# 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 -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci
and are generally taken to be fixed at and
, respectively, on the -axis of the Cartesian coordinate system.

## Basic definition

The most common definition of elliptic cylindrical coordinates is

where 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 form hyperbolae.

## Scale factors

The scale factors for the elliptic cylindrical coordinates and are equal

whereas the remaining scale factor . Consequently, an infinitesimal volume element equals

and the Laplacian equals

Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.

## Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates are sometimes used, where and . Hence, the curves of constant are ellipses, whereas the curves of constant are hyperbolae. The coordinate must belong to the interval [-1, 1], whereas the coordinate must be greater than or equal to one.

The coordinates have a simple relation to the distances to the foci and . For any point in the (x,y) plane, the *sum* of its distances to the foci equals , whereas their *difference* equals .
Thus, the distance to is , whereas the distance to is . (Recall that and are located at and , respectively.)

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

## Alternative scale factors

The scale factors for the alternative elliptic coordinates are

and, of course, . Hence, the infinitesimal volume element becomes

and the Laplacian equals

Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.

## Applications

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 and that sum to a fixed vector , where the integrand was a function of the vector lengths and . (In such a case, one would position between the two foci and aligned with the -axis, i.e., .) For concreteness, , and 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).

## Bibliography

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}} Same as Morse & Feshbach (1953), substituting *u*_{k} for ξ_{k}.

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