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Confocal Ellipsoidal Coordinates

ConfocalQuadrics

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

where -c^2<xi<infty, -b^2<eta<-c^2, and -a^2<zeta<-b^2. These coordinates correspond to three confocal quadrics all sharing the same pair of foci. Surfaces of constant xi are confocal ellipsoids, surfaces of constant eta are one-sheeted hyperboloids, and surfaces of constant zeta are two-sheeted hyperboloids (Hilbert and Cohn-Vossen 1999, pp. 22-23). For every (x,y,z), there is a unique set of ellipsoidal coordinates. However, (xi,eta,zeta) specifies eight points symmetrically located in octants.

Solving for x, y, and z gives

The Laplacian is

del ^2Psi=(eta-zeta)f(xi)partial/(partialxi)[f(xi)(partialPsi)/(partialxi)] +(zeta-xi)f(eta)partial/(partialeta)[f(eta)(partialPsi)/(partialeta)]+(xi-eta)f(zeta)partial/(partialzeta)[f(zeta)(partialPsi)/(partialzeta)],

(7)

where

f(x)=sqrt((x+a^2)(x+b^2)(x+c^2)).

(8)

Another definition is

where

lambda<c^2<mu<b^2<nu<a^2

(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 a=0. Equation (9) represents an ellipsoid, (10) represents a one-sheeted hyperboloid, and (11) represents a two-sheeted hyperboloid.

In terms of Cartesian coordinates,

The scale factors are

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

del ^2=(mu^2-nu^2)(partial^2)/(partialalpha^2)+(lambda^2-nu^2)(partial^2)/(partialbeta^2)+(lambda^2-mu^2)(partial^2)/(partialgamma^2),

(20)

where

Here, F is an elliptic integral of the first kind. In terms of alpha, beta, and gamma,

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

See also

Helmholtz Differential Equation--Confocal Ellipsoidal Coordinates

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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 (xi_1,xi_2,xi_3)." §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 (eta,theta,lambda)." 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.

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Confocal Ellipsoidal Coordinates

Cite this as:

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

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