I have this superellipse equation but how can I plot it? because I need the equation to be expressed as a function of x and y, like $x = \ldots$ and $y = \ldots$.
$(x-a)^n + (y-b)^n = r^n$
where $a$ and $b$ are the coordinates of the center, $r$ is the radius and $n$ an integer but how can I have $x = ...$ and $y = ...$, I mean dissociating x and y from the equation.
Let us work as for $n=2$. There, we had - $$x=a+r\cos t$$ $$y=b+r\sin t$$ This was because we exploited the property $\cos^2 t + \sin^2 t=1$, and as the terms $a$, $b$ were used to essentially re-center the ellipse to the origin.
Analogously, let us define - $$x=a+r\cos^c t$$ $$y=b+r\sin^d t$$ Thus, we get the relation $$\cos^{cn}t+\sin^{dn}t=1$$ This looks very similar to the identity mentioned above, and becomes identical if we take $c=\frac{2}{n}$, $d=\frac{2}{n}$. Thus, we get our parameterisation.