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Prolate Spheroidal Coordinates

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ProlateSpheroidalCoords
ProlateSpheroidalCoords3D

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 x-axis, which is relabeled the z-axis. The third set of coordinates consists of planes passing through this axis.

x=asinhxisinetacosphi
(1)
y=asinhxisinetasinphi
(2)
z=acoshxicoseta,
(3)

where xi in [0,infty), eta in [0,pi], and phi in [0,2pi). Note that several conventions are in common use; Arfken (1970) uses (u,v,phi) instead of (xi,eta,phi), and Moon and Spencer (1988, p. 28) use (eta,theta,psi).

In this coordinate system, the scale factors are

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

The Laplacian is

del ^2f=1/(sinetasinhxi(sin^2eta+sinh^2xi)){partial/(partialxi)(sinetasinhxi(partialf)/(partialxi))+partial/(partialeta)(sinetasinhxi(partialf)/(partialeta))+partial/(partialphi)[(cschxisineta+cscetasinhxi)(partialf)/(partialphi)]}.
(7)
=1/(sin^2eta+sinh^2xi)[(csc^2eta+csch^2xi)(partial^2f)/(partialphi^2)+coteta(partialf)/(partialeta)+(partial^2f)/(partialeta^2)+cothxi(partialf)/(partialxi)+(partial^2f)/(partialxi^2)]
(8)

An alternate form useful for "two-center" problems is defined by

xi_1=coshxi
(9)
xi_2=coseta
(10)
xi_3=phi,
(11)

where xi_1 in [1,infty], xi_2 in [-1,1], and xi_3 in [0,2pi) (Abramowitz and Stegun 1972). In these coordinates,

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

In terms of the distances from the two foci,

xi_1=(r_1+r_2)/(2a)
(15)
xi_2=(r_1-r_2)/(2a)
(16)
2a=r_(12).
(17)

The scale factors are

h_(xi_1)=asqrt((xi_1^2-xi_2^2)/(xi_1^2-1))
(18)
h_(xi_2)=asqrt((xi_1^2-xi_2^2)/(1-xi_2^2))
(19)
h_(xi_3)=asqrt((xi_1^2-1)(1-xi_2^2)),
(20)

and the Laplacian is

del ^2f=1/(a^2){1/(xi_1^2-xi_2^2)partial/(partialxi_1)[(xi_1^2-1)(partialf)/(partialxi_1)]+1/(xi_1^2-xi_2^2)partial/(partialxi_2)[(1-xi_2^2)(partialf)/(partialxi_2)]+1/((xi_1^2-1)(1-xi_2^2))(partial^2f)/(partialxi_3^2)}.
(21)

The Helmholtz differential equation is separable in prolate spheroidal coordinates.

See also

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

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Definition of Prolate 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)." §2.10 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107, 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. 243-244, 1959.Moon, P. and Spencer, D. E. "Prolate Spheroidal Coordinates (eta,theta,psi)." Table 1.06 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 28-30, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.Wrinch, D. M. "Inverted Prolate Spheroids." Philos. Mag. 280, 1061-1070, 1932.

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Prolate Spheroidal Coordinates

Cite this as:

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

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