Barycentric coordinates for convex sets Academic Article uri icon

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. 2006 Springer Science+Business Media, Inc.

published proceedings

  • Advances in Computational Mathematics

author list (cited authors)

  • Warren, J., Schaefer, S., Hirani, A. N., & Desbrun, M.

citation count

  • 135

complete list of authors

  • Warren, Joe||Schaefer, Scott||Hirani, Anil N||Desbrun, Mathieu

publication date

  • October 2007