Loci, 2008

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

In this section, we are going to try to understand the limiting behavior of the curves
`p`_{c}
when
`c`
becomes very large or very small. First, let's look at what happens when
`c`
becomes very large.

Recall that we defined
`p`_{c}(1) = `q`_{c}(2π / `c`),
where
`q`_{c}(`s`)
is parameterized by arclength. This means that the arclength of
`p`_{c}
from
`p`_{c}(0)
to
`p`_{c}(1)
is always equal to
2π / `c`
.
Thus, as
`c`
becomes large, this arclength becomes small, so
`d`_{c}
,
the great circle distance between
`p`_{c}(0)
and
`p`_{c}(1)
,
also becomes small. The function
`f`(`c`) = 2π / `d`_{c}
gets very large, and so
`p`_{c}
is of higher and higher period. Recall that the points
`p`_{c}(0)
,
`p`_{c}(.5)
,
and
`p`_{c}(1)
all lie on a common great circle. For large values of
`c`
,
these points lie very close together. The curve, whose
curvature is oscillating between
-1
and
+ 1
,
is not able to get far from a great circle path.

Figure 11 shows a Mathematica plot of
`f`(`c`)
for
1 ≤ `c` ≤ 6
.
Figure 12 shows a picture of
`p`_{4.94691000}
,
which is the curve of period 5 that corresponds to the rightmost red dot in Figure 11. Notice that the curve is already only deviating slightly from a straight great circle path. As
`c`
increases, this effect becomes more pronounced.

**Figure 12.**
`p`_{4.94691000}
,
which is periodic with period 5.

Now, let's consider what happens as
`c`
becomes very small.

Again, the arclength of one period of the curve is going to be
2π / `c`
.
So, as
`c`
gets small, this arclength gets very large relative to the circumference of the sphere. As a result, the curvature of
`p`_{c}
changes very slowly relative to arclength, and so short segments of the curve will be very close to curves of constant curvature-that is, to circles. For small values of
`t`
,
the curvature of
`p`_{c}(`t`)
,
which we know to be equal to
sin(`c`2π / `c``t`)
,
will be very close to zero. Thus,
`p`_{c}
will start out looking like a circle of curvature zero-that is, a great circle in the
`x``y`
plane. Then, as
`t`
increases, the curvature will increase slowly, and so the curve will spiral inward slowly, so that at time
`t`
it looks like a lesser circle with curvature
sin(`c`2π / `c``t`) = sin(2π`t`)
.
The curvature will be greatest when
`t` = 1/4
,
that is, at
`p`_{c}(.25)
.
At this point the curve will look like it is traveling along a circle with geodesic curvature 1.

Where is the center of this circle? The curve started out traveling along the great circle in the
`x``y`
plane; the center of this circle is at the point
(0, 0, 1)
,
which we can think of as being the north pole. As long as
`c`
is small enough, the change in the curvature should be more or less evenly spread out around the sphere (in terms of longitude). This means that we should expect that the centers of the
circles will stay in approximately the same place as
`t`
increases. Thus, the curve around
`p`_{c}(.25)
should approximate a circle
`L`
with curvature 1 centered at the north pole. This circle is the circle in the plane
`z` = 1 / √2
,
which we can think of as the latitude circle at
45°
north latitude. To check that this circle has the correct curvature, note that its extrinsic radius is
1 / √2
;
this is the circle's extrinsic radius of curvature, so its extrinsic curvature is
√2
.
We know that the square of this curvature, 2, must equal the sum of the squares of the geodesic curvature of the circle and the normal curvature of the sphere. Since the normal curvature of the sphere is 1, the geodesic curvature of the circle must also be one, as desired.

The reader can check that the curve in Figure 9 does in fact have all of these characteristics. The curve starts out along the equator, and slowly spirals up until it reaches
`L`
at the point
`p`_{c}(.25)
.
Notice how similar this curve is to the path of the sun relative to the earth. As mentioned
above, the sun's path is (approximately) a generalized baseball curve. The main differences between the sun's path and the curve in Figure 9 are that

- the sun's path travels around the earth many more times (365.25) in the course of one period;
- the highest latitude that the sun's path reaches is lower-only about 23.5° north latitude; and
- the sun's path is (approximately) of period 4, while the curve in Figure 9 is approximately of period one.

We can also think about what happens as
`c`
gets small by considering what happens to
`p`_{c}(.25)
as we vary
`c`
.
Since
`p`_{c}(.25)
always lies approximately on
`L`
,
and the arclength of the curve up to
`p`_{c}(.25)
is continuously increasing as
`c`
gets smaller, we expect that
`p`_{c}(.25)
should travel approximately in a circle along
`L`
as we vary
`c`
.
We can verify this by graphing
`p`_{c}(.25)
parametrically as a function of
`c`
;
as shown in Figure 13, this is indeed what happens.

If
`p`_{c}(.25)
is travelling approximately along a latitude circle, then the perpendicular to the curve at
`p`_{c}(.25)
must be approximately a longitude circle. We know that
`p`_{c}(.5)
is the reflection of
`p`_{c}(1)
through this perpendicular, and that the reflection of
`E` = (1, 0, 0)
through any longitude circle must lie on the equator. So if
`p`_{c}(.25)
is travelling approximately along a latitude circle,
`p`_{c}(.5)
must be traveling approximately along the equator.

This explains several of our observations in previous sections:

- In looking at points of period 2, we noted that the
`x`-component of`p`_{c}(.5) fluctuated from close to −1 to close to 1. We now know that this occurs because`p`_{c}(.5) is approximately following the equator. In fact, it does get all the way up to 1; this happens when we have a curve of period one half, when`p`_{c}(.5) =`E`. We still don't have any reason to think that it has to get all the way down to −1 , but it gets close, because the point (−1, 0, 0) lies on the equator. - This also explains why we have curves that look like they have proper period one, but don't really have proper period one. This happens every time
`p`_{c}(.5) gets close to (−1, 0, 0) , which, for small values of`c`, happens every time it follows the equator around. - In looking at points of period one half, we noted that
cos(
`θ`_{c}) also fluctuated from close to −1 to close to 1 . Notice that if`p`_{c}(.25) really were following`L`exactly, cos(`θ`_{c}) would be ±1 at the points where the tangent great circle goes through`E`; this happens at the points (0,1 / √2,1 / √2) and (0,-1 / √2,1 / √2) . Half way in between these two points along`L`,`θ`_{c}= 90° and we get the curves of period one half.