affine transformation

Definition 1.

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

$$T(f_1(P,Q))=f_2(\alpha (P),\alpha (Q))$$

for any $P\mathrm{,}Q\mathrm{\in }A$.

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:

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

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

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

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 $\alpha \mathrm{:}\mathrm{:}V\mathrm{\to }W$ such that

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

for some linear transformation $T\mathrm{:}V\mathrm{\to }W$ and some vector $w\mathrm{\in }W$.

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 $v_1,\mathrm{\dots },v_n$:

  $$v=k_1{v}_1+\mathrm{\cdots }+k_n{v}_n,$$

  where $k_i\in F\text{ and }{k}_1+\mathrm{\cdots }+k_n=1$ are the corresponding coefficients. Then

  $A(v)$ $=$ $k_{1}L(v_1)+\mathrm{\cdots }+k_{n}L(v_n)+w$

  $=$ $k_1(L(v_1)+w)+\mathrm{\cdots }+k_n(L(v_n)+w)$

  $=$ $k_{1}A(v_1)+\mathrm{\cdots }+k_{n}A(v_n)$

  is the affine combination of $A(v_1),\mathrm{\dots },A(v_n)$ with the same set of coefficients.

Special Affine Transformations

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 $BC\ne CB$. Geometrically, a translation moves a geometric figure along a straight line.

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

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. When $V=W$, the set of affine maps $V\to V$, with function composition as the product, becomes a group, and is denoted by $\mathrm{IGL}(V)$. 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 $\mathrm{IGL}(V)$. 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, $\mathrm{Aut}(T)$ and $\mathrm{Aut}(D)$ are abelian groups (remember: $F$ is assumed to be a field).

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. 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