Ellipsoidal coordinates explained

( λ , \mu, \nu) 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

The Cartesian coordinates

(x, y, z) can be produced from the ellipsoidal coordinates ( λ , \mu, \nu ) by the equationsx ^{2} = (\left( a ^{2} +λ \right) \left( a ^{2} + \mu \right) \left( a ^{2} + \nu \right))/(\left( a ^{2} - b ^{2} \right) \left( a ^{2} - c ^{2} \right)) y ^{2} = (\left( b ^{2} +λ \right) \left( b ^{2} + \mu \right) \left( b ^{2} + \nu \right))/(\left( b ^{2} - a ^{2} \right) \left( b ^{2} - c ^{2} \right)) z ^{2} = (\left( c ^{2} +λ \right) \left( c ^{2} + \mu \right) \left( c ^{2} + \nu \right))/(\left( c ^{2} - b ^{2} \right) \left( c ^{2} - a ^{2} \right))

where the following limits apply to the coordinates

- λ < c ^{2} < - \mu < b ^{2} < - \nu < a ^{2} .

Consequently, surfaces of constant

λ are ellipsoids(x ^{2})/(a ^{2} +λ) + (y ^{2})/(b ^{2} +λ) + (z ^{2})/(c ^{2} +λ) = 1 ,

whereas surfaces of constant

\mu are hyperboloids of one sheet(x ^{2})/(a ^{2} + \mu) + (y ^{2})/(b ^{2} + \mu) + (z ^{2})/(c ^{2} + \mu) = 1 ,

because the last term in the lhs is negative, and surfaces of constant

\nu are hyperboloids of two sheets(x ^{2})/(a ^{2} + \nu) + (y ^{2})/(b ^{2} + \nu) + (z ^{2})/(c ^{2} + \nu) = 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(\sigma) \stackrel{ def }\ \left(a^ + \sigma \right) \left(b^ + \sigma \right) \left(c^ + \sigma \right)

where

\sigma can represent any of the three variables ( λ , \mu, \nu ) . Using this function, the scale factors can be writtenh _{λ} = (1)/(\sqrt{ 2) (\left( λ- \mu \right) \left( λ- \nu\right))/(S( λ )) }h _{\mu} = (1)/(\sqrt{ 2) (\left( \mu -λ \right) \left( \mu - \nu\right))/(S(\mu)) }h _{\nu} = (1)/(\sqrt{ 2) (\left( \nu -λ \right) \left( \nu - \mu\right))/(S(\nu)) }

Hence, the infinitesimal volume element equals

dV = (\left( λ- \mu \right) \left( λ- \nu \right) \left( \mu - \nu\right))/(8 \sqrt{ - S( λ ) S(\mu) S(\nu)) } \, d\lambda \, d\mu \, d\nu

and the Laplacian is defined by

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

Other differential operators such as

\nabla ⋅ F and \nabla x F can be expressed in the coordinates ( λ , \mu, \nu) 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]

x = a s \sin\theta \cos\phi, y = b s \sin\theta \sin\phi, z = c s \cos\theta. Here, s> 0 parametrizes the concentric ellipsoids around the origin and \theta\in[ 0 ,\pi] and \phi\in [ 0 , 2 \pi] are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element isdx dy dz = a b c s ^{2} \sin\theta ds d\theta d\phi.

See also

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

Bibliography

• Book: Morse PM, Feshbach H . 1953 . Methods of Theoretical Physics, Part I . McGraw-Hill . New York . 663.
• Book: Zwillinger D . 1992 . Handbook of Integration . Jones and Bartlett . Boston, MA . 0-86720-293-9 . 114.
• Book: Sauer R, Szabó I . 1967 . Mathematische Hilfsmittel des Ingenieurs . Springer Verlag . New York . 101 - 102 . 67025285.
• Book: . 1961 . Mathematical Handbook for Scientists and Engineers . registration . McGraw-Hill . New York . 176 . 59014456.
• Book: Margenau H, Murphy GM . 1956 . The Mathematics of Physics and Chemistry . limited . D. van Nostrand . New York. 178 - 180 . 55010911 .
• Book: Moon PH, Spencer DE . 1988 . Ellipsoidal Coordinates (η, θ, λ) . Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions . limited . corrected 2nd, 3rd print . Springer Verlag . New York . 0-387-02732-7 . 40 - 44 (Table 1.10).

Unusual convention

• Book: Landau LD, Lifshitz EM, Pitaevskii LP . 1984 . Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) . 2nd . Pergamon Press . New York . 978-0-7506-2634-7 . 19 - 29 . Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

• MathWorld description of confocal ellipsoidal coordinates

Notes and References

1. Web site: Ellipsoid Quadrupole Moment.