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

摘要

We present a full geometric parameterization of generalizedrnbarycentric coordinates on convex polytopes. We show that theserncontinuous and non-negative coefficients ensuring linear precisionrncan be efficiently and exactly computed through a power diagramrnof the polytope’s vertices and the evaluation point. In particular,rnwe point out that well-known explicit coordinates such as Wachspress,rnDiscrete Harmonic, Voronoi, or Mean Value correspond tornsimple choices of power weights. We also present examples ofrnnew barycentric coordinates, and discuss possible extensions suchrnas power coordinates for non-convex polygons and smooth shapes.