I am interested in working on spheroidal objects, and thus need to use spheroidal coordinates (in my case, prolate spheroidal coordinates). However, when doing some bibliography on the subject, I encountered two different ways of defining these coordinates.
The first one is the one found on the Wikipedia article. The prolate spheroidal coordinates $(\nu, \mu, \varphi)$ is defined from the Cartesian coordinates $(x,y,z)$ such that \begin{align} x &= a \sinh(\mu) \sin(\nu) \cos (\varphi), \\ y &= a \sinh(\mu) \sin(\nu) \sin (\varphi), \\ z &= a \cosh(\mu) \cos(\nu), \end{align} where $a$ is a real constant. In particular, they obey the relation $$ \frac{z^2}{a^2 \cosh^2 \mu} + \frac{x^2+y^2}{a^2 \sinh^2 \mu}=1, $$ and $$ \frac{z^2}{a^2 \cos^2 \nu} - \frac{x^2+y^2}{a^2 \sin^2 \nu}=1. $$
The alternative one is the direct limit of the general case of the ellipsoidal coordinates. Here, the spheroidal coordinates $(\lambda, \phi, \nu)$ are defined from the cylindrical coordinates $(R=\sqrt{x^2+y^2}, \phi, z)$ as the roots in $\tau$ of the equation (see, e.g, equation 2.1 of this paper or equation 12 of this one) $$ \frac{R^2}{\tau+\alpha}+\frac{z^2}{\tau+\gamma}=\frac{x^2+y^2}{\tau+\alpha}+\frac{z^2}{\tau+\gamma}=1, $$ where $\alpha$ and $\gamma$ are real constants.
Are the two parametrisations equivalent ? If so, then what is the link between them?