= Elliptic cylindrical coordinates
{title2=$x=a\cosh u\cos v,\quad y=a\sinh u\sin v$}
For $a>0$, $u\geq0$, $0\leq v<2\pi$ and real $z$, these <orthogonal coordinates> have coordinate tangents $r_u=(a\sinh u\cos v,a\cosh u\sin v,0)$, $r_v=(-a\cosh u\sin v,a\sinh u\cos v,0)$ and $r_z=(0,0,1)$. Their <scale factors of orthogonal coordinates> are $h_u=h_v=a\sqrt{\sinh^2u+\sin^2v}$ and $h_z=1$; the ordered <Jacobian determinant> is positive away from the two focal lines. Positive constant $u$ gives confocal elliptical cylinders, while generic constant $v$ gives a hyperbolic-cylinder half-branch selected by the signs of $x$ and $y$. The limiting $u=0$ surface is the strip $y=0$, $|x|\leq a$, with arbitrary $z$. Constant $z$ gives horizontal planes.
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