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

2. Generalized Baseball Curves

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:

Definition 2. A spherical curve S is said to be a generalized baseball curve if it can be given parametrically by a function p(s) : RS2 , where S2 is the unit sphere in R3 and the parameter s is (a non-zero constant multiple of) arclength along the curve, such that
  1. p′(s) is continuous;
  2. p(s) is periodic, so that there exists a number d such that p(s) = p(s + d) for all s ;
  3. S has translation symmetry, so that there exists a (smallest) number t such that the curve can be translated so that p(s) is translated onto p(s + t) ;
  4. S has half-turn symmetry about the points p(nt/2) , for any integer n ; and
  5. S has reflection symmetry about a perpendicular to the curve drawn at any of the points p(nt/2 + t/4) .

Notice that it follows from the definition that d = mt for some number m ; we call m the period of the curve.