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Disk-Cyclide Coordinates

Disk-CyclideCoordinates

A coordinate system defined by the transformation equations

x=a/Lambdacnmucnnucospsi
(1)
y=a/Lambdacnmucnnusinpsi
(2)
z=a/Lambdasnmudnmusnnudnnu,
(3)

where

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

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

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

surfaces of constant nu by the disk cyclides

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

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(7)

See also

Cap-Cyclide Coordinates, Cyclidic Coordinates, Flat-Ring Cyclide Coordinates

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References

Moon, P. and Spencer, D. E. "Disk-Cyclide Coordinates (mu,nu,psi)." Fig. 4.10 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 129-132, 1988.

Referenced on Wolfram|Alpha

Disk-Cyclide Coordinates

Cite this as:

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

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