Note that is uniquely determined by , since is a function onto . and is called the associated linear transformation of . Let us write the associated linear transformation of . Then the definition above can be illustrated by the following commutative diagram:
Here’s an example of an affine transformation. Let be an affine space with the associated vector space. Fix . For each , let be the unique point in such that . Then is a well-defined function. Furthermore, . Thus is affine, with .
An affine transformation is an affine isomorphism if there is an affine transformation such that and . Two affine spaces and are affinely isomorphic, or simply, isomorphic, if there are affine isomorphism .
Below are some basic properties of an affine transformation (see proofs here (http://planetmath.org/PropertiesOfAnAffineTransformation)):
Because of the last property, it is often enough, in practice, to identify itself as the affine space associated with , up to affine isomorphism, with the direction given by . With this in mind, we may reformulate the definition of an affine transformation as a mapping from one vector space to another, , such that there is a linear transformation such that
By fixing , we get the following equation
Let and be left vector spaces over the same division ring . An affine transformation is a mapping such that
for some linear transformation and some vector .
An affine property is a geometry property that is preserved by an affine transformation. The following are affine properties of an affine transformation Let :
linearity. Given an affine subspace of , then is an affine subspace of .
incidence. Suppose . Pick , so where . Since is bijective, there is such that . So . Since , for some , . Therefore, .
coefficients of an affine combination. Given that is an affine combination of :
where are the corresponding coefficients. Then
is the affine combination of with the same set of coefficients.
Special Affine Transformations
translation. An affine transformation of the form is called a translation. Every affine transformation can be decomposed as a product of a linear transformation and a translation: where and . The order of composition is important, since . Geometrically, a translation moves a geometric figure along a straight line.
dilation (map). If has a unique eigenvalue (that is, may be diagonalized as , the diagonal matrix with non-zero diagonal entries ), then the affine transformation is called a dilation. Note that a dilation may be written as the product of a vector with a scalar: , which is why a dilation is also called a scaling. A dilation can be visualized as magnifying or shrinking a geometric figure.
homothetic transformation. The composition of a dilation followed by a translation is called a homothetic transformation. It has the form , .
When , the set of affine maps , with function composition as the product, becomes a group, and is denoted by . The multiplicative identity is the identity map. If , then . IGL is short for of . Translations, dilations, and homothetic transformations all form subgroups of . If is the group of translations, the group of dilations, and the group of homothetic transformations, then is a normal subgroup of . Also, and are abelian groups (remember: is assumed to be a field).
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.
Another way to generalize an affine transformation is to remove the restriction on the invertibility of the linear transformation . In this respect, the set of affine transformations from to has a natural vector space structure. It is easy to see that the set of linear transformations from to forms a subspace of .
|Date of creation||2013-03-22 14:46:08|
|Last modified on||2013-03-22 14:46:08|
|Last modified by||matte (1858)|
|Defines||associated linear transformation|