We know that the Schwarzschild solution pictures the outside region of a spherical distribution of mass $M$, which is also non-rotating, charge-free and spherically symmetric. We can begin to describe this way objects such like stars or black holes. On the other hand, the Kerr solution is an axially symmetric spacetime for black holes only, which written in Boyer-Lindquist coordinates resembles a line element whose coordinate basis when $M\to 0$ is similar to oblate spheroidal coordinates:
$$x=\sqrt{r^2+a^2}\sin\theta\cos\phi,$$ $$y=\sqrt{r^2+a^2}\sin\theta\sin\phi,$$ $$z=r\cos\theta;$$
for $r=a\sinh\mu$ and $\theta=\pi/2-\nu$.
Therefore, for a "sub-special" case, a Kerr BH looks like an oblate spheroid (seed shaped, like an M&M). Is there a general form to look at a Kerr BH as a 3D surface? I think there should be different cases, depending on the value of the spin parameter $a=J/Mc$ and the mass $M $, i.e. on how the 3D sub-line element looks:
$$d\ell^2=\frac{\Sigma}{\Delta}\,dr^2 +\Sigma\, d\theta^2+\left(r^2+a^2+\frac{2GMra^2}{\Sigma\,c^2}\sin^2\theta\right)\sin^2\theta\,d\varphi^2.$$
However, I cannot imagine this the way I imagine a sphere like in the Schwarzschild solution. What am I missing?
