= Hyperbolic parametrization of prolate spheroidal coordinates
{title2=$x=a\sinh\xi\sin\theta\cos\phi,\quad y=a\sinh\xi\sin\theta\sin\phi,\quad z=a\cosh\xi\cos\theta$}
For focal distance $a>0$, $\xi\geq0$ and $0\leq\theta\leq\pi$, 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 $h_\phi=a\sinh\xi\sin\theta$. 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>.
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