Two Dimensions

Abstract

Seminal construction of rational basis functions rested heavily on fundamental divisor theorems in algebraic geometry. More recent analysis rests primarily on generalization of analysis by Moebius relating to barycentric coordinates. For triangles, the normalized barycentric coordinates that sum to unity are known as areal coordinates (Coxeter 1961). Barycentric coordinates are homogeneous. The numerators of degree-one basis functions with the GADJ ratios are generalized barycentric coordinates. For a triangle, the areal coordinates are obtained by dividing them by their sum which is a constant. For polygons of higher order (n > 3) the sum is no longer a constant. It is a polynomial of maximal order n−3. Although it is no longer an area, the degree-one basis functions obtained by dividing the numerators by this sum are generalized areal coordinates. It is customary in current literature to call these degree-one basis functions “barycentric coordinates.” A unique harmonic function is associated with given boundary values over a closed boundary satisfying certain continuity constraints. The algebraic geometry approach discloses how unique rational basis functions achieving any degree of polynomial approximation are inherited from an element boundary satisfying similar constraints, convexity for polygons and “well-set” for elements with curved sides. It is instructive to review this association. In this discussion the term “degree-one basis functions” is synonymous with “areal coordinates” and “normalized barycentric coordinates” where “normalized” is often omitted.