Equivalents definition of trigonometric function for implicit equation $x^{2n}+y^{2n}=1$

Question

Anyone know if there exist an equivalent parametrization in terms of some sort of $\cos$ and $\sin$ functions (but that doesn't involves trigonometric functions) for the family of circle-like implicit curves given by $$x^{n}+y^{n}=1,\quad n=2,4,6,\dots.$$ Those functions must be periodic with period $p_n$ $$p_n=2\int_{-1}^1\mathrm{d}x\,\sqrt{1+\Big(x^{n-1}(1-x^n)^{\frac{1-n}{n}}\Big)^2},$$ for which I'm also asking if there exist some closed form.

Answer 1

I think you're looking for a parametrisation of Lamé curves.