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

Basic formulae

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The Cartesian coordinates   can be produced from the ellipsoidal coordinates   by the equations

 
 
 

where the following limits apply to the coordinates

 

Consequently, surfaces of constant   are ellipsoids

 

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

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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

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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

 

See also

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References

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  1. ^ "Ellipsoid Quadrupole Moment".

Bibliography

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Unusual convention

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  • 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. ISBN 978-0-7506-2634-7. Uses (ξ, η, ζ) coordinates that have the units of distance squared.
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