Properties of Local Coordinates Based on Dirichlet Tesselations

Local coordinates based on the Dirichlet tessellation (Voronoi diagram) provide a means to express a point as a linear combination of certain fixed points by using ratios of areas (or volumes) of certain regions. We add insight into the structure of the local coordinates by proving some of their basic properties by deriving formulas for their gradients from some simple geometry and proving the properties of smoothness and linear precision.