Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinates that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.

Basic formulae

by the equations

where the following limits apply to the coordinates

are ellipsoids

are hyperboloids of one sheet

are hyperboloids of two sheets

because the last two terms in the lhs are negative.

The orthogonal system of quadrics used for the ellipsoidal coordinates are confocal quadrics.

Scale factors and differential operators

For brevity in the equations below, we introduce a function

. Using this function, the scale factors can be written

Hence, the infinitesimal volume element equals

and the Laplacian is defined by

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

Angular parametrization

An alternative (but non-orthogonal) parametrization exists that closely follows the angular parametrization of spherical coordinates:

are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is

See also

• Ellipsoidal latitude

• Focaloid (shell given by two coordinate surfaces)

• Map projection of the triaxial ellipsoid

References

Bibliography

Unusual convention

• Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

• MathWorld description of confocal ellipsoidal coordinates