Discretizing Wachspress kernels is safe

Barycentric coordinates were introduced by Möbius in 1827 as an alternative to Cartesian coordinates. They describe points relative to the vertices of a simplex and are commonly used to express the linear interpolant of data given at these vertices. Generalized barycentric coordinates and kernels extend this idea from simplices to polyhedra and smooth domains. In this paper, we focus on Wachspress coordinates and Wachspress kernels with respect to strictly convex planar domains. Since Wachspress kernels can be evaluated analytically only in special cases, a common way to approximate them is to discretize the domain by an inscribed polygon and to use Wachspress coordinates, which have a simple closed form. We show that this discretization, which is known to converge quadratically, is safe in the sense that the Wachspress coordinates used in this process are well-defined not only over the inscribed polygon, but over the entire original domain.