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Lichtenfels Minimal Surface

LichtenfelsSurface

A minimal surface that contains lemniscates as geodesics which is given by the parametric equations

x=R[sqrt(2)cos(1/3zeta)sqrt(cos(2/3zeta))]
(1)
y=R[-sqrt(2)sin(1/3zeta)sqrt(cos(2/3zeta))]
(2)
z=R[-1/3sqrt(2)iint_0^zeta(dzeta)/(sqrt(cos(2/3zeta)))]
(3)
=R[-isqrt(2)F(sqrt(1/3zeta),2)],
(4)

where F(x,x) is an incomplete elliptic integral of the first kind and zeta=u+iv is a complex number. A given lemniscate is the intersection of the surface with the xy-plane. The surface is periodic in the direction of the axis with period

omega=2int_0^1(dt)/(sqrt(1-t^2)sqrt(1-1/2t^2))=2K(1/2),
(5)

where K(x) is a complete elliptic integral of the first kind.

See also

Lemniscate, Minimal Surface

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References

do Carmo, M. P. "Minimal Surfaces with a Lemniscate as a Geodesic." §3.5F in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 47, 1986.Lichtenfels, O. von. "Notiz über eine transcendente Minimalfläche." Sitzungsber. Kaiserl. Akad. Wiss. Wien 94, 41-54, 1889.

Cite this as:

Weisstein, Eric W. "Lichtenfels Minimal Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LichtenfelsMinimalSurface.html

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