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Flat-Ring Cyclide Coordinates

Flat-RingCyclideCoords

A coordinate system similar to toroidal coordinates but with fourth-degree instead of second-degree surfaces for constant mu so that the toroids of circular cross section are replaced by flattened rings, and the spherical bowls are replaced by cyclides of rotation for constant nu. The transformation equations are

x=a/Lambdasnmudnnucospsi
(1)
y=a/Lambdasnmudnnusinpsi
(2)
z=a/Lambdacnmudnmusnnucnnu,
(3)

where

Lambda=1-dn^2musn^2nu
(4)

and with mu in [0,K], nu in [0,K^'], and psi in [0,2pi). Surfaces of constant mu are given by the flat-ring cyclides

(x^2+y^2+z^2)^2+(a^2)/(k^4)((1-k^2)^2-2(1-k^2)dn^2mu+(1+k^2)dn^4mu)/(dn^2mucn^2mu)z^2-a^2(sn^2mu+1/(sn^2mu))(x^2+y^2)+(a^4)/(k^2)=0,
(5)

surfaces of constant nu by the cyclides of rotation

[(dn^2nu)/(a^2)(x^2+y^2)+(cn^2nu)/(a^2sn^2nu)z^2]^2-(2cn^2nu)/(a^2sn^2nu)z^2-(2dn^2nu)/(a^2)(x^2+y^2)+1=0,
(6)

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(7)

See also

Cyclidic Coordinates, Toroidal Coordinates

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References

Moon, P. and Spencer, D. E. "Flat-Ring Cyclide Coordinates (mu,nu,psi)." Fig. 4.09 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 126-129, 1988.

Referenced on Wolfram|Alpha

Flat-Ring Cyclide Coordinates

Cite this as:

Weisstein, Eric W. "Flat-Ring Cyclide Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Flat-RingCyclideCoordinates.html

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