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

## 6. When is pc periodic?

We will now look more carefully at the family of functions pc . We would like to know for which values of c they are generalized baseball functions. By Corollary 7, we know that this happens exactly when pc is periodic. So, we would like to know: for which values of c is pc periodic?

Figure 4. Some points (c,n) , where pc is periodic of order n .

It isn't immediately clear why there should be any values of c for which the curve pc is periodic. The curve shown in Figure 3 definitely appears to be periodic, however. Furthermore, we can numerically integrate the spherical Frenet equations to plot pc for many different values of c , and, by trial and error, we can find many different values of c for which it looks like pc is periodic of period n . We can then plot the points (c,n) to look for patterns. Such a plot, with .15 < c < 1 , is shown in Figure 4.

Looking at Figure 4, there appears to be some kind of pattern. We'd like to understand what it is, and why there must be values of c for which pc is periodic.

Recall from the previous section that pc has intrinsic translation symmetry along the great circle from pc(0) to pc(1) . This means that, given any value of c , the points pc(n) and the points pc(n + .5) all lie along the same great circle, Gc .

This gives us a really nice way to tell if pc is periodic, at least numerically. Given a value for c , we can compute pc(1) by integrating the spherical Frenet equations through one period. Once we know where pc(1) is, we can find Gc , and the distance dc between pc(0) and pc(1) along Gc . If dc is a rational multiple n / m of (the length of Gc ), where n / m is in lowest terms, then pc(m) = pc(0) , so pc is periodic, and is of period m .

Thus, if we graph f(c) := 2π/dc , we will get a function with the following properties:

1. if f(c) is equal to an integer m , then pc is periodic with period m ;
2. if f(c) = m / n is rational and written in lowest terms, then pc is periodic with period m ;
3. if f(c) has an asymptote at c , then pc is periodic with period 1;
4. if f(c) is irrational, then pc is not periodic.

Thus, the points we found before in Figure 4 should be related to points on the graph of f(c) These are superimposed on one another in Figure 5; as you can see, they match exactly as they are supposed to. There are red dots that lie on the graph of f when f takes on an integer value; red dots at 1 when f has an asymptote; and when there are red dots above the graph of f at a value n , then f takes on a rational value with a denominator of n in lowest terms.

Figure 5. f(c),.15 ≤ c ≤ 1 .

Also notice that since dc is a continuous function of c , f(c) will be continuous except when dc is zero, which means that it really must hit rational values. The continuity of dc as a function of c follows from the well-known fact that solutions of differential equations X′ = F(t,X) vary continuously as functions of the data F(t,X) ; for proof of this fact, see Hirsch et al. [2004], page 399. It follows that the values of c that we found numerically to be of period 3 or higher must be approximations of c values that really are periodic. It isn't clear yet if there are really values of c that have periods 2 or 1, though-period 2 c values would correspond to minima on our graph, which might not really get down as far as 2, and period 1 c values correspond to asymptotes, which might not be real asymptotes. So we need to look at these more closely.