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

:$x^ = \frac$

:$y^ = \frac$

:$z^ = \frac$

where the following limits apply to the coordinates

:$- \lambda < c^ < - \mu < b^ < -\nu < a^.$

Consequently, surfaces of constant $\lambda$ are ellipsoids

:$\frac + \frac + \frac = 1,$

whereas