## Perturbation to free surface in bispherical coordinates

Hi there!

I'm working on a physics problem where there is a liquid droplet (not necessarily spherical) on a plane. Transforming from cylindrical coordinates to bispherical:

$$(r,\phi,z)\mapsto (\xi,\eta,\varphi;a)$$

such that

$$r={\frac {a\sin \left( \eta \right) }{\cosh \left( \eta \right) +\cos \left( \xi \right) }};\phantom{space}z={\frac {a\sinh \left( \xi \right) }{\cosh \left( \eta \right) +\cos \left( \xi \right) }}; \phantom{space}\phi=\varphi$$

thus the $z-$axis is given by $\eta=0$, the contact curve is $\eta=\infty$ and the base of the drop is $\xi=0$. The spherical cap free surface is then described by $\xi=\alpha$.

I'm looking to describe deformations to the sphere's surface in this coordinate system: has anyone tried to describe surfaces other than planes, spheres, apples or lemons? Or, can someone point out why this would be a bad choice of coordinate system to perform deformations to free surfaces in?

Thanks!
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