An Explicit Formula for the Spherical Curves with Constant Torsion

TL;DR

This paper derives explicit formulas using hypergeometric functions for all spherical curves with constant torsion, providing a new analytical approach to understanding their properties.

Contribution

It introduces explicit formulas for constant torsion curves on the sphere, expressed in terms of hypergeometric functions, which were not previously available.

Findings

Explicit formulas for curves with constant torsion on the sphere

Examples computed for various torsion values

Discussion of basic properties of these curves

Abstract

The purpose of this article is to give an explicit formula for all curves of constant torsion $\tau$ in the unit two-sphere $S^2(1)$. These curves and their basic properties have been known since the 1890's, and some of these properties are discussed in the Appendix. Some example curves, computed with a standard ODE package, with $\tau=.1,.5,1,2$ are shown in Figure \ref{peter}. Though their existence and some of their general properties were known, our explicit formulas for them, in terms of hypergeometric functions, are new.