Positive Geometries for Barycentric Interpolation

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.