The hanging chain problem in the sphere and in the hyperbolic plane

TL;DR

This paper introduces and characterizes catenary curves in spherical and hyperbolic geometries, extending classical notions to curved spaces and exploring their properties relative to geodesics and horocycles.

Contribution

It defines catenaries in curved spaces based on potential energy, providing new characterizations involving curvature and normal vectors, and extends the concept in hyperbolic geometry.

Findings

Catenaries are characterized by curvature and normal vector angles.

In hyperbolic space, catenaries are extended to horocycles and horocycle distances.

The paper establishes multiple geometric characterizations of these curves.

Abstract

In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its potential energy is determined by the distance to a given geodesic of the space. Several characterizations of the catenary are established in terms of the curvature of the curve and of the angle that its unit normal makes with a vector field of the ambient space. Furthermore, in the hyperbolic plane, we extend the concept of catenary substituting the reference geodesic by a horocycle or the hyperbolic distance by the horocycle distance.