Affine coordinate system

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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, … Rn) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula:

(x1, … xn) ↦ O + x1 (R1 − O) + … + xn (Rn − O) .

Note that Rj − O are difference vectors with the origin in O and ends in Rj.

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, … , xn−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.

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