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A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals

Ju, T. and Schaefer, S. and Warren, J. and Desbrun, M. (2005) A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals. In: Eurographics Symposium on Geometry Processing 2005. The Eurographics Association , Aire-la-Ville, Switzerland, pp. 181-186. ISBN 3-905673-24-X.

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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.

Item Type:Book Section
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Desbrun, M.0000-0003-3424-6079
Additional Information:© The Eurographics Association 2005.
Classification Code:I.3.5 [Computer Graphics]: Geometric algorithms, languages and systems
Record Number:CaltechAUTHORS:20161108-143902274
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71821
Deposited On:09 Nov 2016 00:52
Last Modified:11 Nov 2021 04:51

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