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

2. Generalized Baseball Curves

So what are these mathematical properties that any reasonable baseball curve must have?

• Baseball curves are smooth. That is, they don't have any kinks where they change direction abruptly. Mathematically, this means that they can be parametrized as continuously differentiable curves, whose velocity vector is never zero. They can therefore be smoothly parametrized by arclength.
• Baseball curves are periodic. That is, they end up back where they started, closing up on themselves smoothly.

This is just the beginning, however. You may not have ever noticed it before, but baseball curves also have many symmetries-so many, in fact, that it might seem surprising that there exist any spherical curves at all with these symmetries. Notice that:

• Baseball curves have translation symmetry. This means that we can slide a copy of the curve along the surface of the sphere along a straight (great circle) path, and have it line up with another part of the original curve. Notice that if we slide it the shortest distance we can to make it line up, and then slide it again in the same way, the curve is back where it started. For this reason, we say that the curve is periodic with period 2. In Figure 1, we can see that the baseball has translation symmetry along the great circle shown taking point A to point C . This figure, like all of the spherical plots in this paper, is an interactive java applet which should allow you rotate the sphere in any way that you'd like by dragging your mouse over the graphic. These figures were created using Mathematica and LiveGraphics3D. You can also zoom in and out by dragging vertically while holding down the shift key, and you can reset the graphic by pressing the home key.

Figure 1. The seam of this baseball has translation symmetry from point A to point C , and half-turn rotation symmetry about point B .

In describing this as a translation, we are adopting an intrinsic point of view-that is, the point of view of a creature living on the surface of the the sphere. If such a creature were to drag the curve in a straight line along the surface of the sphere from point A to point C , it would line up with its original position. Extrinsically--that is, from the point of view of an external observer-this can be seen as a rotation of the sphere about an axis through the poles opposite from this great circle, but intrinsically, it also makes sense to call it a translation. We will generally take an intrinsic viewpoint whenever possible. For a more complete explanation of the difference between an intrinsic and an extrinsic viewpoint, as well as a detailed description of spherical symmetries, see [Henderson and Taimina, 2005] or [Henderson, 1998].
• Baseball curves have half-turn symmetry. There is a sequence of 4 different points on the curves about which the curves have half-turn symmetry. Each of these points is located half a period away from the previous one. For example, the baseball in Figure 1 has half-turn rotation symmetry about point B . Our intrinsic half-turn symmetry about B can also be viewed extrinsically as a half-turn symmetry about an axis passing through B and the center of the sphere.
• Baseball curves have reflection symmetry about a perpendicular. The reflection symmetry is about a perpendicular drawn to the curve at the points exactly half way between the points at which it has half turn symmetry. The baseball shown in Figure 2 has reflection symmetry about the great circle drawn perpendicular to the curve at point D . Extrinsically, we would view this as a reflection through the plane that contains this great circle.

Figure 2. The seam also has reflection symmetry about the great circle that is perpendicular to it at point D .

These are the properties that we want to generalize. To that end, we make the following definition:

Definition 2. A spherical curve S is said to be a generalized baseball curve if it can be given parametrically by a function p(s) : RS2 , where S2 is the unit sphere in R3 and the parameter s is (a non-zero constant multiple of) arclength along the curve, such that
1. p′(s) is continuous;
2. p(s) is periodic, so that there exists a number d such that p(s) = p(s + d) for all s ;
3. S has translation symmetry, so that there exists a (smallest) number t such that the curve can be translated so that p(s) is translated onto p(s + t) ;
4. S has half-turn symmetry about the points p(nt/2) , for any integer n ; and
5. S has reflection symmetry about a perpendicular to the curve drawn at any of the points p(nt/2 + t/4) .

Notice that it follows from the definition that d = mt for some number m ; we call m the period of the curve.