On the Denominator of Wachspress Basis Functions for Polycons of Order Six

Abstract

Eugene Wachspress's (1975) rational basis functions allow function approximation over regions bounded by lines and conics, called polycons [1]. It is still an open conjecture that the denominator of a Wachspress basis function does not equal zero at any point on the polycon. Proof of this conjecture is necessary to ensure the basis functions are well-defined. In this thesis, we construct the Wachspress basis functions in a more streamlined fashion than [1] and then explain efforts to prove Wachspress's conjecture for polycons bounded by exactly three conics, the simplest case where a counterexample may occur. We make some progress toward a continuity argument that would allow the problem to be split into finitely many cases and provide MATLAB code to test these cases.