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Confocal Ellipsoidal Coordinates

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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

(x^2)/(a^2+xi)+(y^2)/(b^2+xi)+(z^2)/(c^2+xi)=1
(1)
(x^2)/(a^2+eta)+(y^2)/(b^2+eta)+(z^2)/(c^2+eta)=1
(2)
(x^2)/(a^2+zeta)+(y^2)/(b^2+zeta)+(z^2)/(c^2+zeta)=1,
(3)

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

x^2=((a^2+xi)(a^2+eta)(a^2+zeta))/((b^2-a^2)(c^2-a^2))
(4)
y^2=((b^2+xi)(b^2+eta)(b^2+zeta))/((a^2-b^2)(c^2-b^2))
(5)
z^2=((c^2+xi)(c^2+eta)(c^2+zeta))/((a^2-c^2)(b^2-c^2)).
(6)

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

(x^2)/(a^2-lambda)+(y^2)/(b^2-lambda)+(z^2)/(c^2-lambda)=1
(9)
(x^2)/(a^2-mu)+(y^2)/(b^2-mu)+(z^2)/(c^2-mu)=1
(10)
(x^2)/(a^2-nu)+(y^2)/(b^2-nu)+(z^2)/(c^2-nu)=1,
(11)

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,

x^2=((a^2-lambda)(a^2-mu)(a^2-nu))/((a^2-b^2)(a^2-c^2))
(13)
y^2=((b^2-lambda)(b^2-mu)(b^2-nu))/((b^2-a^2)(b^2-c^2))
(14)
z^2=((c^2-lambda)(c^2-mu)(c^2-nu))/((c^2-a^2)(c^2-b^2)).
(15)

The scale factors are

h_lambda=sqrt(((mu-lambda)(nu-lambda))/(4(a^2-lambda)(b^2-lambda)(c^2-lambda)))
(16)
h_mu=sqrt(((nu-mu)(lambda-mu))/(4(a^2-mu)(b^2-mu)(c^2-mu)))
(17)
h_nu=sqrt(((lambda-nu)(mu-nu))/(4(a^2-nu)(b^2-nu)(c^2-nu))).
(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

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

alpha=cint_c^lambda(dlambda)/(sqrt((lambda^2-b^2)(lambda^2-c^2)))
(21)
=F(b/c,pi/2)-F(b/c,sin^(-1)(c/lambda))
(22)
beta=cint_b^mu(dmu)/(sqrt((c^2-mu^2)(mu^2-b^2)))
(23)
=F[sqrt(1-(b^2)/(c^2)),sin^(-1)(sqrt((1-(b^2)/(mu^2))/(1-(b^2)/(c^2))))]
(24)
gamma=cint_0^nu(dnu)/(sqrt((b^2-nu^2)(c^2-nu^2)))
(25)
=F(b/c,sin^(-1)(nu/b)).
(26)

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

lambda=cdc(alpha,b/c)
(27)
mu=bnd(beta,sqrt(1-(b^2)/(c^2)))
(28)
nu=bsn(gamma,b/c),
(29)

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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