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)

References

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.