| Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals (2008) | |||||||||||||||
Abstract | |||||||||||||||
| A 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 | |||||||||||||||
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