The unit circle under the $L_2$ norm can be described parametrically by
$x = \cos(t), y = \sin(t), 0 \le t \le 2\pi$
How do we parametrically describe the unit circle in general $L_p$ space (for simplicity, assume $1 < p < \infty$)? Is it reasonable to view the appropriate functions $x = \cos_p(t), y = \sin_p(t)$ as generalizations of the usual definitions of $\sin, \cos$?
[I think this was an easy question that I asked too quickly, and I can actually answer my own question.]
I think the right analog is $$\cos_p(t) := \cos^{2/p}(t), \sin_p(t) := \sin^{2/p}(t)$$
We then have $\cos_p(t)^p + \sin_p(t)^p = \cos(t)^2 + \sin(t)^2 = 1$, which means that we have correctly described the unit circle.