# affine transformation

###### Definition 1.

Let $(A_{i},f_{i})$ be affine spaces associated with a left (right) vector spaces $V_{i}$ (over some division ring $D$), where $i=1,2$. An affine transformation from $A_{1}$ to $A_{2}$ is a function $\alpha:A_{1}\to A_{2}$ such that there is a linear transformation $T:V_{1}\to V_{2}$ such that

 $T(f_{1}(P,Q))=f_{2}(\alpha(P),\alpha(Q))$

for any $P,Q\in A$.

Note that $T$ is uniquely determined by $\alpha$, since $f_{1}$ is a function onto $V_{1}$. $T$ and is called the associated linear transformation of $\alpha$. Let us write $[\alpha]$ the associated linear transformation of $\alpha$. Then the definition above can be illustrated by the following commutative diagram:

 $\xymatrix@+=2cm{A_{1}\times A_{1}\ar[r]^{-}{f_{1}}\ar[d]_{(\alpha,\alpha)}&V_{% 1}\ar[d]^{[\alpha]}\\ A_{2}\times A_{2}\ar[r]_{-}{f_{2}}&V_{2}}$

Here’s an example of an affine transformation. Let $(A,f)$ be an affine space with $V$ the associated vector space. Fix $v\in V$. For each $P\in A$, let $\alpha(P)$ be the unique point in $A$ such that $f(P,\alpha(P))=v$. Then $\alpha:A\to A$ is a well-defined function. Furthermore, $f(\alpha(P),\alpha(Q))=v+f(\alpha(P),\alpha(Q))-v=f(P,\alpha(P))+f(\alpha(P),% \alpha(Q))+f(\alpha(Q),Q)=f(P,Q)=1_{V}(f(P,Q))$. Thus $\alpha$ is affine, with $[\alpha]=1_{V}$.

An affine transformation $\alpha:A_{1}\to A_{2}$ is an affine isomorphism if there is an affine transformation $\beta:A_{2}\to A_{1}$ such that $\beta\circ\alpha=1_{A_{1}}$ and $\alpha\circ\beta=1_{A_{2}}$. Two affine spaces $A_{1}$ and $A_{2}$ are affinely isomorphic, or simply, isomorphic, if there are affine isomorphism $\alpha:A_{1}\to A_{2}$.

Below are some basic properties of an affine transformation (see proofs here (http://planetmath.org/PropertiesOfAnAffineTransformation)):

1. 1.

$\alpha$ is onto iff $[\alpha]$ is.

2. 2.

$\alpha$ is one-to-one iff $[\alpha]$ is.

3. 3.

A bijective affine transformation $\alpha$ is an affine isomorphism. In fact, $[\alpha^{-1}]=[\alpha]^{-1}$.

4. 4.

Two affine spaces associated with the same vector space are isomorphic.

Because of the last property, it is often enough, in practice, to identify $V$ itself as the affine space associated with $V$, up to affine isomorphism, with the direction given by $f(v,w)=w-v$. With this in mind, we may reformulate the definition of an affine transformation as a mapping $\alpha$ from one vector space $V$ to another, $W$, such that there is a linear transformation $T:V\to W$ such that

 $T(w-v)=\alpha(w)-\alpha(v).$

By fixing $w\in V$, we get the following equation

 $\alpha(v)=T(v)+(\alpha(w)-T(w)).$
###### Definition 2.

Let $V$ and $W$ be left vector spaces over the same division ring $D$. An affine transformation is a mapping $\alpha:\colon V\to W$ such that

 $\alpha(v)=T(v)+w,\quad v\in V$

for some linear transformation $T\colon V\to W$ and some vector $w\in W$.

An affine property is a geometry property that is preserved by an affine transformation. The following are affine properties of an affine transformation Let $A:V\to W$:

• linearity. Given an affine subspace $S+v$ of $V$, then $A(S+v)=L(S+v)+w=L(S)+(L(v)+w)$ is an affine subspace of $W$.

• incidence. Suppose $S+v\subseteq T+u$. Pick $x\in A(S+v)=L(S)+L(v)+w$, so $x=y+L(v)+w$ where $y\in L(S)$. Since $L$ is bijective, there is $z\in S$ such that $L(z)=y$. So $A(z+v)=L(z)+L(v)+w=x$. Since $z+v\in S+v$, $z+v=t+u$ for some $t\in T$, $x=A(z+v)=A(t+u)\in A(T+u)$. Therefore, $A(S+v)\subseteq A(T+u)$.

• parallelism. Given two parallel affine subspaces $S+a$ and $S+b$, then $A(S+a)=L(S)+(L(a)+w)$ and $A(S+b)=L(S)+(L(b)+w)$ are parallel.

• coefficients of an affine combination. Given that $v$ is an affine combination of $v_{1},\ldots,v_{n}$:

 $v=k_{1}v_{1}+\cdots+k_{n}v_{n},$

where $k_{i}\in F\mbox{ and }k_{1}+\cdots+k_{n}=1$ are the corresponding coefficients. Then

 $\displaystyle A(v)$ $\displaystyle=$ $\displaystyle k_{1}L(v_{1})+\cdots+k_{n}L(v_{n})+w$ $\displaystyle=$ $\displaystyle k_{1}(L(v_{1})+w)+\cdots+k_{n}(L(v_{n})+w)$ $\displaystyle=$ $\displaystyle k_{1}A(v_{1})+\cdots+k_{n}A(v_{n})$

is the affine combination of $A(v_{1}),\ldots,A(v_{n})$ with the same set of coefficients.

## Special Affine Transformations

1. 1.

translation. An affine transformation of the form $A(v)=v+w$ is called a translation. Every affine transformation can be decomposed as a product of a linear transformation and a translation: $A(v)=L(v)+w=BC(v)$ where $C(v)=L(v)$ and $B(v)=v+w$. The order of composition is important, since $BC\neq CB$. Geometrically, a translation moves a geometric figure along a straight line.

2. 2.

dilation (map). If $L$ has a unique eigenvalue $d\neq 0$ (that is, $L$ may be diagonalized as $dI$, the diagonal matrix with non-zero diagonal entries $=d\in F$), then the affine transformation $A(v)=L(v)$ is called a dilation. Note that a dilation may be written as the product of a vector with a scalar: $A(v)=dv$, which is why a dilation is also called a scaling. A dilation can be visualized as magnifying or shrinking a geometric figure.

3. 3.

homothetic transformation. The composition of a dilation followed by a translation is called a homothetic transformation. It has the form $A(v)=dv+w$, $0\neq d\in F$.

4. 4.

Euclidean transformation. In the case when both $V$ and $W$ are Euclidean vector spaces, if the associated linear transformation is orthogonal, then the affine transformation is called a Euclidean transformation.

## Remarks

1. 1.

When $V=W$, the set of affine maps $V\to V$, with function composition as the product, becomes a group, and is denoted by ${\rm IGL}(V)$. The multiplicative identity is the identity map. If $A(v)=L(v)+w$, then $A^{-1}(v)=L^{-1}(v)-L^{-1}(w)$. IGL is short for of $V$. Translations, dilations, and homothetic transformations all form subgroups of ${\rm IGL}(V)$. If $T$ is the group of translations, $D$ the group of dilations, and $H$ the group of homothetic transformations, then $T$ is a normal subgroup of $T$. Also, $\operatorname{Aut}(T)$ and $\operatorname{Aut}(D)$ are abelian groups (remember: $F$ is assumed to be a field).

2. 2.

One can more generally define an affine transformation to be an order-preserving bijection between two affine geometries. It can be shown that this definition coincides with the above one if the underlying field admits no non-trivial automorphisms. When the two affine geometries are the same, the bijective affine transformation is called an affinity.

3. 3.

Another way to generalize an affine transformation is to remove the restriction on the invertibility of the linear transformation $L$. In this respect, the set $A(V,W)$ of affine transformations from $V$ to $W$ has a natural vector space structure. It is easy to see that the set $L(V,W)$ of linear transformations from $V$ to $W$ forms a subspace of $A(V,W)$.

 Title affine transformation Canonical name AffineTransformation Date of creation 2013-03-22 14:46:08 Last modified on 2013-03-22 14:46:08 Owner matte (1858) Last modified by matte (1858) Numerical id 37 Author matte (1858) Entry type Definition Classification msc 51A10 Classification msc 51A15 Classification msc 15A04 Synonym scaling Related topic LinearTransformation Related topic AffineSpace Related topic ComplexLine Related topic AffineCombination Related topic AffineGeometry Related topic Collineation Defines IGL Defines translation Defines dilation Defines dilation map Defines homothetic transformation Defines affine property Defines affine isomorphism Defines associated linear transformation Defines affinely isomorphic Defines affinity