The confocal ellipsoidal coordinates (called simply ellipsoidal coordinates by Morse and Feshbach 1953) are given by the
equations
![\begin{displaymath}
{x^2\over a^2+\xi}+{y^2\over b^2+\xi}+{z^2\over c^2+\xi}=1
\end{displaymath}](c2_1052.gif) |
(1) |
![\begin{displaymath}
{x^2\over a^2+\eta}+{y^2\over b^2+\eta}+{z^2\over c^2+\eta}=1
\end{displaymath}](c2_1053.gif) |
(2) |
![\begin{displaymath}
{x^2\over a^2+\zeta}+{y^2\over b^2+\zeta}+{z^2\over c^2+\zeta}=1,
\end{displaymath}](c2_1054.gif) |
(3) |
where
,
, and
. Surfaces of constant
are confocal
Ellipsoids, surfaces of constant
are one-sheeted Hyperboloids, and
surfaces of constant
are two-sheeted Hyperboloids. For every
, there is a unique set
of ellipsoidal coordinates. However,
specifies eight points symmetrically located in octants. Solving for
,
, and
gives
![\begin{displaymath}
x^2={(a^2+\xi)(a^2+\eta)(a^2+\zeta)\over (b^2-a^2)(c^2-a^2)}
\end{displaymath}](c2_1063.gif) |
(4) |
![\begin{displaymath}
y^2={(b^2+\xi)(b^2+\eta)(b^2+\zeta)\over (a^2-b^2)(c^2-b^2)}
\end{displaymath}](c2_1064.gif) |
(5) |
![\begin{displaymath}
z^2={(c^2+\xi)(c^2+\eta)(c^2+\zeta)\over (a^2-c^2)(b^2-c^2)}.
\end{displaymath}](c2_1065.gif) |
(6) |
The Laplacian is
|
|
|
|
|
(7) |
where
![\begin{displaymath}
f(x)\equiv \sqrt{(x+a^2)(x+b^2)(x+c^2)}.
\end{displaymath}](c2_1068.gif) |
(8) |
Another definition is
![\begin{displaymath}
{x^2\over a^2-\lambda} + {y^2\over b^2-\lambda} + {z^2\over c^2-\lambda} = 1
\end{displaymath}](c2_1069.gif) |
(9) |
![\begin{displaymath}
{x^2\over a^2-\mu} + {y^2\over b^2-\mu} + {z^2\over c^2-\mu} = 1
\end{displaymath}](c2_1070.gif) |
(10) |
![\begin{displaymath}
{x^2\over a^2-\nu} + {y^2\over b^2-\nu} + {z^2\over c^2-\nu} = 1,
\end{displaymath}](c2_1071.gif) |
(11) |
where
![\begin{displaymath}
\lambda < c^2 < \mu < b^2 < \nu < a^2
\end{displaymath}](c2_1072.gif) |
(12) |
(Arfken 1970, pp. 117-118). Byerly (1959, p. 251) uses a slightly different definition in which the Greek variables
are replaced by their squares, and
. Equation (9) represents an Ellipsoid, (10) represents a
one-sheeted Hyperboloid, and (11) represents a two-sheeted Hyperboloid. In terms of
Cartesian Coordinates,
![\begin{displaymath}
x^2 = {(a^2-\lambda)(a^2-\mu)(a^2-\nu)\over (a^2-b^2)(a^2-c^2)}
\end{displaymath}](c2_1074.gif) |
(13) |
![\begin{displaymath}
y^2 = {(b^2-\lambda)(b^2-\mu)(b^2-\nu)\over (b^2-a^2)(b^2-c^2)}
\end{displaymath}](c2_1075.gif) |
(14) |
![\begin{displaymath}
z^2 = {(c^2-\lambda)(c^2-\mu)(c^2-\nu)\over (c^2-a^2)(c^2-b^2)}.
\end{displaymath}](c2_1076.gif) |
(15) |
The Scale Factors are
The Laplacian is
Using the Notation of Byerly (1959, pp. 252-253), this can be reduced to
![\begin{displaymath}
\nabla^2=(\mu^2-\nu^2){\partial^2\over\partial\alpha^2}+(\la...
...\beta^2}
+(\lambda^2-\mu^2){\partial^2\over\partial\gamma^2},
\end{displaymath}](c2_1090.gif) |
(20) |
where
Here,
is an Elliptic Integral of the First Kind. In terms of
,
, and
,
where
,
and
are Jacobi Elliptic Functions. The Helmholtz Differential Equation is separable in
confocal ellipsoidal coordinates.
See also Helmholtz Differential Equation--Confocal Ellipsoidal Coordinates
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Elliptical Coordinates.'' §21.1 in
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 752, 1972.
Arfken, G. ``Confocal Ellipsoidal Coordinates
.'' §2.15 in
Mathematical Methods for Physicists, 2nd ed. New York: Academic Press, pp. 117-118, 1970.
Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical,
Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.
© 1996-9 Eric W. Weisstein
1999-05-26