Loci, 2008

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

In this article we have defined generalized baseball curves to be periodic, closed curves on the surface of a sphere that have certain symmetry properties; namely, translation symmetry along the curve, half-turn symmetry about the points midway through one cycle, and reflection symmetry about a perpendicular at points one-quarter of the way through the cycle. Next, we defined baseball curvature functions to be those periodic functions that satisfy two additional symmetry properties: they must be odd functions about the points midway through one period, and even functions about points one-quarter of the way through the period. We then showed geometrically that any spherical curve whose geodesic curvature function is a baseball curvature function is itself a generalized baseball curve as long as it periodic. This brought us to the question of when a curve whose curvature function is a baseball curvature function will be periodic. We focused most of our discussion on the representative case of curves
`p`_{c}
with curvature
`k`(`s`) = sin(`c``s`)
,
and showed that in this parametrized family of curves there exist values of
`c`
for which the curve is periodic of period
`n`
for
`n` ≥ 2
.
For curves of period 1 the situation was more complicated. Curves that come back to their starting point after half a period, curves of *period one half*, were shown to exist. Furthermore, curves of period one that come back to their starting point after exactly one period, curves of *proper period one*, also appeared to exist upon first numerical evidence. However, we conjectured that these curves only come close to closing up and are not true closed periodic curves, although we did not offer a proof of this conjecture.

Although we have focused throughout this paper on the family of curves
`p`_{c}
whose geodesic curvature is given by
sin(`c``s`)
,
we could have started with another baseball curvature function
`f`(`x`)
and looked at the one-parameter family of curvature functions
`f`_{c}(`x`)
.
Each such family will give rise to many different generalized baseball curves, and every generalized baseball curve is in one of these families.

It is our hope that the ideas contained in this article will have sparked your interest in further exploring this subject. There are many possible areas to explore, but here are some of our ideas about interesting places to start:

- A fun computer activity is to explore the spherical curves that can be generated by choosing a geodesic curvature function. We have provided Mathematica code for this purpose that you can download. Choosing
`k`(`s`) = sin(`c``s`) for various values of`c`is a good place to start before trying other families of periodic functions. However, you should be able to find generalized baseball curves starting with any baseball curvature function that strikes your fancy. - There is a lot of material that you can look at on the web related to baseball seams and particular generalized baseball curves; we have collected some of these materials.
- We chose to concentrate on curves with all of the symmetries of a baseball seam, but another area for further exploration would be to look at curvature functions that have fewer symmetries than baseball curvature functions. For example, what changes if you only require your curvature function to have half turn symmetry? Or just to be periodic? This question is discussed at greater length in the companion article [Allison et al., n.d.], which explores many of the same ideas as this article, but from a very different, analytic viewpoint.
- A good exercise in elementary differential geometry is to verify that the closed form expressions for baseball curves that have been offered in the literature and which are referenced in the section on Examples of Generalized Baseball Curves are indeed generalized baseball curves. In other words, compute the curvature functions for these examples and check whether they have the required symmetries. Once you've computed their curvature functions, you can then vary them just as we varied the values of
`c`in the function sin(`c``s`) to obtain whole new families of generalized baseball curves. - Here is an exercise in modeling. As mentioned above, the curve given by the path of the sun in the sky is a generalized baseball curve. Verify this by finding the parametric equations of this curve relative to a spinning geocentric coordinate system.
- A fun activity is to print a planar curve with a specified geodesic curvature, cut out a ribbon around the curve, and then lay the ribbon flat on an appropriate sphere, such as a baseball. This activity gives a nice illustration of the Generalized Ribbon Test (Theorem 3). Our experiments found that a geodesic curvature of
`k`(`s`) = 1.17704 sin(1.169402`s`) came very close to matching that of a real baseball. You can use our Mathematica code to produce and print your own ribbon, although you may have to scale the printout so that it comes out the right size on your printer; you need to scale it so that the baseball that you are putting it on is a unit sphere according to your printed scale. - Similarly, you could try to come up with a ribbon that will fit on the curve of a tennis ball. Tennis ball curves are slightly different from baseball seams, although they have the same symmetries and are therefore generalized baseball curves.
- Another related activity is to roll a real baseball on an ink pad so as to color the seam, and then roll the baseball on a sheet of white paper without slipping or spinning about the point of tangency. This activity, the inverse of the activity described in the previous two items, will produce a planar curve whose plane curvature is the geodesic curvature of the baseball seam.

Have fun!