1. Affine transformations

Definition. $T:$   R$^n \rightarrow$   R$^n$ is an affine transformation if $T$ is a homogeneous linear transformation followed by a translation. In other words, there are a matrix $A$ and vector b such that $T($x$) =$   x$A +$   b for all x.

Example 1.1 . $T($x$) =$   x${ \left [\begin{array}{cc} 2&1\\ 3&4 \end{array} \right ] } + (5,-1)$.

In other words, $T(x,y) = (2x + 3y + 5, x + 4y - 1)$.

Example 1.2 . Any translation: $T($x$) =$   x$+$   b (the case $A = I$).

Example 1.3 . Any homogeneous linear transformation $T($x$) =$   x$A$ (the case b$=$   0).

Example 1.4 . A rotation about an arbitrary center in R$^2$, say by $90 ^\circ$ about the center $(2,1)$. (It will be explained later how to represent such a rotation as affine. Another example is a rotation about an arbitrary axis in R$^3$.)

Example 1.5 . A reflection in an arbitrary mirror line in R$^2$, say in the mirror $x+y = 2$. (It will be explained later how to represent such a reflection as affine. Another example is a reflection in an arbitrary plane mirror in R$^3$.)

Example 1.6 . If you want to transform world coordinates with a window $-2 \leq x _ {world} \leq 6$ and $-1 \leq y _ {world} \leq 5$ to device coordinates with $0 \leq x _ {dev} \leq 400$ and $0 \leq y _ {dev} \leq 300$, you can see by fiddling that the appropriate conversion is $x _ {dev} = 50 (x _ {world} + 2)$ and $y _ {dev} = 50 (y _ {world} + 1)$. This is the same as the affine transformation x$_ {dev} =$   x$_ {world} { \left [\begin{array}{cc} 50&0\\ 0&50 \end{array} \right ] } + (100,50)$. (For more complicated problems of this kind it will be better to use affine transformation methods directly.)

Example 1.7 If one affine transformation is followed by another, the result (their composition) is still affine: If $T($x$) =$   x$A +$   b and $U($x$) =$   x$C +$   d, then $T(U($x$)) = ($x$C +$   d$)A +$   b$=$   x$(CA) + ($d$A +$   b$)$, the form of an affine transformation. (Notice that d$A +$   b is constant. The resulting composition is called $T \circ U$.)

Remarks 1.8

(1)

In the definition of an affine transformation, it is important to realize that in the expression x$A +$   b, the homogeneous linear transformation is applied before the translation. If you did the translation first, you would be evaluating $($x$+$   b$)A$, which equals xA$+$   b$'$ for b$' =$   b$A$. Thus the result is affine but is not the same affine transformation.

(2)

In a transformation $T($x$) =$   x$A +$   b, let us call $A$ the homogeneous part and b the translational part. Almost all of the possible complexity is in the homogeneous part: The determinant of $A$ tells the expansion factor for $T$, $T$ is one-to-one if $A$ is nonsingular, and so on. Thus affine transformations are easy to understand if you already know the properties of homogeneous linear transformations. This is why linear algebra courses usually do not mention affine transformations.