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Henneberg's Minimal Surface

A double algebraic surface of 15th order and fifth class which can be given by parametric equations

$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle 2\sinh u\cos v-{\textstyle{2\over 3}}\sinh(3u)\cos(3v)$ (1)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle 2\sinh u\sin v-{\textstyle{2\over 3}}\sinh(3u)\sin(3v)$ (2)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle 2\cosh(2u)\cos(2v).$ (3)

It can also be obtained from the Enneper-Weierstraß Parameterization with
$\displaystyle f$ $\textstyle =$ $\displaystyle 2-2z^{-4}$ (4)
$\displaystyle g$ $\textstyle =$ $\displaystyle z.$ (5)

See also Minimal Surface


Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, p. 267, 1960.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 446-448, 1993.

Nitsche, J. C. C. Introduction to Minimal Surfaces. Cambridge, England: Cambridge University Press, p. 144, 1989.

© 1996-9 Eric W. Weisstein