Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular

Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae.

-axis of the Cartesian coordinate system.

The most common definition of elliptic cylindrical coordinates

is a nonnegative real number and

These definitions correspond to ellipses and hyperbolae.

The trigonometric identity shows that curves of constant

form ellipses, whereas the hyperbolic trigonometric identity shows that curves of constant

The scale factors for the elliptic cylindrical coordinates

are equal whereas the remaining scale factor

Consequently, an infinitesimal volume element equals and the Laplacian equals Other differential operators such as

by substituting the scale factors into the general formulae found in orthogonal coordinates.

An alternative and geometrically intuitive set of elliptic coordinates

σ = cosh ⁡ μ

τ = cos ⁡ ν

are ellipses, whereas the curves of constant

coordinate must be greater than or equal to one.

have a simple relation to the distances to the foci

For any point in the (x,y) plane, the sum

of its distances to the foci equals

Hence, the infinitesimal volume element becomes and the Laplacian equals Other differential operators such as

by substituting the scale factors into the general formulae found in orthogonal coordinates.

The classic applications of elliptic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a separation of variables.

A typical example would be the electric field surrounding a flat conducting plate of width

The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations.

The geometric properties of elliptic coordinates can also be useful.

A typical example might involve an integration over all pairs of vectors

that sum to a fixed vector

, where the integrand was a function of the vector lengths

could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta).