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 .

fig4.jpg

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.

If you would like to play around with these curves a bit yourself to get a feel for how the curves change as we vary c , you can download a Mathematica notebook written for this purpose. You need to have your own working copy of Mathematica to fully use it. However, even if you don't, you can still read the notebook and see how our curves change as we vary c using the free Mathematica Player software.Depending on the settings of your web browser, the Mathematica notebook may download as a text file, in which case you should save it on your computer and reopen it with Mathematica or Mathematica Player.

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 .

fig5.jpg

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.