information:classification
Affine coordinates

An affine frame of an affine space consists of a point, called the origin, and a linear basis of the associated vector space. More precisely, for an affine space A with associated vector space A → {\displaystyle {\overrightarrow {A}}}, the origin o belongs to A, and the linear basis is a basis (v,..., v) of A → {\displaystyle {\overrightarrow {A}}} (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar).

For each point p of A, there is a unique sequence λ 1, …, λ n {\displaystyle \lambda _{1},\dots,\lambda _{n}} of elements of the ground field such that

or equivalently

The λ i {\displaystyle \lambda _{i}} are called the affine coordinates of p over the affine frame (o, v,..., v).

Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame (o, v,..., v) such that (v,..., v) is an orthonormal basis.