Loci, 2008
Generalized Baseball Curves: Three Symmetries and You're In! Allison, Diaz, and Miller

11. The closed curve problem

We'd like to conclude this article by discussing how our results relate to some much larger open questions.

In this article we have been studying a certain class of periodic or closed spherical curves. The existence of periodic spherical curves with a prescribed geodesic curvature is related to a long-standing open question from classical differential geometry. Independently, Fenchel [1951] and Efimov [1947] formulated the following problem:

Problem 8 [The Closed Curve Problem]. Find (explicit) necessary and sufficient conditions for the curvature and torsion of a space curve as periodic functions of arc length in order that the curve be closed.

This problem remains open although implicit solutions have been found [Hwang, 1981] and most evidence indicates that there is not an effective solution.

A natural subquestion of the Closed Curve Problem is to ask: given a curve on a sphere, can we find necessary and sufficient conditions depending on the geodesic curvature of the curve for the curve to be closed? Nikolaevsky [1994] provided further evidence that even this special case of the Closed Curve Problem likely has no effective solution. An obvious necessary condition for the closure of a spherical curve is that its geodesic curvature be periodic. However, examples in this article make it clear that periodic geodesic curvature is not sufficient for closure. A natural place to start to explore this question is with numerical experimentation of curves with simple periodic geodesic curvature, and in fact, such experimentation was the original impetus to explore the curves we are calling generalized baseball curves. We began by asking for which values of c is a spherical curve with geodesic curvature sin(cs) closed? (As an aside, an interesting article [Scofield, 1995] on curves of constant precession was the result of a similar effort to investigate closure conditions for space curves whose curvature and torsion are given by simple periodic functions, such as κ(s) = ωcos(μs) and τ(s) = ωsin(μs) .) We have shown in the present article that, given a family of periodic geodesic curvature functions with some particular symmetries, there exist many different closed spherical curves whose curvature functions lie in the given family. However, we have not found explicit sufficient conditions for the existence of closed spherical curves and the open question remains.