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 Kⁿ, 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, R₁, … 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:

(x₁, … x_n) ↦ O + x₁ (R₁ − 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, x₁, … , x_n−1). In the latter case the x₀ 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