Enneper surface

 ( Algebraic Surfaces ) 

In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parameter by: $$\backslash begin\{align\}$$

x &= \tfrac{1}{3} u \left(1 - \tfrac{1}{3}u^2 + v^2\right), \\
y &= \tfrac{1}{3} v \left(1 - \tfrac{1}{3}v^2 + u^2\right), \\
z & = \tfrac{1}{3} \left(u^2 - v^2\right). \end{align}
     

It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory.J.C.C. Nitsche, "Vorlesungen über Minimalflächen", Springer (1975) Francisco J. López, Francisco Martín, Complete minimal surfaces in R3Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. .

The Weierstrass–Enneper parameterization is very simple, $f(z)=1,\; g(z)=z$, and the real parametric form can easily be calculated from it. The surface is Associate family to itself.

Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation

Dually, the tangent plane at the point with given parameters is $a\; +\; b\; x\; +\; c\; y\; +\; d\; z\; =\; 0,\backslash$ where Its coefficients satisfy the implicit degree-6 polynomial equation

The Jacobian, Gaussian curvature and mean curvature are $$\backslash begin\{align\}$$

J &= \frac{1}{81}(1 + u^2 + v^2)^4, \\
K &= -\frac{4}{9}\frac{1}{J}, \\
H &= 0.
     

\end{align} The total curvature is $-4\backslash pi$. Robert Osserman proved that a complete minimal surface in $\backslash R^3$ with total curvature $-4\backslash pi$ is either the catenoid or the Enneper surface.R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).

The curves obtained by setting $u=0$ or $v=0$ are of the Enneper surface, and are copies of the Tschirnhausen cubic.

Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.Cosín, C., Monterde, Bézier surfaces of minimal area. In Computational Science — ICCS 2002, eds. J., Sloot, Peter, Hoekstra, Alfons, Tan, C., Dongarra, Jack. Lecture Notes in Computer Science 2330, Springer Berlin / Heidelberg, 2002. pp. 72-81

It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization $f(z)\; =\; 1,\; g(z)\; =\; z^k$ for integer k>1. It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in $\backslash R^n$ for n up to 7.Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569

/ref> /ref> for higher order algebraic Enneper surfaces.

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External links

• Mathematical Sciences Research Institute - The Generalized Enneper's Surfaces (Archived)
• target="_blank" rel="nofollow"> Harvey Mudd College - Enneper's Surface (Archived)

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