The confocal ellipsoidal coordinates (called simply ellipsoidal coordinates by Morse and Feshbach 1953) are given by the
equations

(1) 

(2) 

(3) 
where
,
, and
. Surfaces of constant are confocal
Ellipsoids, surfaces of constant are onesheeted Hyperboloids, and
surfaces of constant are twosheeted Hyperboloids. For every , there is a unique set
of ellipsoidal coordinates. However,
specifies eight points symmetrically located in octants. Solving for
, , and gives

(4) 

(5) 

(6) 
The Laplacian is





(7) 
where

(8) 
Another definition is

(9) 

(10) 

(11) 
where

(12) 
(Arfken 1970, pp. 117118). 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
onesheeted Hyperboloid, and (11) represents a twosheeted Hyperboloid. In terms of
Cartesian Coordinates,

(13) 

(14) 

(15) 
The Scale Factors are
The Laplacian is
Using the Notation of Byerly (1959, pp. 252253), this can be reduced to

(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 EquationConfocal 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. 117118, 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: McGrawHill, p. 663, 1953.
© 19969 Eric W. Weisstein
19990526