### Affine maps

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 .