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

9. What happens when c gets very large or very small?

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

Recall that we defined pc(1) = qc(2π / c), where qc(s) is parameterized by arclength. This means that the arclength of pc from pc(0) to pc(1) is always equal to 2π / c . Thus, as c becomes large, this arclength becomes small, so dc , the great circle distance between pc(0) and pc(1) , also becomes small. The function f(c) = 2π / dc gets very large, and so pc is of higher and higher period. Recall that the points pc(0) , pc(.5) , and pc(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 p4.94691000 , 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 11. f(c),1 ≤ c ≤ 6 .

fig11.jpg

Figure 12. p4.94691000 , 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 pc 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 pc(t) , which we know to be equal to sin(c2π / ct) , will be very close to zero. Thus, pc will start out looking like a circle of curvature zero-that is, a great circle in the xy 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(c2π / ct) = sin(2πt) . The curvature will be greatest when t = 1/4 , that is, at pc(.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 xy 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 pc(.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 pc(.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

  1. the sun's path travels around the earth many more times (365.25) in the course of one period;
  2. the highest latitude that the sun's path reaches is lower-only about 23.5° north latitude; and
  3. 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 pc(.25) as we vary c . Since pc(.25) always lies approximately on L , and the arclength of the curve up to pc(.25) is continuously increasing as c gets smaller, we expect that pc(.25) should travel approximately in a circle along L as we vary c . We can verify this by graphing pc(.25) parametrically as a function of c ; as shown in Figure 13, this is indeed what happens.

Figure 13. pc(.25) plotted as function of c , with .04 ≤ c ≤ .05 .

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

This explains several of our observations in previous sections: