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Inverse Prolate Spheroidal Coordinates

InverseProlateSpheroidal

A system of coordinates obtained by inversion of the prolate spheroids and two-sheeted hyperboloids in prolate spheroidal coordinates. The inverse prolate spheroidal coordinates (eta,theta,psi) are given by the transformation equations

x=(asinhetasinthetacospsi)/(cosh^2eta-sin^2theta)
(1)
y=(asinhetasinthetasinpsi)/(cosh^2eta-sin^2theta)
(2)
z=(acoshetacostheta)/(cosh^2eta-sin^2theta),
(3)

with eta>=0, theta in [0,pi], and psi in [0,2pi). Surfaces of constant eta are given by the cyclides of rotation

x^2+y^2+z^2=asqrt((x^2+y^2)/(sinh^2eta)+(z^2)/(cosh^2eta)),
(4)

surfaces of constant theta by the cyclides of rotation

x^2+y^2+z^2=asqrt(-(x^2+y^2)/(sin^2theta)+(z^2)/(cos^2theta)),
(5)

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(6)

The metric coefficients are given by

g_(etaeta)=(a^2(sinh^2eta+sin^2theta))/((cosh^2eta-sin^2theta)^2)
(7)
g_(thetatheta)=(a^2(sinh^2eta+sin^2theta))/((cosh^2eta-sin^2theta)^2)
(8)
g_(psipsi)=(a^2sinh^2etasin^2theta)/((cosh^2eta-sin^2theta)^2).
(9)

See also

Inverse Oblate Spheroidal Coordinates, Oblate Spheroidal Coordinates

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References

Moon, P. and Spencer, D. E. "Inverse Prolate Spheroidal Coordinate (eta,theta,psi)." Fig.4.05 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 115-118, 1988.

Referenced on Wolfram|Alpha

Inverse Prolate Spheroidal Coordinates

Cite this as:

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

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