Ju, TaoSchaefer, ScottWarren, JoeDesbrun, MathieuMathieu Desbrun and Helmut Pottmann2014-01-292014-01-2920053-905673-24-X1727-8384https://doi.org/10.2312/SGP/SGP05/181-186A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and 3D deformation. A related problem is to express a vector lying in a convex cone as a non-negative combination of edge rays of this cone. This problem also arises in many applications such as planar graph embedding and spherical parameterization. In this paper, we present a unified geometric construction for building these weighted combinations using the notion of polar duals. We show that our method yields a simple geometric construction for Wachspress's barycentric coordinates, as well as for constructing Colin de Verdière matrices from convex polyhedra - a critical step in Lovasz's method with applications to parameterizations.Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Geometric algorithms, languages and systems