The most common definition of prolate spheroidal coordinates is where is a nonnegative real number and. The azimuthal angle belongs to the interval. The trigonometric identity shows that surfaces of constant form prolate spheroids, since they are ellipses rotated about the axis joining their foci. Similarly, the hyperbolic trigonometric identity shows that surfaces of constant form hyperboloids of revolution. The distances from the foci located at are
Scale factors
The scale factors for the elliptic coordinates are equal whereas the azimuthal scale factor is resulting in a metric of Consequently, an infinitesimal volume element equals and the Laplacian can be written 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 prolate spheroidal coordinates are sometimes used, where and. Hence, the curves of constant are prolate spheroids, whereas the curves of constant are hyperboloids of revolution. The coordinate belongs to the interval , whereas the coordinate must be greater than or equal to one. The coordinates and 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. This gives the following expressions for,, and : Unlike the analogous oblate spheroidal coordinates, the prolate spheroid coordinates are not degenerate; in other words, there is a unique, reversible correspondence between them and the Cartesian coordinates
Alternative scale factors
The scale factors for the alternative elliptic coordinates are while the azimuthal scale factor is now 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. As is the case with spherical coordinates, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of prolate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate.
No angles convention
Uses ξ1 = a cosh μ, ξ2 = sin ν, and ξ3 = cos φ.
Same as Morse & Feshbach, substituting uk for ξk.
Uses coordinates ξ = cosh μ, η = sin ν, and φ.
Angle convention
Korn and Korn use the coordinates, but also introduce the degenerate coordinates.
Similar to Korn and Korn, but uses colatitude θ = 90° - ν instead of latitude ν.
Moon and Spencer use the colatitude convention θ = 90° − ν, and rename φ as ψ.
Unusual convention
Treats the prolate spheroidal coordinates as a limiting case of the general ellipsoidal coordinates. Uses coordinates that have the units of distance squared.
