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Oblate Spheroidal Coordinates

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OblateSpheroidalCoords3D
OblateSpheroidalCoordinate

Oblate spheroidal coordinates are a system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the y-axis which is relabeled the z-axis. The third set of coordinates consists of planes passing through this axis.

x=acoshxicosetacosphi
(1)
y=acoshxicosetasinphi
(2)
z=asinhxisineta,
(3)

where a>0, xi in [0,infty), eta in [-pi/2,pi/2], and phi in [0,2pi). Arfken (1970) uses (u,v,phi) instead of (xi,eta,phi). The scale factors are

h_xi=asqrt(sinh^2xi+sin^2eta)
(4)
h_eta=asqrt(sinh^2xi+sin^2eta)
(5)
h_phi=acoshxicoseta.
(6)

The Laplacian is

del ^2f=1/(a^2M)[1/(coshxi)partial/(partialxi)(coshxi(partialf)/(partialxi))+1/(coseta)partial/(partialeta)(coseta(partialf)/(partialeta))]+1/(a^2cosh^2xicos^2eta)(partial^2f)/(partialphi^2)
(7)
=1/(a^2M)((partial^2f)/(partialxi^2)+tanhxi(partialf)/(partialxi)+(partial^2f)/(partialeta^2)-taneta(partialf)/(partialeta))+1/(a^2cosh^2xicos^2eta)(partial^2f)/(partialphi^2),
(8)

where M=sinh^2xi+sin^2eta. These expressions apply away from the coordinate singularities, with xi>0 and -pi/2<eta<pi/2.

An alternate form uses the coordinates

xi_1=sinhxi
(9)
xi_2=sineta
(10)
xi_3=phi,
(11)

where xi_1 in [0,infty), xi_2 in [-1,1], and xi_3 in [0,2pi). The corresponding Cartesian coordinates are

x=asqrt((1+xi_1^2)(1-xi_2^2))cosxi_3
(12)
y=asqrt((1+xi_1^2)(1-xi_2^2))sinxi_3
(13)
z=axi_1xi_2,
(14)

with scale factors

h_(xi_1)=asqrt((xi_1^2+xi_2^2)/(1+xi_1^2))
(15)
h_(xi_2)=asqrt((xi_1^2+xi_2^2)/(1-xi_2^2))
(16)
h_(xi_3)=asqrt((1+xi_1^2)(1-xi_2^2)).
(17)

Away from the coordinate singularities, the Laplacian is

del ^2f=1/(a^2(xi_1^2+xi_2^2))[partial/(partialxi_1)((1+xi_1^2)(partialf)/(partialxi_1))+partial/(partialxi_2)((1-xi_2^2)(partialf)/(partialxi_2))]+1/(a^2(1+xi_1^2)(1-xi_2^2))(partial^2f)/(partialxi_3^2).
(18)

The Helmholtz differential equation is separable.

See also

Helmholtz Differential Equation--Oblate Spheroidal Coordinates, Latitude, Longitude, Prolate Spheroidal Coordinates, Spherical Coordinates

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Definition of Oblate Spheroidal Coordinates." §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.Arfken, G. "Prolate Spheroidal Coordinates (u, v, phi)." §2.11 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107-109, 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, p. 242, 1959.Moon, P. and Spencer, D. E. "Oblate Spheroidal Coordinates (eta,theta,psi)." Table 1.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 31-34, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.National Institute of Standards and Technology. "Wave Equation in Oblate Spheroidal Coordinates." §30.14 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/30.14.

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Oblate Spheroidal Coordinates

Cite this as:

Weisstein, Eric W. "Oblate Spheroidal Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OblateSpheroidalCoordinates.html

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