2016/08/12 by Steven Finch, Steven R. Finch, Finch, Steven
Mathematics · Physics and Astronomy · #51M16 (Primary) 65K10 #68W30 (Secondary) #FOS: Mathematics #Historical Astronomy and Related Studies #Metric Geometry (math.MG) #math.MG #msc:51M16 #msc:65K10 #msc:68W30
paper · pdf · doi:10.48550/arxiv.1608.04299
3 pages, 1 figure
arxiv created 2016/08/12 · openalex publication_date 2016/08/12 · arxiv updated 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let J denote a simple closed curve in the plane. Let points a, b, c, d ∈ 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 & Klén [1] provided bounds on P(J) when J is an ellipse or rectangle of eccentricity ε. We nonrigorously give formulas for P(J) here, in the hope that someone else can fill gaps in our reasoning.