Ellipsoidal coordinates

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system $(\lambda ,\mu ,\nu )$ 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 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 $(\lambda ,\mu ,\nu )$ 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

$\lambda >-c^{2}>\mu >-b^{2}>\nu >-a^{2}$

Consequently, surfaces of constant $\lambda$ 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 $\mu$ 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 $\nu$ are hyperboloids of two sheets

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

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 $\sigma$ can represent any of the three variables $(\lambda ,\mu ,\nu )$. 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 $(\lambda ,\mu ,\nu )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. p. 663. {{cite book}}: |pages= has extra text (help)

• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. pp. p. 114. ISBN 0-86720-293-9. {{cite book}}: |pages= has extra text (help)

• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. pp. 101–102. LCCN 67-0 – 0. {{cite book}}: |pages= has extra text (help)

• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. p. 176. LCCN 59-0 – 0. {{cite book}}: |pages= has extra text (help)

• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. pp. 178–180. LCCN 55-0 – 0. {{cite book}}: |pages= has extra text (help)

• 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. pp. 40–44 (Table 1.10). ISBN 0-387-02732-7. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)

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. pp. 19–29. ISBN 978-0750626347. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)CS1 maint: multiple names: authors list (link) Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

• MathWorld description of confocal ellipsoidal coordinates