Loci, 2008

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

We would like to check that the values of
`c`
that appear to have period 2 are genuine. We'd like to do this by using continuity, as we did for the
`c`
values of 3 and higher. We are especially interested in these since any candidate for the "correct" baseball curve must have period 2.

Consider what happens if we have a curve of period 2. For this to happen,
`p`_{c}(1)
must be exactly half way around
`G`_{c}
.
Because the curves all start at
(1, 0, 0)
,
this means that
`p`_{c}(1) = (−1, 0, 0)
in this case. Furthermore, because of the half-turn symmetry about
`p`_{c}(.5)
,
this means that
`p`_{c}(.5)
must lie exactly a quarter of the way around
`G`_{c}
,
which means that it must lie in the plane
`x` = 0
.
Conversely, if
`p`_{c}(.5)
lies in this plane, then
`p`_{c}(1) = (−1, 0, 0)
,
and
`p`_{c}
is therefore periodic with period 2.

So, to show that
`p`_{c}
is genuinely periodic with period 2, we just need to show that the
`x`
component of
`p`_{c}(.5)
is zero.Figure 6 shows a plot of the
`x`
component of
`p`_{c}(.5)
,
plotted over the same range of
`c`
values as above. As you can see, the
`x`
value fluctuates from at least very close to 1 to very close to
-1
;
each time it does this, it has to pass through zero, and we get a
`c`
value of period 2 each time it does so. The reader can check
that each minimum in Figure 5 corresponds to a zero in Figure 6. So, the curves that appear to be of period 2 are genuine in this case.

We can use the existence of these curves of period two to look for curves that are good approximations to the "correct" baseball curve. As already noted,
`p`_{0.99342800}
,
shown in Figure 3 looks like a good candidate, but it isn't quite right-in particular, the curve comes too close to itself as it curves back around for an actual baseball seam. But we should be able to find a corresponding period two curve starting with *any* baseball curvature function. The easiest way to start doing this is to simply change the amplitude of our curvature function to try to find a baseball curve of period two whose distance of closest self-approach matches that of a real baseball. We find numerically that the curve that does this is the curve
`B`
whose curvature is given as a function of arclength as
1.17704 sin (1.169402 `s`)
.
And how close does this curve come to matching an actual baseball seam? Surprisingly close. We know from the Generalized Ribbon Test Theorem 3) described above that if we print a copy of the planar curve with this curvature, cut out a ribbon around it, and then lay this ribbon flat on a sphere, it will trace out the path with the corresponding geodesic curvature on the sphere. The result of trying this experiment is shown in the photograph in Figure 7. As you can see, the ribbon matches the baseball seam surprisingly well.