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affine.go
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affine.go
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package graphics2d
import (
"fmt"
"math"
"github.com/jphsd/graphics2d/util"
)
// Aff3 is a 3x3 affine transformation matrix in row major order, where the
// bottom row is implicitly [0 0 1].
//
// m[3*r+c] is the element in the r'th row and c'th column.
type Aff3 [6]float64
// NewAff3 creates the identity transform.
func NewAff3() *Aff3 {
var res Aff3
res[3*0+0] = 1
res[3*1+1] = 1
return &res
}
// Translate creates a translation transform.
func Translate(x, y float64) *Aff3 {
xfm := NewAff3()
xfm.Translate(x, y)
return xfm
}
// Rotate creates a rotation transform.
func Rotate(th float64) *Aff3 {
xfm := NewAff3()
xfm.Rotate(th)
return xfm
}
// RotateAbout creates a rotation transform about a point.
func RotateAbout(th, ax, ay float64) *Aff3 {
xfm := NewAff3()
xfm.RotateAbout(th, ax, ay)
return xfm
}
// Scale creates a scale transform.
func Scale(sx, sy float64) *Aff3 {
xfm := NewAff3()
xfm.Scale(sx, sy)
return xfm
}
// ScaleAbout creates a scale transform about a point.
func ScaleAbout(sx, sy, ax, ay float64) *Aff3 {
xfm := NewAff3()
xfm.ScaleAbout(sx, sy, ax, ay)
return xfm
}
// Shear creates a shear transform.
func Shear(shx, shy float64) *Aff3 {
xfm := NewAff3()
xfm.Shear(shx, shy)
return xfm
}
// ShearAbout creates a shear transform about a point.
func ShearAbout(shx, shy, ax, ay float64) *Aff3 {
xfm := NewAff3()
xfm.ShearAbout(shx, shy, ax, ay)
return xfm
}
// Reflect creates a reflection transform.
func Reflect(x1, y1, x2, y2 float64) *Aff3 {
xfm := NewAff3()
xfm.Reflect(x1, y1, x2, y2)
return xfm
}
// Determinant calculates the transform's matrix determinant.
func (a *Aff3) Determinant() float64 {
return a[3*0+0]*a[3*1+1] - a[3*0+1]*a[3*1+0]
}
// Translate adds a translation to the transform.
func (a *Aff3) Translate(x, y float64) *Aff3 {
a[3*0+2] = x*a[3*0+0] + y*a[3*0+1] + a[3*0+2]
a[3*1+2] = x*a[3*1+0] + y*a[3*1+1] + a[3*1+2]
return a
}
// Rotate adds a rotation to the transform. The rotation is about {0, 0}.
func (a *Aff3) Rotate(th float64) *Aff3 {
sin, cos := math.Sin(th), math.Cos(th)
m0, m1 := a[3*0+0], a[3*0+1]
a[3*0+0] = cos*m0 + sin*m1
a[3*0+1] = -sin*m0 + cos*m1
m0, m1 = a[3*1+0], a[3*1+1]
a[3*1+0] = cos*m0 + sin*m1
a[3*1+1] = -sin*m0 + cos*m1
return a
}
// RotateAbout adds a rotation about a point to the transform.
func (a *Aff3) RotateAbout(th, ax, ay float64) *Aff3 {
// Reverse order
a.Translate(ax, ay)
a.Rotate(th)
a.Translate(-ax, -ay)
return a
}
// QuadrantRotate adds a rotation (n * 90 degrees) to the transform. The rotation is about {0, 0}.
// It avoids rounding issues with the trig functions.
func (a *Aff3) QuadrantRotate(n int) *Aff3 {
n %= 4
switch n {
case 0: // 360
break
case 1: // 90
a[3*0+0], a[3*0+1], a[3*1+0], a[3*1+1] = a[3*0+1], -a[3*0+0], a[3*1+1], -a[3*1+0]
case 2: // 180
a[3*0+0], a[3*0+1], a[3*1+0], a[3*1+1] = -a[3*0+0], -a[3*0+1], -a[3*1+0], -a[3*1+1]
case 3: // 270
a[3*0+0], a[3*0+1], a[3*1+0], a[3*1+1] = -a[3*0+1], a[3*0+0], -a[3*1+1], a[3*1+0]
}
return a
}
// QuadrantRotateAbout adds a rotation (n * 90 degrees) about a point to the transform.
// It avoids rounding issues with the trig functions.
func (a *Aff3) QuadrantRotateAbout(n int, ax, ay float64) *Aff3 {
// Reverse order
a.Translate(ax, ay)
a.QuadrantRotate(n)
a.Translate(-ax, -ay)
return a
}
// Scale adds a scaling to the transform centered on {0, 0}.
func (a *Aff3) Scale(sx, sy float64) *Aff3 {
a[3*0+0] *= sx
a[3*1+1] *= sy
a[3*0+1] *= sy
a[3*1+0] *= sx
return a
}
// ScaleAbout adds a scale about a point to the transform.
func (a *Aff3) ScaleAbout(sx, sy, ax, ay float64) *Aff3 {
// Reverse order
a.Translate(ax, ay)
a.Scale(sx, sy)
a.Translate(-ax, -ay)
return a
}
// Shear adds a shear to the transform centered on {0, 0}.
func (a *Aff3) Shear(shx, shy float64) *Aff3 {
m0, m1 := a[3*0+0], a[3*0+1]
a[3*0+0] = m0 + m1*shy
a[3*0+1] = m0*shx + m1
m0, m1 = a[3*1+0], a[3*1+1]
a[3*1+0] = m0 + m1*shy
a[3*1+1] = m0*shx + m1
return a
}
// ShearAbout adds a shear about a point to the transform.
func (a *Aff3) ShearAbout(shx, shy, ax, ay float64) *Aff3 {
// Reverse order
a.Translate(ax, ay)
a.Shear(shx, shy)
a.Translate(-ax, -ay)
return a
}
// Concatenate concatenates a transform to the transform.
func (a *Aff3) Concatenate(aff Aff3) *Aff3 {
m00, m01, m10, m11 := a[3*0+0], a[3*0+1], a[3*1+0], a[3*1+1]
t00, t01, t02, t10, t11, t12 := aff[3*0+0], aff[3*0+1], aff[3*0+2], aff[3*1+0], aff[3*1+1], aff[3*1+2]
a[3*0+0] = t00*m00 + t10*m01
a[3*0+1] = t01*m00 + t11*m01
a[3*0+2] += t02*m00 + t12*m01
a[3*1+0] = t00*m10 + t10*m11
a[3*1+1] = t01*m10 + t11*m11
a[3*1+2] += t02*m10 + t12*m11
return a
}
// PreConcatenate preconcatenates a transform to the transform.
func (a *Aff3) PreConcatenate(aff Aff3) *Aff3 {
m00, m01, m02, m10, m11, m12 := a[3*0+0], a[3*0+1], a[3*0+2], a[3*1+0], a[3*1+1], a[3*1+2]
t00, t01, t02, t10, t11, t12 := aff[3*0+0], aff[3*0+1], aff[3*0+2], aff[3*1+0], aff[3*1+1], aff[3*1+2]
t02 += m02*t00 + m12*t01
t12 += m02*t10 + m12*t11
a[3*0+2] = t02
a[3*1+2] = t12
a[3*0+0] = m00*t00 + m10*t01
a[3*1+0] = m00*t10 + m10*t11
a[3*0+1] = m01*t00 + m11*t01
a[3*1+1] = m01*t10 + m11*t11
return a
}
// InverseOf returns the inverse of the transform.
func (a *Aff3) InverseOf() (*Aff3, error) {
res := a.Copy()
if err := res.Invert(); err != nil {
return nil, err
}
return res, nil
}
// Invert inverts the transform.
func (a *Aff3) Invert() error {
det := a.Determinant()
if util.Equals(math.Abs(det), 0) {
return fmt.Errorf("Determinant is zero => non-invertible")
}
m00, m01, m02, m10, m11, m12 := a[3*0+0], a[3*0+1], a[3*0+2], a[3*1+0], a[3*1+1], a[3*1+2]
a[3*0+0], a[3*0+1], a[3*0+2] = m11/det, -m01/det, (m01*m12-m02*m11)/det
a[3*1+0], a[3*1+1], a[3*1+2] = -m10/det, m00/det, (m02*m10-m00*m12)/det
return nil
}
// String converts the transform into a string.
func (a *Aff3) String() string {
return fmt.Sprintf("{{%f, %f, %f}, {%f, %f, %f}, {0, 0, 1}}",
a[3*0+0], a[3*0+1], a[3*0+2],
a[3*1+0], a[3*1+1], a[3*1+2])
}
// Identity returns true if the transform is the identity.
func (a *Aff3) Identity() bool {
if !util.Equals(a[3*0+0], 1) {
return false
}
if !util.Equals(a[3*0+1], 0) {
return false
}
if !util.Equals(a[3*0+2], 0) {
return false
}
if !util.Equals(a[3*1+0], 0) {
return false
}
if !util.Equals(a[3*1+1], 1) {
return false
}
if !util.Equals(a[3*1+2], 0) {
return false
}
return true
}
// Copy returns a copy of the transform.
func (a *Aff3) Copy() *Aff3 {
res := *a
return &res
}
// Reflect performs a reflection along the axis defined by the two non-coincident points.
func (a *Aff3) Reflect(x1, y1, x2, y2 float64) *Aff3 {
dx, dy := x2-x1, y2-y1
if util.Equals(dy, 0) {
// Horizontal - no rotation required
a.Translate(0, y1)
a.Scale(1, -1)
a.Translate(0, -y1)
return a
}
if util.Equals(dx, 0) {
// Vertical - no rotation required
a.Translate(x1, 0)
a.Scale(-1, 1)
a.Translate(-x1, 0)
return a
}
th := math.Atan2(dy, dx)
if th < 0 {
th += TwoPi
}
a.Translate(x1, y1)
a.Rotate(th)
a.Scale(1, -1)
a.Rotate(-th)
a.Translate(-x1, -y1)
return a
}
// LineTransform produces a transform that maps the line {p1, p2} to {p1', p2'}.
// Assumes neither of the lines are degenerate.
func LineTransform(x1, y1, x2, y2, x1p, y1p, x2p, y2p float64) *Aff3 {
// Calculate the offset, the rotation and the scale
ox, oy := x1p-x1, y1p-y1
dx, dy, dxp, dyp := x2-x1, y2-y1, x2p-x1p, y2p-y1p
th := math.Atan2(dyp, dxp) - math.Atan2(dy, dx)
s := math.Hypot(dxp, dyp) / math.Hypot(dx, dy)
xfm := NewAff3()
// Reverse order
xfm.RotateAbout(th, x1p, y1p)
xfm.ScaleAbout(s, s, x1p, y1p)
xfm.Translate(ox, oy)
return xfm
}
// BoxTransform produces a transform that maps the line {p1, p2} to {p1', p2'} and
// scales the perpendicular by hp / h. Assumes neither of the lines nor h are degenerate.
func BoxTransform(x1, y1, x2, y2, h, x1p, y1p, x2p, y2p, hp float64) *Aff3 {
// Calculate the offset, the rotation and the scale
ox, oy := x1p-x1, y1p-y1
dx, dy, dxp, dyp := x2-x1, y2-y1, x2p-x1p, y2p-y1p
th := math.Atan2(dyp, dxp) - math.Atan2(dy, dx)
s := math.Hypot(dxp, dyp) / math.Hypot(dx, dy)
xfm := NewAff3()
// Reverse order
xfm.RotateAbout(th, x1p, y1p)
xfm.ScaleAbout(s, hp/h, x1p, y1p)
xfm.Translate(ox, oy)
return xfm
}
// CreateTransform returns a transform that performs the requested translation,
// scaling and rotation based on {0, 0}.
func CreateTransform(x, y, scale, rotation float64) *Aff3 {
xfm := NewAff3()
xfm.Translate(x, y)
xfm.Scale(scale, scale)
xfm.Rotate(rotation)
return xfm
}
// Apply implements the Transform interface.
func (a *Aff3) Apply(pts ...[]float64) [][]float64 {
npts := make([][]float64, len(pts))
for i, pt := range pts {
d := len(pt)
npt := make([]float64, d)
x, y := pt[0], pt[1]
// x' = a[3*0+0]*x + a[3*0+1]*y + a[3*0+2]
// y' = a[3*1+0]*x + a[3*1+1]*y + a[3*1+2]
npt[0] = a[0]*x + a[1]*y + a[2]
npt[1] = a[3]*x + a[4]*y + a[5]
// Preserve other values
for i := 2; i < d; i++ {
npt[i] = pt[i]
}
npts[i] = npt
}
return npts
}
// ScaleAndInset produces a transform that will scale and translate a set of points bounded by bb so they fit inside the
// inset box described by width, height, iwidth, iheight located at {0, 0}. If fix is true then the aspect ratio is maintained.
func ScaleAndInset(width, height, iwidth, iheight float64, fix bool, bb [][]float64) *Aff3 {
ox, oy := bb[0][0], bb[0][1]
dx, dy := bb[1][0]-ox, bb[1][1]-oy
w := width - 2*iwidth
h := height - 2*iheight
xfm := NewAff3()
xfm.Translate(width/2, height/2)
if fix {
s := dx
if dy > s {
s = dy
}
xfm.Scale(w/s, h/s)
} else {
xfm.Scale(w/dx, h/dy)
}
xfm.Translate(-(ox + dx/2), -(oy + dy/2))
return xfm
}
// FlipY is a convenience function to create a transform that has +ve Y point up rather than down.
func FlipY(height float64) *Aff3 {
yoffs := height / 2
xfm := NewAff3()
xfm.Translate(0, yoffs)
xfm.Scale(1, -1)
xfm.Translate(0, -yoffs)
return xfm
}