Loci, 2008

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

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`) = `a`cos(π / 2 − `c`) cos(`t`)` `cos[`t` / 2 + `c`sin(2`t`)]

`y`(`y`) = `a`cos(π / 2 − `c`) cos(`t`) sin[`t` / 2 + `c`sin(2`t`)]

`z`(`t`) = `a`sin(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(3`t`)

`y`(`t`) = `a` cos(`t`) − `b` cos(3`t`)

`z`(`t`) = √(4`a``b`)

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.