ABSTRACT
Barycentric coordinates are commonly used in computer graphics and computational mechanics to represent a point inside a simplex as an affine combination of the simplex's vertices. The authors show how they can be generalized to arbitrary polytopes and present the most known constructions of these generalized barycentric coordinates in 2D and 3D. The derivation of discrete harmonic coordinates suggests that different properties of harmonic functions can be exploited to derive other generalized barycentric coordinates. For a better comparison the authors group all coordinates by the first two properties. The first group includes Wachspress, discrete harmonic, and the complete family of coordinates. These coordinates are well-defined for convex polygons only and have a closed form. The second group consists of mean value, positive mean value, metric, Poisson, Gordon–Wixom, and positive Gordon–Wixom coordinates that are well-defined for arbitrary simple polygons and have a closed form.
