Ellipsoidal coordinates

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system [math]\displaystyle{ (\lambda, \mu, \nu) }[/math] 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 [math]\displaystyle{ (x, y, z) }[/math] can be produced from the ellipsoidal coordinates [math]\displaystyle{ ( \lambda, \mu, \nu ) }[/math] by the equations

[math]\displaystyle{ x^{2} = \frac{\left( a^{2} + \lambda \right) \left( a^{2} + \mu \right) \left( a^{2} + \nu \right)}{\left( a^{2} - b^{2} \right) \left( a^{2} - c^{2} \right)} }[/math]

[math]\displaystyle{ y^{2} = \frac{\left( b^{2} + \lambda \right) \left( b^{2} + \mu \right) \left( b^{2} + \nu \right)}{\left( b^{2} - a^{2} \right) \left( b^{2} - c^{2} \right)} }[/math]

[math]\displaystyle{ z^{2} = \frac{\left( c^{2} + \lambda \right) \left( c^{2} + \mu \right) \left( c^{2} + \nu \right)}{\left( c^{2} - b^{2} \right) \left( c^{2} - a^{2} \right)} }[/math]

where the following limits apply to the coordinates

[math]\displaystyle{ \- \lambda \lt c^{2} \lt - \mu \lt b^{2} \lt -\nu \lt a^{2}. }[/math]

Consequently, surfaces of constant [math]\displaystyle{ \lambda }[/math] are ellipsoids

[math]\displaystyle{ \frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1, }[/math]

whereas surfaces of constant [math]\displaystyle{ \mu }[/math] are hyperboloids of one sheet

[math]\displaystyle{ \frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1, }[/math]

because the last term in the lhs is negative, and surfaces of constant [math]\displaystyle{ \nu }[/math] are hyperboloids of two sheets

[math]\displaystyle{ \frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1 }[/math]

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

[math]\displaystyle{ S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right) }[/math]

where [math]\displaystyle{ \sigma }[/math] can represent any of the three variables [math]\displaystyle{ (\lambda, \mu, \nu ) }[/math]. Using this function, the scale factors can be written

[math]\displaystyle{ h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}} }[/math]

[math]\displaystyle{ h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}} }[/math]

[math]\displaystyle{ h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}} }[/math]

Hence, the infinitesimal volume element equals

[math]\displaystyle{ dV = \frac{\left( \lambda - \mu \right) \left( \lambda - \nu \right) \left( \mu - \nu\right)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \, d\lambda \, d\mu \, d\nu }[/math]

and the Laplacian is defined by

[math]\displaystyle{ \begin{align} \nabla^{2} \Phi = {} & \frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)} \frac{\partial}{\partial \lambda} \left[ \sqrt{S(\lambda)} \frac{\partial \Phi}{\partial \lambda} \right] \\\\[1ex] & \+ \frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)} \frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right] \\\\[1ex] & \+ \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)} \frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right] \end{align} }[/math]

Other differential operators such as [math]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\lambda, \mu, \nu) }[/math] 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]

[math]\displaystyle{ x = a s \sin\theta \cos\phi, }[/math]
[math]\displaystyle{ y = b s \sin\theta \sin\phi, }[/math]
[math]\displaystyle{ z = c s \cos\theta. }[/math]

Here, [math]\displaystyle{ s\gt 0 }[/math] parametrizes the concentric ellipsoids around the origin and [math]\displaystyle{ \theta\in[0,\pi] }[/math] and [math]\displaystyle{ \phi\in [0,2\pi] }[/math] are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is

[math]\displaystyle{ dx \, dy \, dz = a b c \, s^2 \sin\theta \, ds \, d\theta \, d\phi. }[/math]

## See also

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

## References

1. ↑ "Ellipsoid Quadrupole Moment". 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

* v
* 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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