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Oblate Spheroidal Coordinates


A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the Elliptic Cylindrical Coordinates about the y-Axis which is relabeled the z-Axis. The third set of coordinates consists of planes passing through this axis.

$\displaystyle x$ $\textstyle =$ $\displaystyle a\cosh\xi\cos\eta\cos\phi$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle a\cosh\xi\cos\eta\sin\phi$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle a\sinh\xi\sin\eta,$ (3)

where $\xi \in [0,\infty)$, $\eta \in [-\pi/2, \pi/2]$, and $\phi \in [0,2\pi)$. Arfken (1970) uses $(u, v, \varphi)$ instead of $(\xi, \eta, \phi)$. The Scale Factors are
$\displaystyle h_\xi$ $\textstyle =$ $\displaystyle a\sqrt{\sinh^2\xi+\sin^2\eta}$ (4)
$\displaystyle h_\eta$ $\textstyle =$ $\displaystyle a\sqrt{\sinh^2\xi+\sin^2\eta}$ (5)
$\displaystyle h_\phi$ $\textstyle =$ $\displaystyle a \cosh \xi \cos\eta.$ (6)

The Laplacian is

$\nabla^2 f = {1\over a^3(\sinh^2\xi+\sin^2\eta)\cosh\xi\cos\eta}$
$ \times \left[{{\partial f\over\partial\xi}\left({a\cosh\xi\cos\eta{\partial f\...
...i+\sin^2\eta)\over a\cosh\xi\cos\eta}{\partial^2 f\over\partial\phi^2 }}\right]$
$={1\over a^3(\sinh^2\xi+\sin^2\eta)\cosh\xi\cos\eta} \left[{a\sinh\xi\cos\eta{\partial f\over\partial\xi}}\right.$
$ +a\cosh\xi\cos\eta{\partial^2 f\over\partial\xi^2}+a\sinh\xi\cos\eta{\partial f\over\partial\eta}$
$ \left.{+a\cosh\xi\cos\eta{\partial^2 f\over\partial\eta^2}}\right]+{1\over a^2(\sinh^2\xi+\sin^2\eta)}{\partial^2 f\over\partial\phi^2}$
$={1\over a^2 (\sinh^2\xi+\sin^2\eta)}\left[{{1\over\cosh\xi}{\partial\over\part...
...}\right]+ {1\over a^2(\cosh^2\xi +\cos^2\eta)}{\partial^2 f\over\partial\phi^2}$ (7)
$={1\over\sin^2\eta+\sinh^2\xi}\left[{(\mathop{\rm sech}\nolimits ^2\xi\tan^2\et...
...ver\partial\xi^2} -\tan\eta{\partial\over\eta}+{\partial^2\over\eta^2}}\right].$ (8)

An alternate form useful for ``two-center'' problems is defined by

$\displaystyle \xi_1$ $\textstyle =$ $\displaystyle \sinh\xi$ (9)
$\displaystyle \xi_1'$ $\textstyle =$ $\displaystyle \cosh\xi$ (10)
$\displaystyle \xi_2$ $\textstyle =$ $\displaystyle \cos\eta$ (11)
$\displaystyle \xi_3$ $\textstyle =$ $\displaystyle \phi,$ (12)

where $\xi_1 \in [1,\infty]$, $\xi_2\in [-1,1]$, and $\xi_3\in [0,2\pi)$. In these coordinates,
$\displaystyle y$ $\textstyle =$ $\displaystyle a\xi_1'\xi_2\sin\xi_3$ (13)
$\displaystyle z$ $\textstyle =$ $\displaystyle a\sqrt{({\xi_1'}^2-1)(1-{\xi_2}^2)}$ (14)
$\displaystyle x$ $\textstyle =$ $\displaystyle a\xi_1'\xi_2\cos\xi_3$ (15)

(Abramowitz and Stegun 1972). The Scale Factors are
$\displaystyle h_{\xi_1}$ $\textstyle =$ $\displaystyle a\sqrt{{\xi_1}^2-{\xi_2}^2\over{\xi_1}^2-1}$ (16)
$\displaystyle h_{\xi_2}$ $\textstyle =$ $\displaystyle a\sqrt{{\xi_1}^2-{\xi_2}^2\over 1-{\xi_2}^2}$ (17)
$\displaystyle h_{\xi_3}$ $\textstyle =$ $\displaystyle a\xi\eta,$ (18)

and the Laplacian is

\nabla^2 f= {1\over a^2}\left\{{{1\over{\xi_1}^2+{\xi_2}^2}{...
...(1-{\xi_2}^2)} {\partial^2 f\over\partial {\xi_3}^2}}\right\}.
\end{displaymath} (19)

The Helmholtz Differential Equation is separable.

See also Helmholtz Differential Equation--Oblate Spheroidal Coordinates, Latitude, Longitude, Prolate Spheroidal Coordinates, Spherical Coordinates


Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Oblate Spheroidal Coordinates.'' §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.

Arfken, G. ``Prolate Spheroidal Coordinates ($u$, $v$, $\phi$).'' §2.11 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107-109, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.

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© 1996-9 Eric W. Weisstein