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For the terrestrial coordinates, see Ellipsoidal coordinates (geodesy).
Ellipsoidal coordinates are a three-dimensional orthogonalcoordinate system that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonalcoordinate systems that feature quadraticcoordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.
whereas surfaces of constant are hyperboloids of one sheet
because the last term in the lhs is negative, and surfaces of constant 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
where can represent any of the three variables .
Using this function, the scale factors can be written
Hence, the infinitesimal volume element equals
and the Laplacian is defined by
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.
Angular parametrization[]
An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates:[1]
Here, parametrizes the concentric ellipsoids around the origin and and are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is
Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. pp. 19–29. ISBN978-0-7506-2634-7. Uses (ξ, η, ζ) coordinates that have the units of distance squared.