The geometry of cyclic hyperbolic polygons

TL;DR

This paper explores the geometric properties of cyclic hyperbolic polygons, providing formulas, bounds, and behavioral insights into their area and radii, with distinctions based on centeredness.

Contribution

It introduces a parametrization of convex cyclic hyperbolic polygons and analyzes the derivatives and monotonicity of their area and radii functions.

Findings

Formulas and bounds for derivatives of area and radii functions.

Distinct monotonicity behaviors between centered and non-centered polygons.

Insights into the geometric behavior of hyperbolic cyclic polygons.

Abstract

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces of $(0,\infty)^n$ that contain their side length collections, and area and circumcircle or "collar" radius determine symmetric, smooth functions on these spaces. We give formulas for and bounds on the derivatives of these functions, and make some observations on their behavior. Notably, the monotonicity properties of area and circumcircle radius exhibit qualitative differences on the collection of centered vs non-centered cyclic polygons, where a cyclic polygon is "centered" if it contains the center of its circumcircle in its interior.