Loci, 2008
Generalized Baseball Curves: Three Symmetries and You're In!
Allison, Diaz, and Miller
Now, let's consider what curves of period 1 could look like. The curve p_{c} is of period 1 if and only if p_{c}(1) = p_{c}(0) . There are actually two ways that this could happen:
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 p_{c}(t) has reflection symmetry about a perpendicular at p_{c}(.25) . Because the curve has period one half, p_{c}(.5) = p_{c}(0) , but p_{c}(.5) is the reflection of p_{c}(0) over the perpendicular at p_{c}(.25) . This can only happen if that perpendicular goes through p_{c}(0) . Conversely, if the perpendicular goes through p_{c}(0) , p_{c}(.5) will equal p_{c}(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 p_{c}(.25) goes through p_{c}(0) . Equivalently, we can check that the great circle connecting p_{c}(0) and p_{c}(.25) , which we'll call Q_{c} , is perpendicular to the velocity vector of p_{c} at p_{c}(.25) . If we let θ_{c} denote the angle this vector makes with Q_{c} , then we can plot cos(θ_{c}) , as is done in Figure 10; and we know that cos(θ_{c}) = 0 if and only if θ_{c} = 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.)
Next, consider a curve of proper period one. If p_{c} has proper period one, then p_{c}(.5) = (−1, 0, 0) . We know that p_{c}(.5) is also the reflection over the perpendicular at p_{c}(.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, p_{c}(.25) must lie on this great circle, and p_{c}(t) must cross the great circle perpendicularly. This means that the velocity vector at p_{c}(.25) must lie in the same direction as Q_{c} ; which means that θ_{c} must be equal to 0° or 180° , and cos(θ_{c}) 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 p_{c} 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 p_{c}(.5) must hit (−1, 0, 0) exactly-and there is nothing to force this to happen. It would happen only if p_{c}(.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.