Could someone help me formulate the circumference of a superellipse? $$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$ If it makes things easier, I'm considering only the cases $n>2$, and $n\in\mathbb{Q}$. Since the formula for an ellipse ($n=2$) involves special functions, I assume the general case won't be any simpler, but I can't seem to find any sources on this.
I've also seen the circumference for an ellipse approximated by: $$C \sim \frac{2\pi}{\sqrt{2}}\sqrt{a^2 + B^2}$$ Can this approximation be extended to the case of a superellipse, i.e: $$C \sim A\sqrt{a^n + B^n}$$ Any help greatly appreciated.
Edit: Someone by the name of Maher Izzedin Aldaher posted some approximate numerical formulae online, one of them being:
$$4 \left[a+b \frac{ b\left(\frac{2.5}{n+0.5} \right)^{1/n} + \frac{0.566 a(n-1)}{n^2} }{b + \frac{4.5 a}{0.5+n^2}} \right]$$ I have no idea how this was derived or how good it is.
The question is taken up in detail in multiple sections here. See all the sections having to do with 'arc length'. This paper also goes through formulas for the area of such a figure, and other interesting properties. Enjoy. (Note: its not very easy stuff; as you pointed out, even the regular old ellipse's perimeter cannot be expressed in an elementary way, see here.)