Ellipsoidal coordinates

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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 coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

1 Basic formulae
2 Scale factors and differential operators
3 See also
4 References
5 External links

[edit] 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>

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>

[edit] 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>

\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] \ + \ </math>

<math>

\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] </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.