A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals

Ju, Tao; Schaefer, Scott; Warren, Joe; Desbrun, Mathieu

A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals

Files

181-186.pdf (377.88 KB)

Date

2005

Authors

Ju, Tao
Schaefer, Scott
Warren, Joe
Desbrun, Mathieu

Publisher

The Eurographics Association

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.

        @inproceedings{:10.2312/SGP/SGP05/181-186
,
booktitle = {Eurographics Symposium on Geometry Processing 2005
},
editor = {Mathieu Desbrun and Helmut Pottmann
},
title = {{A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals
}},
author = {Ju, Tao and 
Schaefer, Scott and 
Warren, Joe and 
Desbrun, Mathieu
},
year = {2005
},
publisher = {The Eurographics Association
},
ISSN = {1727-8384
},
ISBN = {3-905673-24-X
},
DOI = {/10.2312/SGP/SGP05/181-186
}
}