Construction of polygonal interpolants: a maximum entropy approach

In this paper, we establish a link between maximizing (information-theoretic) entropy and the construction of polygonal interpolants. The determination of shape functions on n-gons (n > 3) leads to a non-unique under-determined system of linear equations. The barycentric co-ordinates phi(i), which form a partition of unity, are associated with discrete probability measures, and the linear reproducing conditions are the counterpart of the expectations of a linear function. The phi(i) are computed by maximizing the uncertainty H (phi(1), phi(2), (...) phi(n)) = -Sigma(i=1)(n) phi(i) log phi(i), subject to the above constraints. The description is expository in nature, and the numerical results via the maximum entropy (MAXENT) formulation are compared to those obtained from a few distinct polygonal interpolants. The maximum entropy formulation leads to a feasible solution for phi(i) in any convex or non-convex polygon. This study is an instance of the application of the maximum entropy principle, wherein least-biased inference is made on the basis of incomplete information. Copyright (C) 2004 John Wiley Sons, Ltd.