affine transformation

Definition 1.

Let (Ai,fi) be affine spacesPlanetmathPlanetmath associated with a left (right) vector spacesMathworldPlanetmath Vi (over some division ring D), where i=1,2. An affine transformationMathworldPlanetmath from A1 to A2 is a function α:A1A2 such that there is a linear transformation T:V1V2 such that


for any P,QA.

Note that T is uniquely determined by α, since f1 is a function onto V1. T and is called the associated linear transformation of α. Let us write [α] the associated linear transformation of α. Then the definition above can be illustrated by the following commutative diagramMathworldPlanetmath:


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

An affine transformation α:A1A2 is an affine isomorphism if there is an affine transformation β:A2A1 such that βα=1A1 and αβ=1A2. Two affine spaces A1 and A2 are affinely isomorphic, or simply, isomorphicPlanetmathPlanetmathPlanetmathPlanetmath, if there are affine isomorphism α:A1A2.

Below are some basic properties of an affine transformation (see proofs here (

  1. 1.

    α is onto iff [α] is.

  2. 2.

    α is one-to-one iff [α] is.

  3. 3.

    A bijectiveMathworldPlanetmathPlanetmath affine transformation α is an affine isomorphism. In fact, [α-1]=[α]-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 α from one vector space V to another, W, such that there is a linear transformation T:VW such that


By fixing wV, we get the following equation

Definition 2.

Let V and W be left vector spaces over the same division ring D. An affine transformation is a mapping α::VW such that


for some linear transformation T:VW and some vector wW.

An affine property is a geometryMathworldPlanetmath property that is preserved by an affine transformation. The following are affine properties of an affine transformation Let A:VW:

  • 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+vT+u. Pick xA(S+v)=L(S)+L(v)+w, so x=y+L(v)+w where yL(S). Since L is bijective, there is zS such that L(z)=y. So A(z+v)=L(z)+L(v)+w=x. Since z+vS+v, z+v=t+u for some tT, x=A(z+v)=A(t+u)A(T+u). Therefore, A(S+v)A(T+u).

  • parallelismPlanetmathPlanetmathPlanetmath. 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 v1,,vn:


    where kiF and k1++kn=1 are the corresponding coefficients. Then

    A(v) = k1L(v1)++knL(vn)+w
    = k1(L(v1)+w)++kn(L(vn)+w)
    = k1A(v1)++knA(vn)

    is the affine combination of A(v1),,A(vn) 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 productPlanetmathPlanetmathPlanetmath 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 compositionMathworldPlanetmath is important, since BCCB. Geometrically, a translation moves a geometric figure along a straight line.

  2. 2.

    dilation (map). If L has a unique eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath d0 (that is, L may be diagonalized as dI, the diagonal matrixMathworldPlanetmath with non-zero diagonalMathworldPlanetmath entries =dF), 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, 0dF.

  4. 4.

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


  1. 1.

    When V=W, the set of affine maps VV, with function composition as the product, becomes a group, and is denoted by IGL(V). The multiplicative identityPlanetmathPlanetmath is the identity mapMathworldPlanetmath. 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 subgroupsMathworldPlanetmathPlanetmath of 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 subgroupMathworldPlanetmath of T. Also, Aut(T) and Aut(D) are abelian groupsMathworldPlanetmath (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 automorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath. 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 restrictionPlanetmathPlanetmathPlanetmath 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 structureMathworldPlanetmath. It is easy to see that the set L(V,W) of linear transformations from V to W forms a subspacePlanetmathPlanetmath 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 CollineationMathworldPlanetmath
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