Wachspress' Conjecture and the Geometry of Polycons Bounded by Three Conics

Polycons - first introduced by Wachspress in 1975 - are generalizations of polygons in that they allow conic boundary components and were initially introduced as a tool in finite element methods. We are interested in the adjoint curve of a given polycon, i.e. the unique curve of minimal degree vanishing in the so-called residual arrangement. It was conjectured by Wachspress that under some regularity assumptions this curve does not vanish in the interior of its defining polycon. However, until recently the only class of polycons for which this was proven were convex polygons. We present a polycon bounded by three conics that constitutes a counterexample to Wachspress' conjecture. While this counterexample is easy to state, its origin reveals some beautiful geometry of polycons. There are two underlying results: 1) Replacing one degree two boundary component of a polycon with a line produces a new polycon. We show that the adjoint of the latter is a contact curve to the adjoint of the former. 2) A recent direction of research considers fibers of the maps that assign to a polycon with boundary components of fixed degree its adjoint curve. We give an explicit description of the fibers of the adjoint map in the case of polycons bounded by three conics, as well as some rather immediate corollaries.