Billiards in a circle with trajectories circumscribing a triangle

TL;DR

This paper investigates the geometric properties of large triangles in hyperbolic geometry, reformulating conditions in Euclidean terms, introducing a size measure, and providing explicit formulas for special cases.

Contribution

It reformulates hyperbolic triangle conditions in Euclidean geometry, introduces a size measure, and derives explicit formulas for isosceles triangles.

Findings

Conditions for large triangles are expressed in Euclidean terms.

A new measure of Euclidean size for large triangles is proposed.

Explicit formula provided for isosceles triangles.

Abstract

Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a large triangle, there exist two triangles that are circumscribed around it and inscribed within the unit circle. In the Klein-Beltrami model of hyperbolic geometry, we reformulate the conditions for a triangle to be classified as 'large' in a more Euclidean geometric manner. A proposed measure of its Euclidean geometric size when the triangle is considered 'large' is introduced, and an evaluation of this measure is conducted. We also provide an explicit formula for an isosceles triangle.