Example #1
0
// curviest2 returns the value of t for which the quadratic parametric curve
// (1-t)²*a + 2*t*(1-t).b + t²*c has maximum curvature.
//
// The curvature of the parametric curve f(t) = (x(t), y(t)) is
// |x′y″-y′x″| / (x′²+y′²)^(3/2).
//
// Let d = b-a and e = c-2*b+a, so that f′(t) = 2*d+2*e*t and f″(t) = 2*e.
// The curvature's numerator is (2*dx+2*ex*t)*(2*ey)-(2*dy+2*ey*t)*(2*ex),
// which simplifies to 4*dx*ey-4*dy*ex, which is constant with respect to t.
//
// Thus, curvature is extreme where the denominator is extreme, i.e. where
// (x′²+y′²) is extreme. The first order condition is that
// 2*x′*x″+2*y′*y″ = 0, or (dx+ex*t)*ex + (dy+ey*t)*ey = 0.
// Solving for t gives t = -(dx*ex+dy*ey) / (ex*ex+ey*ey).
func curviest2(a, b, c fixed.Point26_6) fixed.Int52_12 {
	dx := int64(b.X - a.X)
	dy := int64(b.Y - a.Y)
	ex := int64(c.X - 2*b.X + a.X)
	ey := int64(c.Y - 2*b.Y + a.Y)
	if ex == 0 && ey == 0 {
		return 2048
	}
	return fixed.Int52_12(-4096 * (dx*ex + dy*ey) / (ex*ex + ey*ey))
}
Example #2
0
// Add2 adds a quadratic segment to the stroker.
func (k *stroker) Add2(b, c fixed.Point26_6) {
	ab := b.Sub(k.a)
	bc := c.Sub(b)
	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))
	} else {
		k.jr.Join(k.p, &k.r, k.u, k.a, k.anorm, abnorm)
	}

	// Approximate nearly-degenerate quadratics by linear segments.
	abIsSmall := pDot(ab, ab) < epsilon
	bcIsSmall := pDot(bc, bc) < epsilon
	if abIsSmall || bcIsSmall {
		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
		return
	}

	// 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 || 4096 <= t {
		k.addNonCurvy2(b, c)
		return
	}

	// Otherwise, we perform a de Casteljau decomposition at the point of
	// maximum curvature and process the two straighter parts.
	mab := interpolate(k.a, b, t)
	mbc := interpolate(b, c, t)
	mabc := interpolate(mab, mbc, t)

	// 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 := pRot90CCW(pNorm(bc, k.u))
	if pDot(abnorm, bcnorm) < -fixed.Int52_12(k.u)*fixed.Int52_12(k.u)*2047/2048 {
		pArc := pDot(abnorm, bc) < 0

		k.p.Add1(mabc.Add(abnorm))
		if pArc {
			z := pRot90CW(abnorm)
			addArc(k.p, mabc, abnorm, z)
			addArc(k.p, mabc, z, bcnorm)
		}
		k.p.Add1(mabc.Add(bcnorm))
		k.p.Add1(c.Add(bcnorm))

		k.r.Add1(mabc.Sub(abnorm))
		if !pArc {
			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))

		k.a, k.anorm = c, bcnorm
		return
	}

	// Process the decomposed parts.
	k.addNonCurvy2(mab, mabc)
	k.addNonCurvy2(mbc, c)
}
Example #3
0
// addNonCurvy2 adds a quadratic segment to the stroker, where the segment
// defined by (k.a, b, c) achieves maximum curvature at either k.a or c.
func (k *stroker) addNonCurvy2(b, c fixed.Point26_6) {
	// We repeatedly divide the segment at its middle until it is straight
	// enough to approximate the stroke by just translating the control points.
	// ds and ps are stacks of depths and points. t is the top of the stack.
	const maxDepth = 5
	var (
		ds [maxDepth + 1]int
		ps [2*maxDepth + 3]fixed.Point26_6
		t  int
	)
	// Initially the ps stack has one quadratic segment of depth zero.
	ds[0] = 0
	ps[2] = k.a
	ps[1] = b
	ps[0] = c
	anorm := k.anorm
	var cnorm fixed.Point26_6

	for {
		depth := ds[t]
		a := ps[2*t+2]
		b := ps[2*t+1]
		c := ps[2*t+0]
		ab := b.Sub(a)
		bc := c.Sub(b)
		abIsSmall := pDot(ab, ab) < fixed.Int52_12(1<<12)
		bcIsSmall := pDot(bc, bc) < fixed.Int52_12(1<<12)
		if abIsSmall && bcIsSmall {
			// Approximate the segment by a circular arc.
			cnorm = pRot90CCW(pNorm(bc, k.u))
			mac := midpoint(a, c)
			addArc(k.p, mac, anorm, cnorm)
			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)
			mbc := midpoint(b, c)
			t++
			ds[t+0] = depth + 1
			ds[t-1] = depth + 1
			ps[2*t+2] = a
			ps[2*t+1] = mab
			ps[2*t+0] = midpoint(mab, mbc)
			ps[2*t-1] = mbc
			continue
		} else {
			// Translate the control points.
			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))
		}
		if t == 0 {
			k.a, k.anorm = c, cnorm
			return
		}
		t--
		anorm = cnorm
	}
	panic("unreachable")
}
Example #4
0
// interpolate returns the point (1-t)*a + t*b.
func interpolate(a, b fixed.Point26_6, t fixed.Int52_12) fixed.Point26_6 {
	s := 1<<12 - t
	x := s*fixed.Int52_12(a.X) + t*fixed.Int52_12(b.X)
	y := s*fixed.Int52_12(a.Y) + t*fixed.Int52_12(b.Y)
	return fixed.Point26_6{fixed.Int26_6(x >> 12), fixed.Int26_6(y >> 12)}
}
Example #5
0
// 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 (or
// any later version), both of which can be found in the LICENSE file.

package raster

import (
	"github.com/c-darwin/dcoin-go/vendor/src/golang.org/x/image/math/fixed"
)

// Two points are considered practically equal if the square of the distance
// between them is less than one quarter (i.e. 1024 / 4096).
const epsilon = fixed.Int52_12(1024)

// A Capper signifies how to begin or end a stroked path.
type Capper interface {
	// Cap adds a cap to p given a pivot point and the normal vector of a
	// terminal segment. The normal's length is half of the stroke width.
	Cap(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6)
}

// The CapperFunc type adapts an ordinary function to be a Capper.
type CapperFunc func(Adder, fixed.Int26_6, fixed.Point26_6, fixed.Point26_6)

func (f CapperFunc) Cap(p Adder, halfWidth fixed.Int26_6, pivot, n1 fixed.Point26_6) {
	f(p, halfWidth, pivot, n1)
}

// A Joiner signifies how to join interior nodes of a stroked path.
type Joiner interface {
Example #6
0
// pDot returns the dot product p·q.
func pDot(p, q fixed.Point26_6) fixed.Int52_12 {
	px, py := int64(p.X), int64(p.Y)
	qx, qy := int64(q.X), int64(q.Y)
	return fixed.Int52_12(px*qx + py*qy)
}