Axisymmetric coordinates
Below for visualization purposes we demonstrate three different axisymmetric systems: a regular spherical \((r,\theta,\phi)\), an equal area and a "quasi-spherical" \((\xi,\eta,\phi)\), where
\[
\text{equal area} =
\begin{cases}
\xi = \log{(r)}, \\
\eta = -\cos{\theta}, \\
\phi = \phi
\end{cases}
~~~~
\text{quasi-spherical} =
\begin{cases}
\xi = \log{(r - r_0)}, \\
\eta:~\theta = \eta + 2h \eta (\pi - 2 \eta) (\pi - \eta) / \pi^2, \\
\phi = \phi
\end{cases}
\]
with \(r_0\) and \(h\) being user-controlled parameters. The interactive plot below demonstrates the difference between grids, uniformly discretized in each of these coordinate systems.