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Algebraic Construction of Smooth Interpolants on Polygonal Domains

Elisabeth Anna Malsch
Organization: Columbia University
Gautam Dasgupta
Organization: Columbia University
Department: Department of Civil Engineering and Engineering Mechanics

2003 International Mathematica Symposium
Conference location

Imperial College, London

Given a set of vertices for a convex or concave polygon, a smooth and bounded interpolant (valid in the interior of the domain) may be constructed in an explicit algebraic form. Rational polynomial interpolants are suitable for convex polygons; the present treatment addresses concave domains. To resolve the concavity, square roots appear in the numerator and denominator of the rational interpolant. Finite element text books do not generally discuss tessellation into concave domains. Procedural programming languages, which are predominantly used in conventional finite elements, pose formidable challenges to the implementation of abstract geometrical constructs necessary to address convexity issues. Concepts of projective geometry are utilized within Mathematica in symbolic form to yield shape functions on concave polygons.

*Mathematics > Geometry > Surfaces

algebraic construction, concave, convex, polygons
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