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

Period 1 c values

Now, let's consider what curves of period 1 could look like. The curve pc is of period 1 if and only if pc(1) = pc(0) . There are actually two ways that this could happen:

Figure 8. p.64757361 , which has period one half.

Figure 9. p.110732 , which appears to have proper period one. (It doesn't really come precisely back to (1, 0, 0) , so it doesn't really have proper period one.)

We'll now show that the curves that appear to be of period one half are generally real, but that the curves that appear to be of period one are generally fakes.

First, consider a curve of period one half. Recall that pc(t) has reflection symmetry about a perpendicular at pc(.25) . Because the curve has period one half, pc(.5) = pc(0) , but pc(.5) is the reflection of pc(0) over the perpendicular at pc(.25) . This can only happen if that perpendicular goes through pc(0) . Conversely, if the perpendicular goes through pc(0) , pc(.5) will equal pc(0) , and the curve will have period one half. So, to check if a curve has period one half, we can check that the perpendicular at pc(.25) goes through pc(0) . Equivalently, we can check that the great circle connecting pc(0) and pc(.25) , which we'll call Qc , is perpendicular to the velocity vector of pc at pc(.25) . If we let θc denote the angle this vector makes with Qc , then we can plot cos(θc) , as is done in Figure 10; and we know that cos(θc) = 0 if and only if θc = 90° . As before, the graph fluctuates from at least very close to 1 to very close to −1 ; and again, each time it does this, it has to pass through zero, and we get a c value of period one half each time it does so. The reader can check that each apparent asymptote in Figure 5 corresponds to a zero in Figure 10. So, these are genuine curves of period one half. (See [Allison et al., n.d.] for an alternative derivation from a very different perspective of these necessary and sufficient conditions to have a curve of proper period one half.)

Figure 10. A plot of cos(θc) ; the zeros of this graph correspond to curves of period one half.

fig10.jpg

Next, consider a curve of proper period one. If pc has proper period one, then pc(.5) = (−1, 0, 0) . We know that pc(.5) is also the reflection over the perpendicular at pc(.25) of (1, 0, 0) . Since (1, 0, 0) and (−1, 0, 0) are antipodal, the only line that reflects one on the other is the great circle in the plane x = 0 . Thus, pc(.25) must lie on this great circle, and pc(t) must cross the great circle perpendicularly. This means that the velocity vector at pc(.25) must lie in the same direction as Qc ; which means that θc must be equal to or 180° , and cos(θc) must be equal to 1 or −1 . Thus, curves of proper period one can only happen for c values that are maxima or minima of the graph in Figure 10, and only if the graph really gets all the way up to one or down to negative one. So, so far, we don't have assurance that there really are such curves. In fact, we don't think that there are any of the curves pc that really have proper period one-there are just curves that come close to being of proper period one, like the one in Figure 9.

In each of the previous cases, the symmetries force the existence of periodic points. In this case, they don't seem to. To have a curve of proper period one, the point pc(.5) must hit (−1, 0, 0) exactly-and there is nothing to force this to happen. It would happen only if pc(.25) lay on the great circle in the plane x = 0 , and only if the velocity vector at that point was perpendicular to that great circle. There is no reason that these two conditions must be satisfied at the same time. Of course, we can intentionally construct generalized baseball curves of proper period one. But if we start with a random baseball curvature function f(s) and look at the family of curves with geodesic curvature given by f(cs) there is no reason why there should be curves of proper period one in this family, in general.

We might wonder, though, why then are there curves that come so close to being of proper period one in all of these families, like the curve shown in Figure 9? In order to understand this, we need to look more closely at what happens when c gets very large or very small.