What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

Question

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set

$T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$

Answer 1

It turns out the set $T_p$ goes by the name (superellipse) or Lamé curve.

The parametrization is:

$$ x(\theta) = \pm \cos^{2/n} \theta \\ y(\theta) = \pm \sin^{2/n} \theta \\ \text{ for } 0 \le \theta < \frac{\pi}{2} $$