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Confocal Paraboloidal Coordinates

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(x^2)/(a^2-lambda)+(y^2)/(b^2-lambda)=z-lambda
(1)
(x^2)/(a^2-mu)+(y^2)/(b^2-mu)=z-mu
(2)
(x^2)/(a^2-nu)+(y^2)/(b^2-nu)=z-nu,
(3)

where lambda in (-infty,b^2), mu in (b^2,a^2), and nu in (a^2,infty).

x^2=((a^2-lambda)(a^2-mu)(a^2-nu))/((b^2-a^2))
(4)
y^2=((b^2-lambda)(b^2-mu)(b^2-nu))/((a^2-b^2))
(5)
z=lambda+mu+nu-a^2-b^2.
(6)

The scale factors are

h_lambda=sqrt(((mu-lambda)(nu-lambda))/(4(a^2-lambda)(b^2-lambda)))
(7)
h_mu=sqrt(((nu-mu)(lambda-mu))/(4(a^2-mu)(b^2-mu)))
(8)
h_nu=sqrt(((lambda-nu)(mu-nu))/(16(a^2-nu)(b^2-nu))).
(9)

The Laplacian is

del ^2=(2(a^2+b^2-2nu))/((mu-nu)(nu-lambda))partial/(partialnu)+(4(a^2-nu)(nu-b^2))/((mu-nu)(nu-lambda))(partial^2)/(nu^2)+(2(a^2+b^2-2mu))/((mu-lambda)(nu-mu))partial/(partialmu)+(4(a^2-mu)(mu-b^2))/((mu-lambda)(nu-mu))(partial^2)/(partialmu^2)+(2(2lambda-a^2-b^2))/((mu-lambda)(nu-lambda))partial/(partiallambda)+(4(lambda-a^2)(lambda-b^2))/((mu-lambda)(nu-lambda))(partial^2)/(partiallambda^2).
(10)

The Helmholtz differential equation is separable.

See also

Helmholtz Differential Equation--Confocal Paraboloidal Coordinates, Paraboloidal Coordinates

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References

Arfken, G. "Confocal Parabolic Coordinates (xi_1, xi_2, xi_3)." §2.17 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 119-120, 1970.Moon, P. and Spencer, D. E. "Paraboloidal Coordinates (mu,nu,lambda)." Table 1.11 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 44-48, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 664, 1953.

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Confocal Paraboloidal Coordinates

Cite this as:

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

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