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f32: implement 2D affine transformations
Implements 2D affine transformations. This commit is a step towards full affine transformations for drawing operations. Heavily based on the work by Péter Szilágyi in patch 9212 Signed-off-by: Viktor <viktor.ogeman@gmail.com>
This commit is contained in:
Viktor
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// SPDX-License-Identifier: Unlicense OR MIT
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package f32
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import (
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"math"
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)
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// Affine2D represents an affine 2D transformation. The zero value if Affine2D
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// represents the identity transform.
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type Affine2D struct {
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// in order to make the zero value of Affine2D represent the identity
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// transform we store it with the identity matrix subtracted, that is
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// if the actual transformaiton matrix is:
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// [sx, hx, ox]
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// [hy, sy, oy]
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// [ 0, 0, 1]
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// we store a = sx-1 and e = sy-1
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a, b, c float32
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d, e, f float32
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}
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// NewAffine2D creates a new Affine2D transform from the matrix elements
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// in row major order. The rows are: [sx, hx, ox], [hy, sy, oy], [0, 0, 1].
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func NewAffine2D(sx, hx, ox, hy, sy, oy float32) Affine2D {
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return Affine2D{
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a: sx, b: hx, c: ox,
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d: hy, e: sy, f: oy,
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}.encode()
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}
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// Offset the transformation.
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func (a Affine2D) Offset(offset Point) Affine2D {
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return Affine2D{
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a.a, a.b, a.c + offset.X,
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a.d, a.e, a.f + offset.Y,
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}
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}
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// Scale the transformation around the given origin.
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func (a Affine2D) Scale(origin, factor Point) Affine2D {
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if origin == (Point{}) {
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return a.scale(factor)
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}
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a = a.Offset(origin.Mul(-1))
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a = a.scale(factor)
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return a.Offset(origin)
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}
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// Rotate the transformation by the given angle (in radians) counter clockwise around the given origin.
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func (a Affine2D) Rotate(origin Point, radians float32) Affine2D {
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if origin == (Point{}) {
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return a.rotate(radians)
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}
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a = a.Offset(origin.Mul(-1))
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a = a.rotate(radians)
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return a.Offset(origin)
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}
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// Shear the transformation by the given angle (in radians) around the given origin.
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func (a Affine2D) Shear(origin Point, radiansX, radiansY float32) Affine2D {
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if origin == (Point{}) {
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return a.shear(radiansX, radiansY)
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}
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a = a.Offset(origin.Mul(-1))
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a = a.shear(radiansX, radiansY)
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return a.Offset(origin)
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}
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// Mul returns A*B.
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func (A Affine2D) Mul(B Affine2D) (r Affine2D) {
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A, B = A.decode(), B.decode()
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r.a = A.a*B.a + A.b*B.d
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r.b = A.a*B.b + A.b*B.e
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r.c = A.a*B.c + A.b*B.f + A.c
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r.d = A.d*B.a + A.e*B.d
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r.e = A.d*B.b + A.e*B.e
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r.f = A.d*B.c + A.e*B.f + A.f
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return r.encode()
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}
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// Invert the transformation. Note that if the matrix is close to singular
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// numerical errors may become large or infinity.
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func (a Affine2D) Invert() Affine2D {
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if a.a == 0 && a.b == 0 && a.d == 0 && a.e == 0 {
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return Affine2D{a: 0, b: 0, c: -a.c, d: 0, e: 0, f: -a.f}
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}
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a = a.decode()
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det := a.a*a.e - a.b*a.d
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a.a, a.e = a.e/det, a.a/det
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a.b, a.d = -a.b/det, -a.d/det
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temp := a.c
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a.c = -a.a*a.c - a.b*a.f
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a.f = -a.d*temp - a.e*a.f
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return a.encode()
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}
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// Transform p by returning a*p.
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func (a Affine2D) Transform(p Point) Point {
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a = a.decode()
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return Point{
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X: p.X*a.a + p.Y*a.b + a.c,
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Y: p.X*a.d + p.Y*a.e + a.f,
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}
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}
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// Elems returns the matrix elements of the transform in row-major order. The
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// rows are: [sx, hx, ox], [hy, sy, oy], [0, 0, 1].
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func (a Affine2D) Elems() (sx, hx, ox, hy, sy, oy float32) {
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a = a.decode()
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return a.a, a.b, a.c, a.d, a.e, a.f
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}
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func (a Affine2D) encode() Affine2D {
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// since we store with identity matrix subtracted
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a.a -= 1
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a.e -= 1
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return a
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}
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func (a Affine2D) decode() Affine2D {
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// since we store with identity matrix subtracted
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a.a += 1
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a.e += 1
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return a
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}
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func (a Affine2D) scale(factor Point) Affine2D {
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a = a.decode()
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return Affine2D{
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a.a * factor.X, a.b * factor.X, a.c * factor.X,
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a.d * factor.Y, a.e * factor.Y, a.f * factor.Y,
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}.encode()
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}
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func (a Affine2D) rotate(radians float32) Affine2D {
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sin, cos := math.Sincos(float64(radians))
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s, c := float32(sin), float32(cos)
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a = a.decode()
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return Affine2D{
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a.a*c - a.d*s, a.b*c - a.e*s, a.c*c - a.f*s,
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a.a*s + a.d*c, a.b*s + a.e*c, a.c*s + a.f*c,
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}.encode()
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}
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func (a Affine2D) shear(radiansX, radiansY float32) Affine2D {
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tx := float32(math.Tan(float64(radiansX)))
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ty := float32(math.Tan(float64(radiansY)))
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a = a.decode()
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return Affine2D{
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a.a + a.d*tx, a.b + a.e*tx, a.c + a.f*tx,
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a.a*ty + a.d, a.b*ty + a.e, a.f*ty + a.f,
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}.encode()
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}
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@@ -0,0 +1,121 @@
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// SPDX-License-Identifier: Unlicense OR MIT
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package f32
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import (
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"math"
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"testing"
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)
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func eq(p1, p2 Point) bool {
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tol := 1e-5
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dx, dy := p2.X-p1.X, p2.Y-p1.Y
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return math.Abs(math.Sqrt(float64(dx*dx+dy*dy))) < tol
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}
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func TestTransformOffset(t *testing.T) {
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p := Point{X: 1, Y: 2}
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o := Point{X: 2, Y: -3}
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r := Affine2D{}.Offset(o).Transform(p)
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if !eq(r, Pt(3, -1)) {
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t.Errorf("offset transformation mismatch: have %v, want {3 -1}", r)
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}
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i := Affine2D{}.Offset(o).Invert().Transform(r)
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if !eq(i, p) {
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t.Errorf("offset transformation inverse mismatch: have %v, want %v", i, p)
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}
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}
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func TestTransformScale(t *testing.T) {
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p := Point{X: 1, Y: 2}
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s := Point{X: -1, Y: 2}
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r := Affine2D{}.Scale(Point{}, s).Transform(p)
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if !eq(r, Pt(-1, 4)) {
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t.Errorf("scale transformation mismatch: have %v, want {-1 4}", r)
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}
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i := Affine2D{}.Scale(Point{}, s).Invert().Transform(r)
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if !eq(i, p) {
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t.Errorf("scale transformation inverse mismatch: have %v, want %v", i, p)
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}
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}
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func TestTransformRotate(t *testing.T) {
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p := Point{X: 1, Y: 0}
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a := float32(math.Pi / 2)
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r := Affine2D{}.Rotate(Point{}, a).Transform(p)
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if !eq(r, Pt(0, 1)) {
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t.Errorf("rotate transformation mismatch: have %v, want {0 1}", r)
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}
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i := Affine2D{}.Rotate(Point{}, a).Invert().Transform(r)
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if !eq(i, p) {
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t.Errorf("rotate transformation inverse mismatch: have %v, want %v", i, p)
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}
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}
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func TestTransformShear(t *testing.T) {
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p := Point{X: 1, Y: 1}
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r := Affine2D{}.Shear(Point{}, math.Pi/4, 0).Transform(p)
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if !eq(r, Pt(2, 1)) {
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t.Errorf("shear transformation mismatch: have %v, want {2 1}", r)
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}
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i := Affine2D{}.Shear(Point{}, math.Pi/4, 0).Invert().Transform(r)
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if !eq(i, p) {
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t.Errorf("shear transformation inverse mismatch: have %v, want %v", i, p)
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}
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}
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func TestTransformMultiply(t *testing.T) {
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p := Point{X: 1, Y: 2}
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o := Point{X: 2, Y: -3}
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s := Point{X: -1, Y: 2}
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a := float32(-math.Pi / 2)
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r := Affine2D{}.Offset(o).Scale(Point{}, s).Rotate(Point{}, a).Shear(Point{}, math.Pi/4, 0).Transform(p)
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if !eq(r, Pt(1, 3)) {
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t.Errorf("complex transformation mismatch: have %v, want {1 3}", r)
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}
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i := Affine2D{}.Offset(o).Scale(Point{}, s).Rotate(Point{}, a).Shear(Point{}, math.Pi/4, 0).Invert().Transform(r)
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if !eq(i, p) {
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t.Errorf("complex transformation inverse mismatch: have %v, want %v", i, p)
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}
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}
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func TestTransformScaleAround(t *testing.T) {
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p := Pt(-1, -1)
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target := Pt(-6, -13)
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pt := Affine2D{}.Scale(Pt(4, 5), Pt(2, 3)).Transform(p)
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if !eq(pt, target) {
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t.Log(pt, "!=", target)
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t.Error("Scale not as expected")
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}
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}
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func TestTransformRotateAround(t *testing.T) {
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p := Pt(-1, -1)
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pt := Affine2D{}.Rotate(Pt(1, 1), -math.Pi/2).Transform(p)
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target := Pt(-1, 3)
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if !eq(pt, target) {
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t.Log(pt, "!=", target)
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t.Error("Rotate not as expected")
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}
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}
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func TestMulOrder(t *testing.T) {
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A := Affine2D{}.Offset(Pt(100, 100))
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B := Affine2D{}.Scale(Point{}, Pt(2, 2))
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_ = A
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_ = B
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T1 := Affine2D{}.Offset(Pt(100, 100)).Scale(Point{}, Pt(2, 2))
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T2 := B.Mul(A)
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if T1 != T2 {
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t.Log(T1)
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t.Log(T2)
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t.Error("multiplication / transform order not as expected")
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}
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}
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