I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have been looking for the answer but keep ending up with the Gilies formula:
$$ r(\theta)=\left[\left |{\frac{cos\left( \frac{1}{4}m\theta \right )^{n_2}}{a}} \right | +\left |{\frac{sin\left( \frac{1}{4}m\theta \right )^{n_3}}{b}} \right |\right]^{\frac{-1}{n_1}} $$
which is nice but with this one it is not possible to relate $a$ and $b$ directly to the semi-major and semi-minor axis. I guess it could be converted to the top form if: $$n_1=f(n_2,n_3)$$ but have not found the correct relation yet.
I also found this page: Polar form of a superellipse? but it does not cover the answer completely.