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{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:

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. $$

Unusual convention

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