# Affine coordinate system

In mathematics, an **affine coordinate system** is a coordinate system on an affine space where each coordinate is an affine map to the number line. In other words, it is an injective affine map from an affine space A to the coordinate space *K*^{n}, where K is the field of scalars, for example, the real numbers **R**.

The most important case of affine coordinates in Euclidean spaces is real-valued Cartesian coordinate system. Orthogonal affine coordinate systems are rectangular, and others are referred to as **oblique**.

A system of n coordinates on n-dimensional space is defined by a (n+1)-tuple (*O*, *R*_{1}, … *R*_{n}) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula:

- (
*x*_{1}, …*x*_{n}) ↦*O*+*x*_{1}(*R*_{1}−*O*) + … +*x*_{n}(*R*_{n}−*O*) .

Note that *R*_{j} − *O* are *difference* vectors with the origin in O and ends in R_{j} .

An affine space cannot have a coordinate system with n less than its dimension, but n may indeed be greater, which means that the coordinate map is not necessary surjective.
Examples of n-coordinate system in an (n−1)-dimensional space are barycentric coordinates and affine "homogeneous" coordinates (1, *x*_{1}, … , *x*_{n−1}). In the latter case the x_{0} coordinate is equal to 1 on all space, but this "reserved" coordinate allows for matrix representation of affine maps similar to one used for projective maps.

## See also

- Convex combination
- Centroid, can be calculated in affine coordinates
- Homogeneous coordinates, a similar concept but without uniqueness of values