Power Coordinates: A Geometric Construction of Barycentric Coordinates on Convex Polytopes

Abstract

We present a full geometric parameterization of generalized barycentric coordinates on convex polytopes. We show that these continuous and non-negative coefficients ensuring linear precision can be efficiently and exactly computed through a power diagram of the polytope's vertices and the evaluation point. In particular, we point out that well-known explicit coordinates such as Wachspress, Discrete Harmonic, Voronoi, or Mean Value correspond to simple choices of power weights. We also present examples of new barycentric coordinates, and discuss possible extensions such as power coordinates for non-convex polygons and smooth shapes.

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