Loci, 2008

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

We will now look more carefully at the family of functions
`p`_{c}
.
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
`p`_{c}
is periodic. So, we would like to know: for which values of
`c`
is
`p`_{c}
periodic?

It isn't immediately clear why there should be *any* values of
`c`
for which the curve
`p`_{c}
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
`p`_{c}
for many different values of
`c`
,
and, by trial and error, we can find many different values of
`c`
for which it looks like
`p`_{c}
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
`p`_{c}
is periodic.

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

This gives us a really nice way to tell if
`p`_{c}
is periodic, at least numerically. Given a value for
`c`
,
we can compute
`p`_{c}(1)
by integrating the spherical Frenet equations through one period. Once we know where
`p`_{c}(1)
is, we can find
`G`_{c}
,
and the distance
`d`_{c}
between
`p`_{c}(0)
and
`p`_{c}(1)
along
`G`_{c}
.
If
`d`_{c}
is a rational multiple
`n` / `m`
of
2π
(the length of
`G`_{c}
), where
`n` / `m`
is in lowest terms, then
`p`_{c}(`m`) = `p`_{c}(0)
,
so
`p`_{c}
is periodic, and is of period
`m`
.

Thus, if we graph
`f`(`c`) := 2π/`d`_{c}
,
we will get a function with the following properties:

- if
`f`(`c`) is equal to an integer`m`, then`p`_{c}is periodic with period`m`; - if
`f`(`c`) =`m`/`n`is rational and written in lowest terms, then`p`_{c}is periodic with period`m`; - if
`f`(`c`) has an asymptote at`c`, then`p`_{c}is periodic with period 1; - if
`f`(`c`) is irrational, then`p`_{c}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.

Also notice that since
`d`_{c}
is a continuous function of
`c`
,
`f`(`c`)
will be continuous except when
`d`_{c}
is zero, which means that it
really must hit rational values. The continuity of
`d`_{c}
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.