Transfinite barycentric interpolation via Dirichlet energy minimization for conical surfaces

In this theoretical work, we analyze general constructions for transfinite (also known as continuous and integral-based) barycentric coordinates and consider a simple variational principle to arrive at a transfinite version of the Laplace barycentric coordinates. We demonstrate how our approach leads to a general description of the transfinite barycentric coordinates and establish links with Dirichlet energy minimization problems for conical surfaces. Both the 2D and 3D cases are studied. Links with the Minkowski and Christoffel inverse problem in differential geometry are discussed.