# How do trig functions change in an $L_p$ norm?

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 suspect this might be a preference question, but Generalized spherical and simplicial coordinates [W.D.Richter 2007] Suggests that the correct unit circle identity would actually be:

$\mid\cos_p(t)\mid^p + \mid\sin_p(t)\mid^p = 1$

He defines the trig functions as:

$\cos_p(t) = \dfrac{\cos(t)}{N_p(t)}$ and $\sin_p(t) = \dfrac{\sin(t)}{N_p(t)}$ with $N_p(t)=(\mid\cos(t)\mid^p + \mid\sin(t)\mid^p)^{\frac{1}{p}}$

My sense is that his unit circle identity seems more analogous to other expressions involving p-norms than yours is.

[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.