Ellipsoidal coordinates
# Ellipsoidal coordinates

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

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

Other differential operators such as $\nabla \cdot \mathbf {F}$ 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:

    $x=as\sin \theta \cos \phi ,$
    $y=bs\sin \theta \sin \phi ,$
    $z=cs\cos \theta .$

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

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