 Hyperbolic parametrization of prolate spheroidal coordinates

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.