{{Short description|Three-dimensional coordinate system}} {{No footnotes|date=April 2021}} {{for|the terrestrial coordinates|Ellipsoidal coordinates (geodesy)}} '''Ellipsoidal coordinates''' are a three-dimensional orthogonal coordinate system <math>(\lambda, \mu, \nu)</math> that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinates that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.
==Basic formulae==
The Cartesian coordinates <math>(x, y, z)</math> can be produced from the ellipsoidal coordinates <math>( \lambda, \mu, \nu )</math> by the equations
:<math> 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)} </math>
:<math> 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)} </math>
:<math> 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)} </math>
where the following limits apply to the coordinates
:<math> - \lambda < c^{2} < - \mu < b^{2} < -\nu < a^{2}. </math>
Consequently, surfaces of constant <math>\lambda</math> are ellipsoids
:<math> \frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1, </math>
whereas surfaces of constant <math>\mu</math> are hyperboloids of one sheet
:<math> \frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1, </math>
because the last term in the lhs is negative, and surfaces of constant <math>\nu</math> are hyperboloids of two sheets :<math> \frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1 </math>
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
:<math> S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right) </math>
where <math>\sigma</math> can represent any of the three variables <math>(\lambda, \mu, \nu )</math>. Using this function, the scale factors can be written
:<math> h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}} </math>
:<math> h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}} </math>
:<math> h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}} </math>
Hence, the infinitesimal volume element equals
:<math> 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 </math>
and the Laplacian is defined by
:<math>\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}</math>
Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\lambda, \mu, \nu)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.
== Angular parametrization ==
An alternative (but non-orthogonal) parametrization exists that closely follows the angular parametrization of spherical coordinates:<ref>{{Cite web|url=https://photonics101.com/multipole-moments-electric/quadrupole-multipole-moments-homogeneously-charged-ellipsoid#hints |title = Ellipsoid Quadrupole Moment| date=9 October 2013 }}</ref> :<math> x = a s \sin\theta \cos\phi, </math> :<math> y = b s \sin\theta \sin\phi, </math> :<math> z = c s \cos\theta. </math> Here, <math>s>0</math> parametrizes the concentric ellipsoids around the origin and <math>\theta\in[0,\pi]</math> and <math>\phi\in [0,2\pi]</math> are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is :<math> dx \, dy \, dz = a b c \, s^2 \sin\theta \, ds \, d\theta \, d\phi. </math>
==See also== * Ellipsoidal latitude * Focaloid (shell given by two coordinate surfaces) * Map projection of the triaxial ellipsoid
==References== {{reflist}}
==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== *[https://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html MathWorld description of confocal ellipsoidal coordinates]
{{Orthogonal coordinate systems}}
Category:Three-dimensional coordinate systems Category:Orthogonal coordinate systems Category:Ellipsoids