Maximum Entropy Coordinates for Arbitrary Polytopes

• Published in: Computer graphics forum 2008-07, Vol.27 (5), p.1513-1520
• Main Authors: Hormann, K., Sukumar, N.
• Format: Article
• Language: English
• Subjects: Entropy; G.1.1 [Numerical Analysis]: Interpolation formulas; G.1.6 [Numerical Analysis]: Constrained optimization; I.3.5 [Computer Graphics]: Geometric algorithms; Polytopes
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/j.1467-8659.2008.01292.x

Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle's vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton's method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higher‐dimensional polytopes.