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Bicyclide Coordinates

Bi-CyclideCoordinates

A coordinate system which is similar to bispherical coordinates but having fourth-degree surfaces instead of second-degree surfaces for constant mu. The coordinates are given by the transformation equations

x=a/Lambdacnmudnmusnnucnnucospsi
(1)
y=a/Lambdacnmudnmusnnucnnusinpsi
(2)
z=a/Lambdasnmudnnu,
(3)

where

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

mu in [0,K], nu in [0,K^'], psi in [0,2pi), and cnx, dnx, and snx are Jacobi elliptic functions. Surfaces of constant mu are given by the bicyclides

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

surfaces of constant nu by the cyclides of rotation

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

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(7)

See also

Bispherical Coordinates, Cap-Cyclide Coordinates, Cyclidic Coordinates

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References

Moon, P. and Spencer, D. E. "Bicyclide Coordinates (mu,nu,psi)." Fig. 4.08 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 124-126, 1988.

Referenced on Wolfram|Alpha

Bicyclide Coordinates

Cite this as:

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

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