Polygons in hyperbolic geometry 2: Maximality of area

TL;DR

This paper investigates the maximal area configurations of polygons in hyperbolic geometry, establishing conditions for cocyclicity and uniqueness, and contrasting hyperbolic and Euclidean cases.

Contribution

It proves that maximal polygons are cocyclic and unique up to hyperbolic motions, highlighting differences from Euclidean geometry and addressing hyperbolic-specific challenges.

Findings

Maximal polygons are cocyclic and oriented-convex or collinear.

Maximal polygons are unique up to hyperbolic motions.

Vertices lie on cycles: circle, line, or horocycle, unlike Euclidean case.

Abstract

This second part on polygons in the hyperbolic plane is based on the first part which deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The topic here is the maximum question for the area of these polygons. It is shown that the maximal copies must be cocyclic and oriented-convex or have a special collinear shape. The maximal copies are unique up to rigid hyperbolic motions. The cocyclicity means that the vertices are situated on a distance circle, a distance line or a horocycle, the three types of cycles in the hyperbolic plane. The phenomenon of different circle types stands in salient contrast to the Euclidean case and pays for various difficulties in the hyperbolic discussion.