Prolate spheroidal coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located.

Rotation about the other axis produces oblate spheroidal coordinates.

Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z-axis.

One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+.

Another example is solving for the electric field generated by two small electrode tips.

Other limiting cases include areas generated by a line segment (μ = 0) or a line with a missing segment (ν=0).

The electronic structure of general diatomic molecules with many electrons can also be solved to excellent precision in the prolate spheroidal coordinate system.

[1] The most common definition of prolate spheroidal coordinates

( μ , ν , φ )

is a nonnegative real number and

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

The distances from the foci located at

are 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

( μ , ν , φ )

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

An alternative and geometrically intuitive set of prolate spheroidal coordinates

( σ , τ , ϕ )

τ = cos ⁡ ν

are prolate spheroids, 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 plane, the sum

: Unlike the analogous oblate spheroidal coordinates, the prolate spheroid coordinates (σ, τ, φ) are not degenerate; in other words, there is a unique, reversible correspondence between them and the Cartesian coordinates 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

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 (See Smythe, 1968).

The three coordinate surfaces of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ = 1, and the blue two-sheet hyperboloid corresponds to ν = 45°. The yellow half-plane corresponds to φ = −60°, which is measured relative to the x -axis (highlighted in green). The black sphere represents the intersection point of the three surfaces, which has Cartesian coordinates of roughly (0.831, −1.439, 2.182).
Prolate spheroidal coordinates μ and ν for a = 1. The lines of equal values of μ and ν are shown on the xz -plane, i.e. for φ = 0. The surfaces of constant μ and ν are obtained by rotation about the z -axis, so that the diagram is valid for any plane containing the z -axis: i.e. for any φ .
In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in Cartesian coordinates ; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at ( x , y , ± z ). However, neither of the definitions presented here are degenerate.