Affine coordinate system

This article is about affine coordinates in affine geometry. For affine coordinate systems in algebraic geometry, see Algebraic variety.

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 $(O, R 1, \dots R n)$ 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, \dots x n) \mapsto O + x 1 (R 1 - O) + \dots + 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 $(1, x 1, \dots , x n -1)$ . 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.

Affine coordinate system

See also