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 , 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 plane, the sum of its distances to the foci equals, whereas their difference equals. Thus, the distance to is, whereas the distance to is. 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. 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.
