Barycentric coordinates for convex sets

Joe Warren, Scott Schaefer, Anil N. Hirani, Mathieu Desbrun

Research output: Contribution to journalArticle

Abstract

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green's functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.

Original languageEnglish (US)
Pages (from-to)319-338
Number of pages20
JournalAdvances in Computational Mathematics
Volume27
Issue number3
DOIs
StatePublished - Oct 2007
Externally publishedYes

Keywords

  • barycentric coordinates
  • convex polyhedra
  • convex sets

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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