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Algebraic Construction of Smooth Interpolants on Polygonal Domains

# Introduction

There are many varied applications for a smooth and bounded interpolant. The most straightforward is for computer graphics, for example in rendering and coloring applications [1]. Smoothly distributing color defined at nodes over a domain requires a bounded interpolating function. Interpolants can also be employed as test functions for computational modeling [2]. For the finite element method, they act as shape functions [3, 4]. Similar interpolants have been applied to the analysis of biomedical growth and shape change [5].

Conventional methods for constructing interpolants do not distribute boundary values smoothly over arbitrary polygonal shapes. Most either require a mesh or apply only to very simple geometries [4, 6]. For example, the conventional finite element development does not specifically address concavity; instead, concave domains are tessellated into convex parts. An exception is the R-function construction that applies to any polygonal domain [7]. Unlike the proposed formulation, it can only represent homogeneous boundary conditions.

In general, any function can be approximated as the sum of the products of the value at given nodes and an interpolant :

The derived interpolants are linearly independent:

Along any given boundary, only the values given at the end nodes prescribe the behavior along the line segment:

Within the domain, the interpolants are bounded, normalized between zero and one, and smooth. The interpolant generated is not unique. Any function that satisfies an elliptic operator and the boundary conditions satisfies the interpolant requirements.

Using the same method to construct general bounded interpolants on any polygon, linear interpolants are constructed. A linear interpolant satisfies the additional requirements of boundary linearity:

and exact reproduction of linear fields:

The constant field is satisfied by construction:

In most cases, the linearity requirements do not prescribe a unique interpolant. Therefore additional global conditions, such as higher-order fields, could also be imposed.