Ellipsoidal coordinates

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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 not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

1 Basic formulae
2 Scale factors and differential operators
3 See also
4 References
5 Bibliography
- 5.1 Unusual convention
6 External links

[edit] Basic formulae

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

$x^{2} = \frac{\left( a^{2} + \lambda \right) \left( a^{2} + \mu \right) \left( a^{2} + \nu \right)}{\left( a^{2} - b^{2} \right) \left( a^{2} - c^{2} \right)}$

$y^{2} = \frac{\left( b^{2} + \lambda \right) \left( b^{2} + \mu \right) \left( b^{2} + \nu \right)}{\left( b^{2} - a^{2} \right) \left( b^{2} - c^{2} \right)}$

$z^{2} = \frac{\left( c^{2} + \lambda \right) \left( c^{2} + \mu \right) \left( c^{2} + \nu \right)}{\left( c^{2} - b^{2} \right) \left( c^{2} - a^{2} \right)}$

where the following limits apply to the coordinates

λ > - c 2 > μ > - b 2 > ν > - a 2

Consequently, surfaces of constant $λ$ are ellipsoids

$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1$

whereas surfaces of constant $μ$ are hyperboloids of one sheet

$\frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1$

and surfaces of constant $ν$ are hyperboloids of two sheets

$\frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1$

[edit] Scale factors and differential operators

For brevity in the equations below, we introduce a function

$S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right)$

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

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}}$

$h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}}$

$h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}}$

Hence, the infinitesimal volume element equals

$dV = \frac{\left( \lambda - \mu \right) \left( \lambda - \nu \right) \left( \mu - \nu\right)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \ d\lambda d\mu d\nu$

and the Laplacian is defined by

$\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] \ + \$

$\frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)} \frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right] \ + \ \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)} \frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right]$

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(λ,μ,ν)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] See also

Orthogonal coordinates
Two dimensional orthogonal coordinate systems

Three dimensional orthogonal coordinate systems

[edit] References

[edit] Bibliography

Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663.

Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett, p. 114. ISBN 0-86720-293-9.

Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, pp. 101–102. LCCN 67-25285.

Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 176. LCCN 59-14456.

Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand, pp. 178–180. LCCN 55-10911.

Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)", Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, corrected 2nd ed., 3rd print ed., New York: Springer Verlag, pp. 40–44 (Table 1.10). ISBN 0-387-02732-7.

[edit] Unusual convention

Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics), 2nd edition, New York: Pergamon Press, pp. 19–29. ISBN 978-0750626347. Uses (ξ, η, ζ) coordinates that have the units of distance squared.

[edit] External links

MathWorld description of confocal ellipsoidal coordinates

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