Overview

The underlying linear space A is used to define affine coordinates in . However, vectors are not necessary to describe points. In 1827 M bius introduced barycentric coordinates which define a point with respect to some basis points. Barycentric coordinates are symmetric relative to these basis points, and they provide excellent insight into affine spaces and their structure.

Literature: Baker, Blaschke, M bius

10.1 Barycentric Coordinates

Affine coordinates in an affine space refer to a basis a ₁, ..., a _n of the underlying linear space A. The vector basis can be suppressed by introducing the points p ₀ = a, p ₁ = a + a ₁, ..., p _n = a + a _n. Thus, an arbitrary point p = a + a ₁ x ₁ + ? + a _nx _n has the representation

or more symmetrically

where x ₀ is defined by

Combining the last two equations one obtains

This representation of p where the coefficients of the points sum to 1 is called an affine combination. The x ₀, ..., x _n are called barycentric co- ordinates of p with respect to the frame p ₀, ..., p _n. One says that is spanned by p ₀, ..., p _n, or is the affine hull of p ₀, ...,