PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Very high
affine transformation (Definition)
Definition 1   Let $ V$ and $ W$ be vector spaces over the same field $ F$. An affine transformation is a mapping $ A\colon V \to W$ such that
$\displaystyle A(v)=L(v)+w, \quad v\in V $
for some invertible linear transformation $ L\colon V\to W$ and some vector $ w\in W$.

$ L$ is called the associated linear transformation of the affine transformation $ A$.

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$:
    $\displaystyle v=k_1v_1+\cdots +k_nv_n,$
    where $ k_i\in F$ and $ k_1+\cdots+k_n=1$ are the corresponding coefficients. Then
    $\displaystyle A(v)$ $\displaystyle =$ $\displaystyle k_1L(v_1)+\cdots+k_nL(v_n)+w$  
      $\displaystyle =$ $\displaystyle k_1(L(v_1)+w)+\cdots+k_n(L(v_n)+w)$  
      $\displaystyle =$ $\displaystyle k_1A(v_1)+\cdots+k_nA(v_n)$  

    is the affine combination of $ A(v_1),\ldots,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\neq CB$. Geometrically, a translation moves a geometric figure along a straight line.
  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. 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. 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 $ {\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 Inhomogenous General Linear group 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. 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.
  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)$.



Anyone with an account can edit this entry. Please help improve it!

"affine transformation" is owned by matte. [ full author list (3) ]
(view preamble)

View style:

See Also: linear transformation, geometrization of $\mathbb{R}^n$, complex line, affine combination, affine geometry

Other names:  scaling
Also defines:  IGL, translation, dilation, dilation map, homothetic transformation, affine property
Log in to rate this entry.
(view current ratings)

Cross-references: subspace, easy to see, structure, restriction, automorphisms, affine geometries, bijection, abelian groups, normal subgroup, subgroups, identity map, multiplicative identity, group, function, orthogonal, Euclidean vector spaces, Euclidean transformation, scalar, diagonal, diagonal matrix, eigenvalue, line, straight, composition, order, product, affine combination, coefficients, parallel, parallelism, bijective, affine subspace, property, geometry, linear transformation, vector, invertible linear transformation, mapping, field, vector spaces
There are 45 references to this entry.

This is version 26 of affine transformation, born on 2004-10-24, modified 2007-07-08.
Object id is 6413, canonical name is AffineTransformation.
Accessed 17618 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
 51A15 (Geometry :: Linear incidence geometry :: Structures with parallelism)
 51A10 (Geometry :: Linear incidence geometry :: Homomorphism, automorphism and dualities)
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy
minimal number of pts. to determine that a transform is affine? by joshsamani on 2008-02-25 20:37:01
I have the following question:

Let $V$ be a finite-dimensional inner product space over a field $K$ with $\dim(V) = n$. Consider a finite set of points $v_1, \dots , v_m \in V$. Let $F:V\to V$. What is the lowest $m$ such that

if for all $i,j\in\{1, \dots, m\}$

$|F(v_i) - F(v_j)| = |v_i - v_j|$

then for all $v,w\in V$

$|F(v) - F(w)| = |v - w|$

???

I feel as though the answer is that the minimum such $m$ is $n+1$ based in Euclidean space intuition and I also feel that in this case the points should be affinely independent. Also, does one need to restrict the field $K$ in any way?
[ reply | up ]
transformation of convex polygons by pesachzon on 2006-01-24 11:56:12
Hello,
If a point P is inside a convex polygon Q and f is an affine transformation, will f(P) be inside f(Q)? i think the answer is yes, but why?
[ reply | up ]
affine transformation requirement by CWoo on 2004-10-24 19:00:36
I think the linear transformation in the definition of affine transformation is required to be invertible. Correct?
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)