Affine generalised barycentric coordinates

Shayne Waldron


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