Positive Geometries for Barycentric Interpolation

TL;DR

This paper introduces a new theoretical framework for barycentric interpolation based on positive geometries, unifying previous methods and enabling applications in 3D line space, mean-value coordinates, and splines.

Contribution

It develops a generalized approach to barycentric coordinates using positive geometries, connecting and extending existing interpolation techniques.

Findings

Unifies various barycentric coordinate methods under a positive geometry framework.

Defines generalized barycentric coordinates using rational functions associated with boundary geometries.

Discusses potential applications in 3D interpolation, mean-value coordinates, and spline constructions.

Abstract

We propose a novel theoretical framework for barycentric interpolation, using concepts recently developed in mathematical physics. Generalized barycentric coordinates are defined similarly to Shepard's method, using positive geometries - subsets which possess a rational function naturally associated to their boundaries. Positive geometries generalize certain properties of simplices and convex polytopes to a large variety of geometric objects. Our framework unifies several previous constructions, including the definition of Wachspress coordinates over polytopes in terms of adjoints and dual polytopes. We also discuss potential applications to interpolation in 3D line space, mean-value coordinates and splines.