Best-fit curves may vary from simple linear equations to more complex quadratic, polynomial, exponential, and periodic curves.
2.
Denoting by \ Lambda M the space of smooth 1-periodic curves on " M ", closed geodesics of period 1 are precisely the critical points of the energy function E : \ Lambda M \ rightarrow \ mathbb R, defined by
3.
I did a quick graph of 2 ^ { x + yi } + 3 ^ { x + yi } = 13 in the OS X grapher application and got what looked like a periodic curve ( but probably isn't ), which shouldn't be terribly surprising.
4.
Thus, in the Brinkmann chart, the world lines of our observers, in the shortwave case, are periodic curves which have the form of sinusoidals with period 2 \ pi / q, modulated by much smaller sinusoidal perturbations in the null direction and having a much shorter period, 2 \ pi / \ omega.
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