Ellipsoidal coordinates

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

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 (but non-orthogonal) parametrization exists that closely follows the angular parametrization of spherical coordinates: : x = a s \sin\theta \cos\phi, : y = b s \sin\theta \sin\phi, : z = c s \cos\theta. Here, s0 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

Unusual convention

• Uses (ξ, η, ζ) coordinates that have the units of distance squared.

References

1. (9 October 2013). "Ellipsoid Quadrupole Moment".