Different definitions of conic sections in hyperbolic geometry

TL;DR

This paper explores how various definitions of conic sections in Euclidean geometry behave in hyperbolic geometry, revealing that they remain meaningful but are no longer equivalent, and analyzes their interrelations.

Contribution

It demonstrates that Euclidean conic definitions extend to hyperbolic geometry but lose their equivalence, providing insights into their relationships in the hyperbolic plane.

Findings

Definitions remain meaningful in hyperbolic geometry

The equivalence of Euclidean conic definitions does not hold in hyperbolic geometry

Relationships among different hyperbolic conic definitions are characterized

Abstract

In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the relationships among them.