{"paper":{"title":"Ptolemy Constants as Described by Eccentricity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Steven Finch","submitted_at":"2016-08-12T16:01:14Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04299","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}