elliptic-cylinder coordinates

The v coordinates are the asymptotic angle of confocal hyperbolic cylinders symmetrical about the x-axis. The u coordinates are confocal elliptic cylinders centered on the origin. x | = | a cosh u cos v y | = | a sinh u sin v z | = | z, where u element [0, ∞), v element [0, 2π), and z element (-∞, ∞). They are related to Cartesian coordinates by x^2/(a^2 cosh^2 u) + y^2/(a^2 sinh^2 u) | = | 1 x^2/(a^2 cos^2 v) - y^2/(a^2 sin^2 v) | = | 1.

cylindrical coordinates | Helmholtz differential equation--elliptic cylindrical coordinates