= Hyperbolic parametrization of oblate spheroidal coordinates
{title2=$h_\rho=h_u=a\sqrt{\sinh^2\rho+\sin^2u},\quad h_v=a\cosh\rho\cos u$}
Set $x=a\cosh\rho\cos u\cos v$, $y=a\cosh\rho\cos u\sin v$, $z=a\sinh\rho\sin u$, with $a>0$, $\rho\geq0$ and $|u|\leq\pi/2$. Differentiating the position vector gives pairwise orthogonal coordinate tangents and the displayed <scale factors of orthogonal coordinates>. Thus the positive volume density is $a^3\cosh\rho\cos u(\sinh^2\rho+\sin^2u)$. The ordered <Jacobian determinant> is negative for the order $(\rho,u,v)$; volume uses its absolute value. Positive constant $\rho$ surfaces are <oblate spheroids>. For $0<|u|<\pi/2$, constant $u$ gives the half of the <one-sheet hyperboloid> $(x^2+y^2)/(a^2\cos^2u)-z^2/(a^2\sin^2u)=1$ with the sign of $z$ fixed by $u$. At $\rho=0$ the coordinates cover the equatorial disk, while $u=0$ covers the exterior equatorial plane.
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