  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.
