September 2008. Article ID 2866
Consider the curve traced out on the surface of a baseball by its seam. Many different authors have tried to give mathematical equations describing this curve in many different ways. There have been so many different descriptions of this curve, in fact, that John Conway proposed the following conjecture [Weisstein, n.d.]:
Conjecture 1 [Conway's Baseball Conjecture]. No two definitions of "the correct baseball curve" will give the same answer unless their equivalence is obvious from the start.
This paper will take a slightly different direction than most past attempts to define baseball curves. Rather than focusing immediately on a particular curve, we will identify several symmetry properties that any reasonable candidate for a baseball curve must have, and call the class of curves on the sphere that have these symmetries generalized baseball curves. This class of curves not only includes just about every curve ever proposed by anyone as a potential baseball curve, but also includes other curves such as Viviani's curve and the path of the sun relative to the earth.
We're going to explore the symmetry properties of these generalized baseball curves using the concept of geodesic curvature, an idea from differential geometry which measures how much a curve on the sphere deviates locally from a great circle. We'll discuss a simple physical characterization of geodesic curvature in terms of planar ribbons which we believe deserves to be more widely known. We will show that curves whose geodesic curvature functions satisfy certain symmetries are in fact generalized baseball curves as long as they close back on themselves. We are then going to focus on the particular family of curves whose geodesic curvature is of the form k_{c}(s) = sin(cs) . We will give geometric arguments showing that, perhaps surprisingly, there are many such curves that do close back on themselves and are therefore generalized baseball curves. Even more surprisingly, we will show that there are such curves that close back on themselves after only one or two periods of the function sin(cs) . Along the way, we'll come up with what we think is a particularly nice, and (we think) previously undiscussed particular curve that fits the actual seam of a baseball surprisingly well.
So what are these mathematical properties that any reasonable baseball curve must have?
To start with, notice that:
This is just the beginning, however. You may not have ever noticed it before, but baseball curves also have many symmetries-so many, in fact, that it might seem surprising that there exist any spherical curves at all with these symmetries. Notice that:
These are the properties that we want to generalize. To that end, we make the following definition:
Notice that it follows from the definition that d = mt for some number m ; we call m the period of the curve.
While this is the first time that this definition of a generalized baseball curve has been given, many previously studied spherical curves are in fact generalized baseball curves.
Unsurprisingly, most of the many published attempts to define the particular curve that determines the seam of a baseball satisfy our definition of a generalized baseball curve. Some authors have given closed form expressions for such curves. One of these, for example, is the parametrized curve of Gray [1998], page 927, defined by
x(t) = acos(π / 2 − c) cos(t) cos[t / 2 + csin(2t)]
y(y) = acos(π / 2 − c) cos(t) sin[t / 2 + csin(2t)]
z(t) = asin(π / 2 − c) cos(t),
where a is the radius of the baseball and b and c are constants. Another example is that of López-López, who raises the question of whether the seam of a baseball has the shape it does due to some physical minimization principle. In López-López [1996], he suggests that the seam is defined by the parametric equations
x(t) = a sin(t) + b sin(3t)
y(t) = a cos(t) − b cos(3t)
z(t) = √(4ab)
where a + b is the radius of the baseball. These equations are the same as those derived by von Seggern [1994]; but they give a different curve from the one defined by Gray's equations.
A third example of a possible baseball seam curve is found in an article by Thompson [1998] about designing the cover for a baseball. In this article, he finds an explicit parametrization of the seam of a baseball that depends on some reasonable choice of a function that is basically the projection of the seam into a suitable plane. His method allows a designer of a baseball cover to choose the function that creates the baseball seam that best satisfies whatever additional criteria the manufacturer and players desire.
Our definition of generalized baseball curves also includes many examples of spherical curves that are not candidates for the seam of a baseball but do satisfy the required symmetry properties. One such curve of historical interest is Viviani's curve. In 1692, Vincenzo Viviani, a disciple of Galileo, asked whether it is possible for a hemisphere to have four windows of equal size such that the remaining surface area can be exactly squared [Gray, 1998, page 201]. One solution to this problem involves Viviani's curve, the curve that lies on the intersection between a sphere and a right circular cylinder passing through the center of the sphere whose diameter is half that of the sphere.
Another family of curves that includes many generalized baseball curves is the family of Seiffert's spherical spirals. Seiffert's spherical spirals are the curves produced when one moves on the surface of a sphere with constant speed maintaining a constant angular velocity with respect to a fixed diameter. In general, these curves will not be closed, but Erdös [2000] obtained conditions on the parameters under which these curves would be closed. Many of the closed curves in this family are generalized baseball curves.
Our final example of a generalized baseball curve is a physical one. Consider the point on the earth where the sun is directly overhead. There is always one such point on the earth, the point closest to the sun. This point circles the earth once a day. At the equinoxes, the point is on the equator; at the June solstice, the point is at its farthest point north of the equator, and at the December solstice, at its furthest point south of he equator. This point circles the earth about 365.25 times in one yearly period. In this example, the symmetries are approximate since the earth's orbit is not a perfect circle, the year isn't exactly 365.25 days long, and there are various complicating wobbles. However, in the idealized case without these complications, the path of the sun in the sky would be a generalized baseball curve of period 4. The path that the sun traces as it moves northward between the spring equinox and the summer solstice is a mirror image of the path it traces as it moves back to the equator between the summer solstice and the fall equinox. The path that it then traces during the other half of the year is the same, but in the southern hemisphere, so that its path has half turn symmetry about the point on the equator hit at the fall equinox. Thus, after one year starting at the spring equinox, the sun will have moved through one period of a function with appropriate symmetries to be a generalized baseball curve. If we assume that the year is exactly 365.25 days long, the sun will have ended up exactly one quarter of the way around the equator from its starting point, and the curve will repeat itself exactly after four years have passed.
We'd now like to explore the properties of generalized baseball curves, but we first need to develop some basic ideas from differential geometry. In particular, we will need a basic understanding of the geodesic curvature of a curve on a surface. Because this isn't the main focus of our article, we will give just enough detail for the following discussion to make sense; the interested reader can find more details in any standard differential geometry reference, such as Millman and Parker [1977], Gray [1998], or Henderson [1998]. Brief summaries of some of these ideas can also be found at Wikipedia and MathWorld.
We will assume that any curve α : I → R^{3} that we discuss is at least C^{3} and regular; that is, that dα / ds ≠ 0 for any s in I . If P(s) is a curve in three-dimensional space (or more generally, in R^{n} ) that is parameterized by arclength s , then we can define the unit tangent vector at any point along the curve by T(s) = dP / ds . This derivative gives a unit vector because the curve is parameterized by arclength, which means that we are essentially moving along the curve with constant unit speed. We can then define the curvature vector κ(s) at a given point along the curve to be the derivative of this unit tangent vector; that is, κ(s) = dT / ds .
What does the curvature of a curve at a point tell us? Because it is the derivative of the unit tangent vector, it is a measure of how fast the curve is turning at the given point.
However, for a curve winding and twisting around in R^{3} , curvature is not sufficient to characterize all of the turning that a curve can do. Another quantity, the torsion, measures the tendency of the curve to twist out of the plane that the curve comes closest to being in near the point at which we are measuring the curving. A central result of differential geometry is the Fundamental Theorem of Space Curves, which essentially says that, up to position in space, a regular curve with nonzero κ is completely determined by its curvature κ and torsion.
Curvature and torsion provide a nice way to talk about how much a curve is turning in three-dimensional space. However, in order to study curves that are constrained to stay on a two-dimensional surface like a sphere, it will be useful to refine these ideas. Consider, for example, a great circle on a sphere. Extrinsically, the great circle has a constant non-zero curvature. However, intrinsically the great circle looks like a straight line on the sphere: it has all of the intrinsic symmetries of a straight line, and a two-dimensional creature that walked along the curve would move the right-hand and left-hand parts of its body the same amount. In other words, all of the turning in the curve comes from the curving of the surface, rather than from the turning of the creature's body. Thus, intrinsically, we would like to say that the curvature of this curve should be zero.
We can capture this notion by defining the intrinsic or geodesic curvature vector at a point P of a curve that lies in a orientable surface S to be the projection of the curvature vector of the curve at P onto the plane tangent to S at P . We will sometimes also call the curvature vector of a curve the extrinsic curvature vector, to distinguish it from the geodesic curvature vector. We can furthermore define a scalar curvature k(s) that gives the length of the geodesic curvature vector. It is conventional to simply refer to the scalar geodesic curvature as the geodesic curvature. In the rest of this paper we will follow that convention and use k(s) for the (scalar) geodesic curvature. We assign the geodesic curvature of the curve at a point a positive or negative sign depending on whether the geodesic curvature vector lies to the right or the left of the curve within the surface relative to its orientation as determined by a choice of a normal vector to the surface.
As an example, again consider the case of a great circle on a sphere. Its extrinsic curvature vector always points in towards the center of the sphere; so at any point on the great circle, the extrinsic curvature vector is orthogonal to the tangent plane. Thus, the geodesic curvature of a great circle is always zero, as we'd like it to be. It should be pointed out that it isn't obvious from the definition that we have given of geodesic curvature that it should be an intrinsic quantity, since it relies on the definition of the extrinsic curvature of a curve, and this will change depending on how the surface in which the curve lies is embedded in R^{3} . However, a standard result of differential geometry shows that the geodesic curvature of a curve on a surface is independent of how the surface is embedded in the larger space, and thus the geodesic curvature is, in fact, an intrinsic quantity. For proof, see Millman and Parker [1977], p.106.
One nice way of understanding geodesic curvature is through the following result, which generalizes the Ribbon Test defined by Henderson [1998], Problem 3.4.
Theorem 3 [The Generalized Ribbon Test]. Let s(t) be a curve on some orientable surface, and let r(t) be a planar curve such that it is possible to cut out a ribbon around r(t) in the plane and lay it flat on the surface so that the moved image of each point r(t) lies on s(t) . (If we denote this moved image by m(r(t)) , we're saying that m(r(t)) = s(t) . By lying flat, we mean that the ribbon is tangent to the surface at each point along the curve.) Then the geodesic curvature of s at the point s(t_{0}) is the same as the (signed) extrinsic curvature of r(t_{0}) in the original plane.
Proof. Since the surface and the ribbon are tangent to one another at the point s(t_{0}) , they have the same tangent plane. Thus, since s(t) = m(r(t)) and the curve can be considered to lie in either the surface or the ribbon, the curve has the same geodesic curvature whether it is measured as a curve on the surface or the ribbon. Since the geodesic curvature of a curve on a surface is independent of how that surface sits in three-dimensional space, the geodesic curvature of m(r(t_{0})) in the curved ribbon is the same as the geodesic curvature of r(t_{0}) in the plane. And since the plane is its own tangent plane, the geodesic curvature of r at r(t_{0}) is the same as its extrinsic curvature. Thus, the geodesic curvature of s at s(t_{0}) is the same as the extrinsic curvature of r at r(t_{0}) .
This result gives us a nice way to understand geodesic curvature-it says that the geodesic curvature of a curve C on a surface is the same as the extrinsic curvature of a curve in the plane that can be cut out and laid along C .
There are some curves in some surfaces along which it is impossible to lay a ribbon; but Henderson [1998], Problem 7.6 shows that it is always possible as long as the normal curvature of the curve is never zero. Since a standard result from differential geometry says that the normal curvature of any curve on the sphere can never be zero, a ribbon can always be laid along any spherical curve.
A third way of thinking about geodesic curvature on the sphere is in terms of rolling. If we have a curve on the sphere, we can imagine laying a ribbon along it, and putting glue on the outside of the ribbon. Then, if we roll the sphere on a plane along the curve, the ribbon will stick to the plane and will trace out the path of the sphere on the plane. Thus, we can also think of the geodesic curvature of a curve on the sphere as being the extrinsic curvature of the path that the sphere takes when rolled on a plane along the curve.
Now, if we have a planar curve and we know its starting point and direction and its curvature (as a function of arclength), then we can reconstruct the whole curve, essentially by integrating the curvature function. That is to say, the curvature function of a curve determines that curve uniquely once we pick a starting point and direction for the curve. Likewise, if we know the starting point and direction of a spherical curve and we also know its geodesic curvature function, then we can again reconstruct the curve. To be precise, we can state the following:
Theorem 4 [The Fundamental Theorem of Spherical Curves]. Let (a,b) be an interval about 0 , k a C^{1} function on (a,b) , x_{0} a fixed point of S^{2} , and V a fixed unit-length vector tangent to S^{2} at the point x_{0} . Then there exists a unique C^{1} regular curve α:(a,b) → S^{2} such that α(0) = x_{0} , T(0) = V , and the geodesic curvature of α is given by k .
To prove this theorem, we would derive the following system of differential equations known as the Spherical Frenet Equations, which are given by
P ′ = 0P + 1T + 0N
T ′ = −1P + 0 T + k(s)N
N ′ = 0P − k(s)T + 0N
where P(s) is the position vector of the unit speed parametrized curve, T = dP / ds is the unit tangent vector, N = P × T is the unit normal vector to the curve, and k(s) is the geodesic curvature of P(s) . (If you are interested, here is a derivation of these equations.)
Once we know that every smooth spherical curve satisfies these equations, Theorem 4 follows almost immediately, using the existence and uniqueness theory for systems of linear differential equations.
One nice way to actually construct this desired spherical curve, however, assuming that we know how to construct the planar curve with the given curvature function, is to cut out a ribbon in the plane around this planar curve, and to then lay the ribbon on the sphere at the prescribed starting point, heading in the prescribed direction. If we have a curve that is a candidate for the seam of a baseball and we know its geodesic curvature function, we can use this method to check how close the candidate comes to fitting an actual baseball by printing out the planar curve with the same geodesic curvature, cutting it out, and seeing how closely it fits the baseball's seam when laid flat on the baseball.
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.
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:
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.
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.
Now, let's consider what curves of period 1 could look like. The curve p_{c} is of period 1 if and only if p_{c}(1) = p_{c}(0) . There are actually two ways that this could happen:
We'll now show that the curves that appear to be of period one half are generally real, but that the curves that appear to be of period one are generally fakes.
First, consider a curve of period one half. Recall that p_{c}(t) has reflection symmetry about a perpendicular at p_{c}(.25) . Because the curve has period one half, p_{c}(.5) = p_{c}(0) , but p_{c}(.5) is the reflection of p_{c}(0) over the perpendicular at p_{c}(.25) . This can only happen if that perpendicular goes through p_{c}(0) . Conversely, if the perpendicular goes through p_{c}(0) , p_{c}(.5) will equal p_{c}(0) , and the curve will have period one half. So, to check if a curve has period one half, we can check that the perpendicular at p_{c}(.25) goes through p_{c}(0) . Equivalently, we can check that the great circle connecting p_{c}(0) and p_{c}(.25) , which we'll call Q_{c} , is perpendicular to the velocity vector of p_{c} at p_{c}(.25) . If we let θ_{c} denote the angle this vector makes with Q_{c} , then we can plot cos(θ_{c}) , as is done in Figure 10; and we know that cos(θ_{c}) = 0 if and only if θ_{c} = 90° . As before, the graph fluctuates from at least very close to 1 to very close to −1 ; and again, each time it does this, it has to pass through zero, and we get a c value of period one half each time it does so. The reader can check that each apparent asymptote in Figure 5 corresponds to a zero in Figure 10. So, these are genuine curves of period one half. (See [Allison et al., n.d.] for an alternative derivation from a very different perspective of these necessary and sufficient conditions to have a curve of proper period one half.)
Next, consider a curve of proper period one. If p_{c} has proper period one, then p_{c}(.5) = (−1, 0, 0) . We know that p_{c}(.5) is also the reflection over the perpendicular at p_{c}(.25) of (1, 0, 0) . Since (1, 0, 0) and (−1, 0, 0) are antipodal, the only line that reflects one on the other is the great circle in the plane x = 0 . Thus, p_{c}(.25) must lie on this great circle, and p_{c}(t) must cross the great circle perpendicularly. This means that the velocity vector at p_{c}(.25) must lie in the same direction as Q_{c} ; which means that θ_{c} must be equal to 0° or 180° , and cos(θ_{c}) must be equal to 1 or −1 . Thus, curves of proper period one can only happen for c values that are maxima or minima of the graph in Figure 10, and only if the graph really gets all the way up to one or down to negative one. So, so far, we don't have assurance that there really are such curves. In fact, we don't think that there are any of the curves p_{c} that really have proper period one-there are just curves that come close to being of proper period one, like the one in Figure 9.
In each of the previous cases, the symmetries force the existence of periodic points. In this case, they don't seem to. To have a curve of proper period one, the point p_{c}(.5) must hit (−1, 0, 0) exactly-and there is nothing to force this to happen. It would happen only if p_{c}(.25) lay on the great circle in the plane x = 0 , and only if the velocity vector at that point was perpendicular to that great circle. There is no reason that these two conditions must be satisfied at the same time. Of course, we can intentionally construct generalized baseball curves of proper period one. But if we start with a random baseball curvature function f(s) and look at the family of curves with geodesic curvature given by f(cs) there is no reason why there should be curves of proper period one in this family, in general.
We might wonder, though, why then are there curves that come so close to being of proper period one in all of these families, like the curve shown in Figure 9? In order to understand this, we need to look more closely at 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 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(c2π / ct) , 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 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 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 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 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
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:
We can also use our observations to explain the shape of the function f(c) as c becomes small. We know that p_{c}(.5) is roughly traveling around the equator as c changes, that it narrowly misses the point (−1, 0, 0) each time it goes around, and that it hits the opposite point E each time it goes around. From our half-turn symmetry, this means that the point p_{c}(1) is also roughly traveling around the equator, but twice as fast. Every other time it passes the point E it hits it; but the other half of the time it (just barely) misses it. Thus, d_{c} fluctuates between 0 and π ; every other time it gets close to zero it turns around before hitting it. f(c) has an asymptote whenever d_{c} is zero, and has a fake asymptote where it gets really big when d_{c} comes close to zero. So fake asymptotes alternate with real asymptotes, and in between, the graph comes down to an attained minimum at 2.
This pattern can be clearly seen in Figure 5 and Figure 11. The first "fake asymptote" is hardly more than a bump and occurs when c is roughly 1.3 in Figure 11. It is followed by a real asymptote at c ≈ .64757361 , which gives rise to the curve of period one half shown in Figure 8. In between them is a minimum at c ≈ 0.99342800 which gives rise to the curve of period 2 shown in Figure 3. The next "fake asymptote" is at c ≈ .425 , and it is followed by a real asymptote at c ≈ 0.30840100 . This pattern continues, with each fake asymptote getting higher as c decreases, as our argument predicts. Figure 14 shows f(c) graphed on the interval .1 ≤ .15 , along with a magnified version of cos(θ_{c}) . By this point the fake asymptotes have gotten so large that we can't distinguish them from the real asymptotes on the graph, but we can still pick them out because they correspond to the maxima and minima of the graph of cos(θ_{c}) ; the real asymptotes correspond to the zeros. The curve shown in Figure 9, which appeared to have proper period one, corresponds to the fake asymptote shown in Figure 14 at c = .110732 .
Next, let's look at the spacing between the asymptotes. It appears that as c gets smaller, the asymptotes get closer and closer together. Why should this be? In order to explain this, we need to investigate how fast p_{c}(.5) is travelling around the equator. Recall that the arclength of p_{c} from p_{c}(0) to p_{c}(.5) is s(c) = π/c . Therefore, the rate of change of the arclength of our curve up to p_{c}(.5) with respect to c is −π / c^{2} . It would make sense for p_{c}(.5) 's velocity with respect to c to be roughly proportional to this rate of change; this would make it approximately inversely proportional to c^{2} . To test this theory, we can plot p_{c}(.5) 's velocity divided by c^{2} as a function of c . This plot is shown in Figure 15.
It appears from the plot that p_{c}(.5) 's velocity is in fact roughly inversely proportional to c^{2} , with a proportionality constant of approximately 3.82. We can therefore approximate it by the function v(c) := 3.82 / c^{2} .
We can actually give a more careful derivation of this formula that explains where the number 3.82 comes from: it is actually π times the average extrinsic curvature of our curve over half a period, which is the same for any value of c and given by
From the symmetries of a baseball curvature function we know that this is the same as the average extrinsic curvature of our curve over any whole number multiple of a quarter of a period. To see how we get the formula for v(c) , let θ_{(c,t)} be the angle that the projection of the point p_{c}(t) onto the xy -plane makes with thepositive x axis. Recall that for small enough fixed values of c , and for s values near s_{0} , p_{c} looks like a latitude circle about the north pole with extrinsic radius r = 1 / κ(c, s_{0}) , where κ(c,s) is the extrinsic curvature of p_{c} at s , which is equal to √[1 + sin^{2}(cs)] . While we're traveling along this latitude circle, if we make our curve longer by the amount ds , then θ will change by the amount dθ = (1/r) ds = κ(c, s) ds . So
Since p_{c}(.5) is approximately traveling around the equator, which is a circle of radius 1, we have v(c) is approximately equal to −dθ_{(c,.5)} / dc = (π / c^{2})κ_{avg} ≈ 3.8202/c^{2} , exactly as observed. The minus sign comes from the fact that θ gets bigger as c gets smaller.
Now that we have an approximation of the velocity of p_{c}(.5) , we can use it to predict how far around the equator it should be for any c value. We need one point to start from; we will use c_{0} = 0.30840100 , which is one of the points of period one half that we have found. Because p_{0.30840100} has period one half, we know that p_{0.30840100}(.5) = E . Then, for any other value c_{1} for c , we can approximate how far p_{c1}(.5) has traveled from p_{c0}(.5) by integrating our velocity function to give us the following function:
This gives us a really nice way to approximate f(c) . Since we know how far p_{c1}(.5) has traveled around the equator from E , we also know how to approximate how far p_{c1}(1) is from E --it has traveled twice as far, because of the half-turn symmetry. We can use this to approximate d_{c} ; we just have to plug 2t(c_{1}) into the following sawtooth function:
Finally, we can define our function h(c) to approximate f(c) as follows:
h(c) := 2π / s[2t(c)].
If our assumptions have been correct, h(c) should be a good approximation to f(c) for small values of c . To check this, we plot f(c) and h(c) together in Figure 16 for .1 ≤ c ≤ .15 . f(c) is plotted in black, while h(c) is plotted in red. As you can see, the two graphs are virtually indistinguishable.
Our assumptions were only supposed to hold for very small values of c , and our constant of proportionality was determined empirically for small values of c , so we would not expect the graphs to match well for larger values of c . However, as shown in Figure 17 and Figure 18, the graphs match astonishingly well, given how h(c) was created.
We'd like to conclude this article by discussing how our results relate to some much larger open questions.
In this article we have been studying a certain class of periodic or closed spherical curves. The existence of periodic spherical curves with a prescribed geodesic curvature is related to a long-standing open question from classical differential geometry. Independently, Fenchel [1951] and Efimov [1947] formulated the following problem:
Problem 8 [The Closed Curve Problem]. Find (explicit) necessary and sufficient conditions for the curvature and torsion of a space curve as periodic functions of arc length in order that the curve be closed.
This problem remains open although implicit solutions have been found [Hwang, 1981] and most evidence indicates that there is not an effective solution.
A natural subquestion of the Closed Curve Problem is to ask: given a curve on a sphere, can we find necessary and sufficient conditions depending on the geodesic curvature of the curve for the curve to be closed? Nikolaevsky [1994] provided further evidence that even this special case of the Closed Curve Problem likely has no effective solution. An obvious necessary condition for the closure of a spherical curve is that its geodesic curvature be periodic. However, examples in this article make it clear that periodic geodesic curvature is not sufficient for closure. A natural place to start to explore this question is with numerical experimentation of curves with simple periodic geodesic curvature, and in fact, such experimentation was the original impetus to explore the curves we are calling generalized baseball curves. We began by asking for which values of c is a spherical curve with geodesic curvature sin(cs) closed? (As an aside, an interesting article [Scofield, 1995] on curves of constant precession was the result of a similar effort to investigate closure conditions for space curves whose curvature and torsion are given by simple periodic functions, such as κ(s) = ωcos(μs) and τ(s) = ωsin(μs) .) We have shown in the present article that, given a family of periodic geodesic curvature functions with some particular symmetries, there exist many different closed spherical curves whose curvature functions lie in the given family. However, we have not found explicit sufficient conditions for the existence of closed spherical curves and the open question remains.
In this article we have defined generalized baseball curves to be periodic, closed curves on the surface of a sphere that have certain symmetry properties; namely, translation symmetry along the curve, half-turn symmetry about the points midway through one cycle, and reflection symmetry about a perpendicular at points one-quarter of the way through the cycle. Next, we defined baseball curvature functions to be those periodic functions that satisfy two additional symmetry properties: they must be odd functions about the points midway through one period, and even functions about points one-quarter of the way through the period. We then showed geometrically that any spherical curve whose geodesic curvature function is a baseball curvature function is itself a generalized baseball curve as long as it periodic. This brought us to the question of when a curve whose curvature function is a baseball curvature function will be periodic. We focused most of our discussion on the representative case of curves p_{c} with curvature k(s) = sin(cs) , and showed that in this parametrized family of curves there exist values of c for which the curve is periodic of period n for n ≥ 2 . For curves of period 1 the situation was more complicated. Curves that come back to their starting point after half a period, curves of period one half, were shown to exist. Furthermore, curves of period one that come back to their starting point after exactly one period, curves of proper period one, also appeared to exist upon first numerical evidence. However, we conjectured that these curves only come close to closing up and are not true closed periodic curves, although we did not offer a proof of this conjecture.
Although we have focused throughout this paper on the family of curves p_{c} whose geodesic curvature is given by sin(cs) , we could have started with another baseball curvature function f(x) and looked at the one-parameter family of curvature functions f_{c}(x) . Each such family will give rise to many different generalized baseball curves, and every generalized baseball curve is in one of these families.
It is our hope that the ideas contained in this article will have sparked your interest in further exploring this subject. There are many possible areas to explore, but here are some of our ideas about interesting places to start:
Have fun!