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

1. Introduction

Consider the curve traced out on the surface of a baseball by its seam. Many different authors have tried to give mathematical equations describing this curve in many different ways. There have been so many different descriptions of this curve, in fact, that John Conway proposed the following conjecture [Weisstein, n.d.]:

Conjecture 1 [Conway's Baseball Conjecture]. No two definitions of "the correct baseball curve" will give the same answer unless their equivalence is obvious from the start.

This paper will take a slightly different direction than most past attempts to define baseball curves. Rather than focusing immediately on a particular curve, we will identify several symmetry properties that any reasonable candidate for a baseball curve must have, and call the class of curves on the sphere that have these symmetries generalized baseball curves. This class of curves not only includes just about every curve ever proposed by anyone as a potential baseball curve, but also includes other curves such as Viviani's curve and the path of the sun relative to the earth.

We're going to explore the symmetry properties of these generalized baseball curves using the concept of geodesic curvature, an idea from differential geometry which measures how much a curve on the sphere deviates locally from a great circle. We'll discuss a simple physical characterization of geodesic curvature in terms of planar ribbons which we believe deserves to be more widely known. We will show that curves whose geodesic curvature functions satisfy certain symmetries are in fact generalized baseball curves as long as they close back on themselves. We are then going to focus on the particular family of curves whose geodesic curvature is of the form kc(s) = sin(cs) . We will give geometric arguments showing that, perhaps surprisingly, there are many such curves that do close back on themselves and are therefore generalized baseball curves. Even more surprisingly, we will show that there are such curves that close back on themselves after only one or two periods of the function sin(cs) . Along the way, we'll come up with what we think is a particularly nice, and (we think) previously undiscussed particular curve that fits the actual seam of a baseball surprisingly well.