Spherical curves whose curvature depends on distance to a great circle

TL;DR

This paper investigates spherical curves with curvature depending on the distance to a great circle, introducing spherical angular momentum to characterize known and new curves, and analyzing their energy functional critical points.

Contribution

It introduces the concept of spherical angular momentum and provides new characterizations and families of spherical curves based on curvature-distance relations.

Findings

Characterization of elastic spherical curves, catenaries, and loxodromic curves

Derivation of new spherical curve families with explicit equations

Identification of curves as critical points of energy functionals

Abstract

Motivated by a problem posed by David A. Singer in 1999 and by the elastic spherical curves, we study the spherical curves whose curvature is expressed in terms of the distance to a great circle (or from a point). By introducing the notion of spherical angular momentum, we provide new characterizations of some well known curves, like the mentioned elastic curves, spherical catenaries, loxodromic-type spherical curves, the Viviani's curve, and the spherical Archimedean spirals curves. Furthermore, we show that they may be obtained as critical points of some energy curvature functionals. We also find out several new families of spherical curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions, and we are able to get arc length parametrizations of them.