Ellipsoidal coordinates

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, 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. ==See also==