Confocal Ellipsoidal Coordinates -- from Wolfram MathWorld

Confocal Ellipsoidal Coordinates

The confocal ellipsoidal coordinates, called simply "ellipsoidal coordinates" by Morse and Feshbach (1953) and "elliptic coordinates" by Hilbert and Cohn-Vossen (1999, p. 22), are given by the equations

			(1)
			(2)
			(3)

where , , and . These coordinates correspond to three confocal quadrics all sharing the same pair of foci. Surfaces of constant are confocal ellipsoids, surfaces of constant are one-sheeted hyperboloids, and surfaces of constant are two-sheeted hyperboloids (Hilbert and Cohn-Vossen 1999, pp. 22-23). 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. 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,

			(13)
			(14)
			(15)

The scale factors are

			(16)
			(17)
			(18)

The Laplacian is

del ^2=2(a^2b^2+a^2c^2+b^2c^2-2nu(a^2+b^2+c^2)+3nu^2)/((mu-nu)(nu-lambda))partial/(partialnu)+4((a^2-nu)(b^2-nu)(c^2-nu))/((mu-nu)(nu-lambda))(partial^2)/(partialnu^2)+2(a^2b^2+a^2c^2+b^2c^2-2mu(a^2+b^2+c^2)+3mu^2)/((nu-mu)(mu-lambda))partial/(partialmu)+4((a^2-mu)(b^2-mu)(c^2-mu))/((mu-lambda)(nu-mu))(partial^2)/(partialmu^2)+2(-(a^2b^2+a^2c^2+b^2c^2)+2lambda(a^2+b^2+c^2)-3lambda^2)/((mu-lambda)(nu-lambda))partial/(partiallambda)+4((a^2-lambda)(b^2-lambda)(c^2-lambda))/((mu-lambda)(nu-lambda))(partial^2)/(partiallambda^2).

(19)

Using the notation of Byerly (1959, pp. 252-253), this can be reduced to

(20)

where

			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

Here, is an elliptic integral of the first kind. In terms of , , and ,

			(27)
			(28)
			(29)

where , and are Jacobi elliptic functions. The Helmholtz differential equation is separable in confocal ellipsoidal coordinates.

REFERENCES:

Abramowitz, M. and Stegun, I. 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, pp. 251-252, 1959.

Hilbert, D. and Cohn-Vossen, S. "The Thread Construction of the Ellipsoid, and Confocal Quadrics." §4 in Geometry and the Imagination. New York: Chelsea, pp. 19-25, 1999.

Moon, P. and Spencer, D. E. "Ellipsoidal Coordinates ." Table 1.10 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 40-44, 1988.

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

CITE THIS AS:

Weisstein, Eric W. "Confocal Ellipsoidal Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html