Prolate spheroidal coordinates ($(\lambda ,\mu ,\varphi )$)

For $0<\alpha <\beta$, define $-\beta \le \mu \le -\alpha \le \lambda$ by $R^2/(\tau +\alpha )+z^2/(\tau +\beta )=1$. Then $R^2=(\lambda +\alpha )(\mu +\alpha )/(\alpha -\beta )$ and $z^2=(\lambda +\beta )(\mu +\beta )/(\beta -\alpha )$. Constant $\lambda$ surfaces are confocal prolate spheroids and constant interior $\mu$ surfaces are two-sheeted hyperboloids, with foci $z=\pm \sqrt{\beta -\alpha }$. The coordinates are orthogonal away from their degeneracies. The squared scale factors are $h_{\lambda }^2=(\lambda -\mu )/[4(\lambda +\alpha )(\lambda +\beta )]$, $h_{\mu }^2=(\mu -\lambda )/[4(\mu +\alpha )(\mu +\beta )]$ and $h_{\varphi }^2=R^2$. The two signs of $z$ are coordinate branches, joined at the equatorial fold $\mu =-\beta$.

For focal distance $a>0$, $\xi \ge 0$ and $0\le \theta \le \pi$, the displayed prolate spheroidal coordinates are orthogonal coordinates. The three coordinate tangent vectors have lengths $h_{\xi }=h_{\theta }=a\sqrt{{\sinh }^2\xi +{\sin }^2\theta }$ and $h_{\varphi }=a\sinh \xi \sin \theta$. Their product gives the Jacobian determinant $a^3({\sinh }^2\xi +{\sin }^2\theta )\sinh \xi \sin \theta$. A positive constant $\xi$ gives a prolate spheroid with equatorial semiaxis $a\sinh \xi$, polar semiaxis $a\cosh \xi$, and foci $(0,0,\pm a)$. The surface degenerates to the focal segment at $\xi =0$; this coordinate degeneracy is distinct from a singularity of the surrounding Euclidean space.