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Bispherical Coordinates

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BisphericalCoordinates
BisphericalCoordinates3D

A system of curvilinear coordinates variously denoted (xi,eta,phi) (Arfken 1970) or (theta,eta,psi) (Moon and Spencer 1988). Using the notation of Arfken, the bispherical coordinates are defined by

x=(asinxicosphi)/(cosheta-cosxi)
(1)
y=(asinxisinphi)/(cosheta-cosxi)
(2)
z=(asinheta)/(cosheta-cosxi).
(3)

Surfaces of constant eta are given by the spheres

x^2+y^2+(z-acotheta)^2=(a^2)/(sinh^2eta),
(4)

surfaces of constant xi by apple surfaces (xi<pi/2) or lemon surfaces (xi>pi/2)

x^2+y^2+z^2-2asqrt(x^2+y^2)cotxi=a^2,
(5)

and surface of constant psi by the half-planes

tanphi=y/x.
(6)

The scale factors are

h_xi=a/(cosheta-cosxi)
(7)
h_eta=a/(cosheta-cosxi)
(8)
h_phi=(asinxi)/(cosheta-cosxi).
(9)

The Laplacian is given by

del ^2f=((cosheta-cosxi)^3)/(a^2sinxi){sinxipartial/(partialeta)(1/(cosheta-cosxi)(partialf)/(partialeta))+partial/(partialxi)((sinxi)/(cosheta-cosxi)(partialf)/(partialxi))}+((cosheta-cosxi)^2)/(a^2sin^2xi)(partial^2f)/(partialphi^2).
(10)

In bispherical coordinates, Laplace's equation is separable (Moon and Spencer 1988), but the Helmholtz differential equation is not.

See also

Bicyclide Coordinates, Laplace's Equation--Bispherical Coordinates, Spherical Coordinates, Toroidal Coordinates

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References

Arfken, G. "Bispherical Coordinates (xi,eta,phi)." §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.Moon, P. and Spencer, D. E. "Bispherical Coordinates (eta,theta,psi)." Fig. 4.03 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 110-112, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.

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Bispherical Coordinates

Cite this as:

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

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