Ellipsoidal coordinates

{{for|the terrestrial coordinates|Ellipsoidal coordinates (geodesy)}}

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 based on confocal quadrics.

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

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

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

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

because the last two terms in the lhs are negative.

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

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

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

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.

Angular parametrization

An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates:{{Cite web|url=https://photonics101.com/multipole-moments-electric/quadrupole-multipole-moments-homogeneously-charged-ellipsoid#hints |title = Ellipsoid Quadrupole Moment| date=9 October 2013 }}

: $x = a s \sin\theta \cos\phi,$

: $y = b s \sin\theta \sin\phi,$

: $z = c s \cos\theta.$

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

: $dx \, dy \, dz = a b c \, s^2 \sin\theta \, ds \, d\theta \, d\phi.$

References

Bibliography

{{cite book | vauthors = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | page = 663}}
{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}}
{{cite book | vauthors = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 101–102 | lccn = 67025285}}
{{cite book | vauthors = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | url = https://archive.org/details/mathematicalhand0000korn | url-access = registration | publisher = McGraw-Hill | location = New York | page = [https://archive.org/details/mathematicalhand0000korn/page/176 176] | lccn = 59014456}}
{{cite book | vauthors = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsphysi00marg_500 | url-access = limited | publisher = D. van Nostrand | location = New York| pages = [https://archive.org/details/mathematicsphysi00marg_500/page/n191 178]–180 | lccn = 55010911 }}
{{cite book | vauthors = Moon PH, Spencer DE | year = 1988 | chapter = Ellipsoidal Coordinates (η, θ, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | url = https://archive.org/details/fieldtheoryhandb00moon | url-access = limited | edition = corrected 2nd, 3rd print | publisher = Springer Verlag | location = New York | isbn = 0-387-02732-7 | pages = [https://archive.org/details/fieldtheoryhandb00moon/page/n42 40]–44 (Table 1.10)}}

=Unusual convention=

{{cite book | vauthors = Landau LD, Lifshitz EM, Pitaevskii LP | year = 1984 | title = Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) | edition = 2nd | publisher = Pergamon Press | location = New York | isbn = 978-0-7506-2634-7 | pages = 19–29 }} Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

[http://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html MathWorld description of confocal ellipsoidal coordinates]

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems

Category:Ellipsoids