Loci, 2008
Generalized Baseball Curves: Three Symmetries and You're In!
Allison, Diaz, and Miller
We would now like to understand what the geodesic curvature function of a generalized baseball curve looks like. From this point onward, we will be working on the surface of the sphere, so when we refer to a curvature function of a curve, we will mean its geodesic curvature. Each of the symmetries of a generalized baseball curve gives rise to a corresponding symmetry of its curvature function. Thus, just as the curve has three symmetries, so must its curvature function k(s) .
For example, because any generalized baseball curve has translation symmetry taking p(s) to p(s + t) , the curve must bend in the same way at these two points, and so its curvature k(s) at these two points must be the same. Thus, k(s) = k(s + t) , which means that k is periodic and has translation symmetry itself.
Likewise, the curve has half-turn symmetry about each point p(nt/2) taking p(nt/2 + a) to p(nt/2 − a) , where n is any integer. So, again, the curve is the same at these two points, except curving in the opposite direction: if it curves to the left at p(nt/2 + a) , then it curves to the right at p(nt/2 − a) . This means that k(nt/2 + a) = −k(nt/2 − a) . In other words, k is an odd function about the point nt/2 , and therefore has half-turn symmetry about this point.
The final symmetry that the curve has is reflection symmetry about a perpendicular to the curve drawn at the point p(nt/2 + t/4) , which takes the point p(nt/2 + t/4 + a) to the point p(nt/2 + t/4 − a) . Thus, k(nt/2 + t/4 + a) = k(nt/2 + t/4 − a) , making k an even function about the point nt/2 + t/4 , and so it also has reflection symmetry about this point.
This leads us to make the following definition:
Definition 5. A baseball curvature function is a function k : R → R such that
It follows from the above discussion that the curvature function of any generalized baseball curve must be a baseball curvature function. On the other hand, according to Theorem 4, if we start with a baseball curvature function k , then there must be a spherical curve whose curvature function is k . This curve may not be a generalized baseball curve though-it may not even close up on itself. In general, there's no reason to think that it will. However, if it does close back on itself, then it turns out that it will be a generalized baseball curve, as we will show below.
As a concrete example, we will consider the family of curvature functions k_{c}(s) = sin(cs) . The reader can check that these are, indeed, baseball curvature functions. We will look at curves with these curvature functions on the unit sphere that are parametrized by arclength, that start at the point E = (1, 0, 0) , and that initially travel parallel to the y -axis, so that the velocity vector at time t = 0 is (0,1,0) . We will let q_{c}(s) denote the curve of this form whose geodesic curvature at time s is k_{c}(s) . Notice that q_{c}(s) has run though one full period of its periodic curvature after arclength 2π / c ; with this in mind, we define p_{c}(t) = q_{c}(t 2π / c) . Thus, p_{c}(t) and q_{c}(t) trace out the same curve, but p_{c}(t) is reparameterized so that p_{c} has run through one full period of its periodic curvature at time t = 1 .
Figure 3 shows one such curve, p_{0.99342800} . This curve certainly appears to be a generalized baseball curve that is periodic with period 2.In this figure, as in subsequent figures, the curve is plotted with a thick dark red line; the points p_{c}(n) , where n is any integer, are plotted as black dots; and the points p_{c}(n + .5) are plotted as magenta dots. The curve was generated by using Mathematica to numerically solve the spherical Frenet equations for the specified curvature function.
Now, let's check that this curve, like any generated from a baseball curvature function, has the symmetries of a generalized baseball curve.
First of all, notice that the curve has intrinsic half-turn rotational symmetry about the point p_{c}(.5) . (This is the first magenta dot.) To see why this must be the case for any p_{c}(t) , imagine two bugs, standing back to back on the point p_{c}(.5) . If they walk away from each other along the curve at a constant speed, their bodies will turn in the same way at the same time. This is because sin(π + a) = −sin(π− a) (i.e., the sine function is odd at this point), and they are facing in opposite directions. This means that p_{c}(t) must have half turn symmetry about the point p_{c}(.5) . By similar reasoning, any curve with a baseball curvature function has half-turn symmetry.
Next, consider what happens if we draw the great circle G_{c} that connects p_{c}(0) and p_{c}(.5) . (This is the thin green circle in Figure 3.) Great circles have half-turn symmetry about any of their points, so G_{c} also has half-turn symmetry about p_{c}(.5) . Thus, since p_{c}(0) lies on G_{c} , its image under the rotation must also lie on G_{c} ; this image is p_{c}(1) . So p_{c}(0) , p_{c}(.5) , and p_{c}(1) all lie on a common great circle. Now notice that since p_{c} and G_{c} both have half-turn rotation symmetry about p_{c}(.5) , the angle between them must be the same at p_{c}(0) and p_{c}(1) . Furthermore, since the curvature of p_{c} is the same at p_{c}(t) and p_{c}(t + 1) , p_{c} looks the same between t = 1 and t = 2 as it does between t = 0 and t = 1 . It follows from these two facts that p_{c} also has intrinsic translation symmetry along the great circle vector 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} . Again, this argument shows that any curve with a baseball curvature function has translation symmetry.
Finally, we'd like to show that these curves have intrinsic reflection symmetry across a line (great circle) drawn perpendicular to the curve at the point p_{c}(.25) . To see that they do, imagine our two bugs again, this time standing back to back at the point p_{c}(.25) . This time, as they again walk away from each other along the curve at a constant speed, they again turn the same amount at the same time, but now in opposite directions. As one bug turns left, the other turns right. This is because sin[c(2π / 4c − t)] = sin[c( 2π / 4c + t)] (sine is an even function at these points), and they are facing in opposite directions. This means that p_{c}(t) must have reflection symmetry about a perpendicular at p_{c}(.25) . Thus, p_{c} has inherited all of the symmetries of the sine function-half turn symmetry about its midpoint, translation symmetry of one period, and reflection symmetry about a perpendicular through the point one quarter of a period in. Again, this argument holds for any curve with a baseball curvature function.
Putting together all of our above observations, we have the following theorem:
Theorem 6. A spherical curve is a generalized baseball curve if and only if it is periodic and its curvature function is a baseball curvature function.
Corollary 7. The curve p_{c} , whose curvature function is given by k_{c}(s) = sin(cs) , is a generalized baseball curve for exactly those values of c for which it is periodic.
In this section, we have given geometric arguments for these results; however, in the companion paper [Allison et al., n.d.], we show how to derive the same results from an analytical point of view, using matrices and differential equations.