Ellipsoidal coordinates

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.

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 < c² < − μ < b² < − ν < a².

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

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

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

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

because the last two terms in the lhs are negative.

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.

See also

• Focaloid (shell given by two coordinate surfaces)

References

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. ed.). New York: Springer Verlag. pp. 40–44 (Table 1.10). ISBN 0-387-02732-7. 

Unusual convention

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

External links

• MathWorld description of confocal ellipsoidal coordinates