Ptolemy Constants as Described by Eccentricity

TL;DR

This paper explores the Ptolemy constant for simple closed curves, providing formulas and bounds related to eccentricity, with the aim of encouraging further rigorous analysis.

Contribution

It offers nonrigorous formulas for the Ptolemy constant of curves like ellipses and rectangles, extending previous bounds based on eccentricity.

Findings

Bounds on P(J) for ellipses and rectangles

Formulas for P(J) related to eccentricity

Encourages rigorous proof development

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.