Ptolemy Constants as Described by Eccentricity

Abstract: Let J denote a simple closed curve in the plane. Let points a, b, c, d \in J occur in this order when traversing J in a counterclockwise direction. Define p(a,b,c,d) to be the ratio of ab*cd+ad*bc to ac*bd, where zw denotes distance between z and w. Define P(J) to be the supremum of p over all such points. Harmaala & Kl\'en [1] provided bounds on P(J) when J is an ellipse or rectangle of eccentricity \epsilon. We nonrigorously give formulas for P(J) here, in the hope that someone else can fill gaps in our reasoning.