TOPICS
Search

Inverse Oblate Spheroidal Coordinates

InverseOblateSpheroidal

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

x=(acoshetasinthetacospsi)/(cosh^2eta-cos^2theta)
(1)
y=(acoshetasinthetasinpsi)/(cosh^2eta-cos^2theta)
(2)
z=(asinhetacostheta)/(cos^2eta-cos^2theta),
(3)

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

See also

Inverse Prolate Spheroidal Coordinates, Prolate Spheroidal Coordinates

Explore with Wolfram|Alpha

References

Moon, P. and Spencer, D. E. "Inverse Oblate Spheroidal Coordinate (eta,theta,psi)." Fig. 4.06 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 119-121, 1988.

Referenced on Wolfram|Alpha

Inverse Oblate Spheroidal Coordinates

Cite this as:

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

Subject classifications