The elliptic cylindrical curvilinear coordinate system is one of the many coordinate systems that make the Laplace and Helmoltz differential equations separable. This system is used when simple boundary conditions on a segment in the - plane are specified, as in the computation of the electric field around an infinite rectangular conducting plate whose intersection with the - plane is the line segment with endpoints and .
The more opaque half of the hyperbolic cylinder has ; the other half has .
From the tutorial on MathematicaVector Analysis Package: The elliptic cylindrical coordinate system , parameterized by , is built around two foci separated by . Holding coordinate constant while varying the other coordinates produces a family of confocal ellipses. Fixing coordinate produces a family of confocal hyperbolas. The coordinate specifies distance along the axis of common focus. The default value for parameter is .