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.

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

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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