Ellipsoidal coordinates

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Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ${\displaystyle (\lambda ,\mu ,\nu )}$ 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 based on confocal quadrics.

The Cartesian coordinates ${\displaystyle (x,y,z)}$ can be produced from the ellipsoidal coordinates ${\displaystyle (\lambda ,\mu ,\nu )}$ by the equations

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

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

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

where the following limits apply to the coordinates

${\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}.}$

Consequently, surfaces of constant ${\displaystyle \lambda }$ are ellipsoids

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

whereas surfaces of constant ${\displaystyle \mu }$ are hyperboloids of one sheet

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

because the last term in the lhs is negative, and surfaces of constant ${\displaystyle \nu }$ are hyperboloids of two sheets

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

because the last two terms in the lhs are negative.

The orthogonal system of quadrics used for the ellipsoidal coordinates are confocal quadrics.

For brevity in the equations below, we introduce a function

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

where ${\displaystyle \sigma }$ can represent any of the three variables ${\displaystyle (\lambda ,\mu ,\nu )}$. Using this function, the scale factors can be written

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

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

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

Hence, the infinitesimal volume element equals

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

and the Laplacian is defined by

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

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

An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates:^[1]

${\displaystyle x=as\sin \theta \cos \phi ,}$

${\displaystyle y=bs\sin \theta \sin \phi ,}$

${\displaystyle z=cs\cos \theta .}$

Here, ${\displaystyle s>0}$ parametrizes the concentric ellipsoids around the origin and ${\displaystyle \theta \in [0,\pi ]}$ and ${\displaystyle \phi \in [0,2\pi ]}$ are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is

${\displaystyle dx\,dy\,dz=abc\,s^{2}\sin \theta \,ds\,d\theta \,d\phi .}$

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

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• 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 67025285.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 176. LCCN 59014456.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 178–180. LCCN 55010911.
• Moon PH, Spencer DE (1988). "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. pp. 40–44 (Table 1.10). ISBN 0-387-02732-7.

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

• MathWorld description of confocal ellipsoidal coordinates