A function , where and are affine spaces is called an affine map, if for any and , such that ,
Equivalently, is an affine map, if for a point and a linear mapping we get
The map does not depend on and is called the derivative of and denoted by .
It is worth noticing that
- for any point and , ,
- for , ,
- if is a point basis of and is a system of points in , there exists exactly one affine map , such that , for all ,
- if is a basic system of and and , there exists exactly one affine map , such that and , for .
Standard examples of affine maps
The map , for a given is called the translation by .
The map such that for a givon and for a given is called a homothety with centre in and scale .
If , then a map is called a projection onto along , if and for and for .
If , then a map is called a reflection across along , if and for and for .