Affine Transformation

Any Transformation preserving Collinearity (i.e., all points lying on a Line initially still lie on a Line after Transformation). An affine transformation is also called an Affinity. An affine transformation of $\Bbb{R}^n$ is a Map $F:\Bbb{R}^n\to\Bbb{R}^n$ of the form

$$F({\bf p})=A{\bf p}+{\bf q}$$ (1)

for all $p\in\Bbb{R}^n$, where $A$ is a linear transformation of $\Bbb{R}^n$. If $\mathop{\rm det}(A)=1$, the transformation is Orientation-Preserving; if $\mathop{\rm det}(A)=-1$, it is Orientation-Reversing.

Dilation (Contraction, Homothecy), Expansion, Reflection, Rotation, and Translation are all affine transformations, as are their combinations. A particular example combining Rotation and Expansion is the rotation-enlargement transformation

$$\left[\begin{array}{c}x'\\ y'\end{array}\right] = s\left[\begin{array}{cc}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha\end{array}\right]\left[\begin{array}{c}x-x_0\\ y-y_0\end{array}\right]$$

$$= s\left[\begin{array}{c}\cos\alpha (x-x_0)+\sin\alpha(y-y_0)\\ -\sin\alpha(x-x_0)+\cos\alpha(y-y_0)\end{array}\right].$$ (2)

Separating the equations,

$$x' = (s\cos\alpha)x+(s\sin\alpha)y-s(x_0\cos\alpha+y_0\sin\alpha)$$ (3)

$$y' = (-s\sin\alpha)x+(s\cos\alpha)y+s(x_0\sin\alpha-y_0\cos\alpha).$$ (4)

This can be also written as

$$x' = ax+by+c$$ (5)

$$y' = bx+ay+d,$$ (6)

where

$$a = s\cos\alpha$$ (7)

$$b = -s\sin\alpha.$$ (8)

The scale factor $s$ is then defined by

$$s\equiv \sqrt{a^2+b^2},$$ (9)

and the rotation Angle by

$$\alpha=\tan^{-1}\left({-{b\over a}}\right).$$ (10)

See also Affine Complex Plane, Affine Connection, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity, Equiaffinity, Euclidean Motion

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 105, 1993.