Ellipsoidal coordinates

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Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ${\displaystyle (\lambda ,\mu ,\nu )}$ that generalizes the two-dimensional

.

Basic formulae

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

.

Scale factors and differential operators

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

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.

Angular parametrization

An alternative parametrization exists that closely follows the angular parametrization of

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

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

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

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

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

See also

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

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 67025285
  .
• Korn GA,
  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
  .

Unusual convention

• Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the
  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