# Affine generalised barycentric coordinates

## Shayne Waldron

## Abstract:

For a given set of points in $\R^d$,
there may be many ways to write a point $x$
in their affine hull as an affine combination of them.
We show there is a *unique* way
which minimises the sum of the squares of the coefficients.
It turns out that these coefficients, which are given by
a simple formula, are affine functions of $x$, and so generalise
the barycentric coordinates.
These *affine* generalised barycentric coordinates have many
nice properties, e.g.,
they depend continuously on the points,
and transform naturally under symmetries
and affine transformations of the points.
Because of this, they are well suited to representing
polynomials on polytopes.
We give a brief discussion of the corresponding Bernstein--B\'ezier
form and potential applications, such as finite elements
and orthogonal polynomials.

**Keywords:**
barycentric coordinates,
Wachspress coordinates,
mean value coordinates,
multivariate Bernstein polynomials,
least squares method

**Math Review Classification:**
Primary 41A65, 65D17, 52B11, 41A10;
Secondary 42C15, 41A36, 65D10

**Length:** 15 pages

**Last Updated:** 2 March 2010

## Availability: