorm)
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addArc(&k.r, mac, pNeg(anorm), pNeg(cnorm))
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} else if depth < maxDepth && angleGreaterThan45(ab, bc) {
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// Divide the segment in two and push both halves on the stack.
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mab := midpoint(a, b)
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mbc := midpoint(b, c)
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t++
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ds[t+0] = depth + 1
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ds[t-1] = depth + 1
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ps[2*t+2] = a
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ps[2*t+1] = mab
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ps[2*t+0] = midpoint(mab, mbc)
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ps[2*t-1] = mbc
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continue
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} else {
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// Translate the control points.
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bnorm := pRot90CCW(pNorm(c.Sub(a), k.u))
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cnorm = pRot90CCW(pNorm(bc, k.u))
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k.p.Add2(b.Add(bnorm), c.Add(cnorm))
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k.r.Add2(b.Sub(bnorm), c.Sub(cnorm))
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}
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if t == 0 {
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k.a, k.anorm = c, cnorm
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return
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}
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t--
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anorm = cnorm
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}
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panic("unreachable")
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}
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// Add1 adds a linear segment to the stroker.
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func (k *stroker) Add1(b fixed.Point26_6) {
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bnorm := pRot90CCW(pNorm(b.Sub(k.a), k.u))
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if len(k.r) == 0 {
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k.p.Start(k.a.Add(bnorm))
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k.r.Start(k.a.Sub(bnorm))
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} else {
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k.jr.Join(k.p, &k.r, k.u, k.a, k.anorm, bnorm)
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}
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k.p.Add1(b.Add(bnorm))
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k.r.Add1(b.Sub(bnorm))
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k.a, k.anorm = b, bnorm
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}
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// Add2 adds a quadratic segment to the stroker.
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func (k *stroker) Add2(b, c fixed.Point26_6) {
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ab := b.Sub(k.a)
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bc := c.Sub(b)
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abnorm := pRot90CCW(pNorm(ab, k.u))
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if len(k.r) == 0 {
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k.p.Start(k.a.Add(abnorm))
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k.r.Start(k.a.Sub(abnorm))
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} else {
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k.jr.Join(k.p, &k.r, k.u, k.a, k.anorm, abnorm)
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}
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// Approximate nearly-degenerate quadratics by linear segments.
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abIsSmall := pDot(ab, ab) < epsilon
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bcIsSmall := pDot(bc, bc) < epsilon
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if abIsSmall || bcIsSmall {
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acnorm := pRot90CCW(pNorm(c.Sub(k.a), k.u))
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k.p.Add1(c.Add(acnorm))
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k.r.Add1(c.Sub(acnorm))
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k.a, k.anorm = c, acnorm
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return
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}
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// The quadratic segment (k.a, b, c) has a point of maximum curvature.
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// If this occurs at an end point, we process the segment as a whole.
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t := curviest2(k.a, b, c)
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if t <= 0 || 4096 <= t {
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k.addNonCurvy2(b, c)
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return
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}
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// Otherwise, we perform a de Casteljau decomposition at the point of
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// maximum curvature and process the two straighter parts.
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mab := interpolate(k.a, b, t)
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mbc := interpolate(b, c, t)
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mabc := interpolate(mab, mbc, t)
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// If the vectors ab and bc are close to being in opposite directions,
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// then the decomposition can become unstable, so we approximate the
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// quadratic segment by two linear segments joined by an arc.
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bcnorm := pRot90CCW(pNorm(bc, k.u))
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if pDot(abnorm, bcnorm) < -fixed.Int52_12(k.u)*fixed.Int52_12(k.u)*2047/2048 {
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pArc := pDot(abnorm, bc) < 0
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k.p.Add1(mabc.Add(abnorm))
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if pArc {
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z := pRot90CW(abnorm)
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addArc(k.p, mabc, abnorm, z)
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addArc(k.p, mabc, z, bcnorm)
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}
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k.p.Add1(mabc.Add(bcnorm))
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k.p.Add1(c.Add(bcnorm))
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k.r.Add1(mabc.Sub(abnorm))
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if !pArc {
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z := pRot90CW(abnorm)
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addArc(&k.r, mabc, pNeg(abnorm), z)
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addArc(&k.r, mabc, z, pNeg(bcnorm))
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}
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k.r.Add1(mabc.Sub(bcnorm))
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k.r.Add1(c.Sub(bcnorm))
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k.a, k.anorm = c, bcnorm
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return
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}
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// Process the decomposed parts.
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k.addNonCurvy2(mab, mabc)
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k.addNonCurvy2(mbc, c)
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}
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// Add3 adds a cubic segment to the stroker.
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func (k *stroker) Add3(b, c, d fixed.Point26_6) {
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panic("freetype/raster: stroke unimplemented for cubic segments")
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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 (k *stroker) stroke(q Path) {
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// Stroking is implemented by deriving two paths each k.u apart from q.
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// The left-hand-side path is added immediately to k.p; the right-hand-side
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// path is accumulated in k.r. Once we've finished adding the LHS to k.p,
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// we add the RHS in reverse order.
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k.r = make(Path, 0, len(q))
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k.a = fixed.Point26_6{q[1], q[2]}
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for i := 4; i < len(q); {
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switch q[i] {
|
||
case 1:
|
||
k.Add1(
|
||
fixed.Point26_6{q[i+1], q[i+2]},
|
||
)
|
||
i += 4
|
||
case 2:
|
||
k.Add2(
|
||
fixed.Point26_6{q[i+1], q[i+2]},
|
||
fixed.Point26_6{q[i+3], q[i+4]},
|
||
)
|
||
i += 6
|
||
case 3:
|
||
k.Add3(
|
||
fixed.Point26_6{q[i+1], q[i+2]},
|
||
fixed.Point26_6{q[i+3], q[i+4]},
|
||
fixed.Point26_6{q[i+5], q[i+6]},
|
||
)
|
||
i += 8
|
||
default:
|
||
panic("freetype/raster: bad path")
|
||
}
|
||
}
|
||
if len(k.r) == 0 {
|
||
return
|
||
}
|
||
// 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(), pNeg(k.anorm))
|
||
addPathReversed(k.p, k.r)
|
||
pivot := q.firstPoint()
|
||
k.cr.Cap(k.p, k.u, pivot, pivot.Sub(fixed.Point26_6{k.r[1], k.r[2]}))
|
||
}
|
||
|
||
// Stroke adds q stroked with the given width to p. The result is typically
|
||
// self-intersecting and should be rasterized with UseNonZeroWinding.
|
||
// cr and jr may be nil, which defaults to a RoundCapper or RoundJoiner.
|
||
func Stroke(p Adder, q Path, width fixed.Int26_6, cr Capper, jr Joiner) {
|
||
if len(q) == 0 {
|
||
return
|
||
}
|
||
if cr == nil {
|
||
cr = RoundCapper
|
||
}
|
||
if jr == nil {
|
||
jr = RoundJoiner
|
||
}
|
||
if q[0] != 0 {
|
||
panic("freetype/raster: bad path")
|
||
}
|
||
s := stroker{p: p, u: width / 2, cr: cr, jr: jr}
|
||
i := 0
|
||
for j := 4; j < len(q); {
|
||
switch q[j] {
|
||
case 0:
|
||
s.stroke(q[i:j])
|
||
i, j = j, j+4
|
||
case 1:
|
||
j += 4
|
||
case 2:
|
||
j += 6
|
||
case 3:
|
||
j += 8
|
||
default:
|
||
panic("freetype/raster: bad path")
|
||
}
|
||
}
|
||
s.stroke(q[i:])
|
||
}
|