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Cap-Cyclide Coordinates

Cap-CyclideCoordinates

A coordinate system obtained by inversion of the bicyclide coordinates. They are given by the transformation equations

x=Lambda/(aUpsilon)snmudnnucospsi
(1)
y=Lambda/(aUpsilon)snmudnnusinpsi
(2)
z=(sqrt(k)Pi)/(2aUpsilon),
(3)

where

Lambda=1-dn^2musn^2nu
(4)
Upsilon=sn^2mudn^2nu+[Lambda/(sqrt(k))+cnmudnmusnnucnnu]^2
(5)
Pi=(Lambda^2)/k-(sn^2mudn^2nu+cn^2mudn^2musn^2nucn^2nu),
(6)

and cnx, dnx, and snx are Jacobi elliptic functions. Surfaces of constant mu are ring cyclides with complicated equations (Moon and Spencer 1988, p. 133), surfaces of constant nu are cap-cyclides with complicated equations (Moon and Spencer 1988, p. 133), and surfaces of constant psi are half-planes

tanpsi=y/x.
(7)

See also

Bicyclide Coordinates, Cyclidic Coordinates, Disk-Cyclide Coordinates, Flat-Ring Cyclide Coordinates

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References

Moon, P. and Spencer, D. E. "Cap-Cyclide Coordinates (mu,nu,psi)." Fig. 4.11 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 132-135, 1988.

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Cap-Cyclide Coordinates

Cite this as:

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

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