Ellipsoidal coordinates

Short description: Three-dimensional coordinate system

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For the terrestrial coordinates, see Ellipsoidal coordinates (geodesy).

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (λ,μ,ν) 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.

## Contents

* 1 Basic formulae
* 2 Scale factors and differential operators
* 3 Angular parametrization
* 4 See also
* 5 References
* 6 Bibliography
* 6.1 Unusual convention
* 7 External links

## Basic formulae

The Cartesian coordinates (x,y,z) can be produced from the ellipsoidal coordinates (λ,μ,ν) by the equations

x2=(a2+λ)(a2+μ)(a2+ν)(a2−b2)(a2−c2)

y2=(b2+λ)(b2+μ)(b2+ν)(b2−a2)(b2−c2)

z2=(c2+λ)(c2+μ)(c2+ν)(c2−b2)(c2−a2)

where the following limits apply to the coordinates

−λ<c2<−μ<b2<−ν<a2.

Consequently, surfaces of constant λ are ellipsoids

x2a2+λ+y2b2+λ+z2c2+λ=1,

whereas surfaces of constant μ are hyperboloids of one sheet

x2a2+μ+y2b2+μ+z2c2+μ=1,

because the last term in the lhs is negative, and surfaces of constant ν are hyperboloids of two sheets

x2a2+ν+y2b2+ν+z2c2+ν=1

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

S(σ) =def (a2+σ)(b2+σ)(c2+σ)

where σ can represent any of the three variables (λ,μ,ν). Using this function, the scale factors can be written

hλ=12(λ−μ)(λ−ν)S(λ)

hμ=12(μ−λ)(μ−ν)S(μ)

hν=12(ν−λ)(ν−μ)S(ν)

Hence, the infinitesimal volume element equals

dV=(λ−μ)(λ−ν)(μ−ν)8−S(λ)S(μ)S(ν)dλdμdν

and the Laplacian is defined by

∇2Φ=4S(λ)(λ−μ)(λ−ν)∂∂λ[S(λ)∂Φ∂λ]+4S(μ)(μ−λ)(μ−ν)∂∂μ[S(μ)∂Φ∂μ]+4S(ν)(ν−λ)(ν−μ)∂∂ν[S(ν)∂Φ∂ν]

Other differential operators such as ∇⋅𝐅 and ∇×𝐅 can be expressed in the coordinates (λ,μ,ν) 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:[1]

x=assin⁡θcos⁡ϕ,
y=bssin⁡θsin⁡ϕ,
z=cscos⁡θ.

Here, s>0 parametrizes the concentric ellipsoids around the origin and θ∈[0,π] and ϕ∈[0,2π] are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is

dxdydz=abcs2sin⁡θdsdθdϕ.

## See also

* Ellipsoidal latitude
* Focaloid (shell given by two coordinate surfaces)
* Map projection of the triaxial ellipsoid

## References

1. ↑ "Ellipsoid Quadrupole Moment". 9 October 2013. https://photonics101.com/multipole-moments-electric/quadrupole-multipole-moments-homogeneously-charged-ellipsoid#hints.

## Bibliography

* Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953. p. 663.
* Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9.
* Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967. pp. 101-102.
* Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961. p. 176. https://archive.org/details/mathematicalhand0000korn.
* The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956. pp. 178-180. https://archive.org/details/mathematicsphysi00marg_500.
* "Ellipsoidal Coordinates (η, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer Verlag. 1988. pp. 40-44 (Table 1.10). ISBN 0-387-02732-7. https://archive.org/details/fieldtheoryhandb00moon.

### Unusual convention

* Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. 1984. pp. 19-29. ISBN 978-0-7506-2634-7. Uses (ξ, η, ζ) coordinates that have the units of distance squared.

## External links

* MathWorld description of confocal ellipsoidal coordinates

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* t
* e

Orthogonal coordinate systems

Two dimensional|

* Cartesian
* Polar
* Parabolic
* Bipolar
* Elliptic

Three dimensional|

* Cartesian coordinate system
* Cylindrical coordinate system
* Spherical coordinate system
* Parabolic cylindrical coordinates
* Paraboloidal coordinates
* Oblate spheroidal coordinates
* Prolate spheroidal coordinates
* Ellipsoidal coordinates
* Elliptic cylindrical coordinates
* Toroidal coordinates
* Bispherical coordinates
* Bipolar cylindrical coordinates
* Conical coordinates
* 6-sphere coordinates
* Flat-ring cyclide coordinates
* Flat-disk cyclide coordinates
* Bi-cyclide coordinates
* Cap-cyclide coordinates
* Concave bi-sinusoidal single-centered coordinates
* Concave bi-sinusoidal double-centered coordinates
* Convex inverted-sinusoidal spherically aligned coordinates
* Quasi-random-intersection cartesian coordinates

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