Enneper's Minimal Surface

A self-intersecting Minimal Surface having zero Mean Curvature and nonconstant Gaussian Curvature. Enneper's minimal surface can be generated using the Enneper-Weierstraß Parameterization with

$\displaystyle f(z)$	$\textstyle =$	$\displaystyle 1$	(1)
$\displaystyle g(z)$	$\textstyle =$	$\displaystyle \zeta.$	(2)

Letting $z=re^{i\phi}$ and taking the Real Part give

$\displaystyle x$	$\textstyle =$	$\displaystyle \Re[r e^{i\phi} - {\textstyle{1\over 3}} r^3 e^{3i\phi}]$	(3)
$\displaystyle y$	$\textstyle =$	$\displaystyle \Re[ir e^{i\phi} + {\textstyle{1\over 3}}ir^3 e^{3i\phi}]$	(4)
$\displaystyle z$	$\textstyle =$	$\displaystyle \Re[r^2 e^{2i\phi}],$	(5)

where $r\in [0,1]$ and $\phi\in[-\pi,\pi)$ . Letting

gives the figure above, with parametrization

$\displaystyle x$	$\textstyle =$	$\displaystyle u-{\textstyle{1\over 3}}u^3+uv^2$	(6)
$\displaystyle y$	$\textstyle =$	$\displaystyle -v-u^2v+{\textstyle{1\over 3}} v^3$	(7)
$\displaystyle z$	$\textstyle =$	$\displaystyle u^2-v^2$	(8)

(do Carmo 1986, Gray 1997, Nordstrand).

References

Dickson, S. ``Minimal Surfaces.'' Mathematica J. 1, 38-40, 1990.

do Carmo, M. P. ``Enneper's Surface.'' §3.5C in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.

Enneper, A. ``Analytisch-geometrische Untersuchungen.'' Z. Math. Phys. 9, 96-125, 1864.

Gray, A. ``Examples of Minimal Surfaces.'' §30.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 358 and 684-685, 1997.

Maeder, R. The Mathematica Programmer. San Diego, CA: Academic Press, pp. 150-151, 1994.

Nordstrand, T. ``Enneper's Minimal Surface.'' http://www.uib.no/people/nfytn/enntxt.htm.