After a long research, I turn to you looking for an answer to my problem consisting in converting an explicit equation to a parametric one.
As we all know $x^2+y^2=1$ is the unit circle and can be written: $$ \begin{cases} x=\cos(t)&\\ y=\sin(t) \end{cases} $$ But I would like to turn $x^{2n}+y^{2n}=1$ to a parametric system in order to plot it on a parametric grapher (grasshopper) with $n \to +∞$ and observe the circle tending towards a square.
After this I would also like to find the parametric system of $x^{2n}+y^{2n}+z^{2n}=1$ to move to 3D and find the cube.
Maybe we can start with $x^4+y^4=1$ and then find our way to $2n$ by mathematical induction.
Thank you!
If you take $x=\cos(t)^{1/n}$ and $y=\sin(t)^{1/n},$ then you'll have $x^{2n}+y^{2n}=\cos^2(t)+\sin^2(t)=1.$
If you take $x=\cos(t)^{1/n}\cos(u)^{1/n}$, $y=\cos(t)^{1/n}\sin(u)^{1/n},$ and $z=\sin(t)^{1/n}$,
then you'll have $x^{2n}+y^{2n}+z^{2n}=1$.