Enneper's Surface

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Introduction:

Enneper's surface is a well-known minimal surface. Though it has a fairly uncomplicated parameterization (first equations below), it is somewhat hard to visualize because of its self-intersections. The plot above suggests the self-intersections exhibited by the surface, but the plot range has been kept small enough that the structure of the surface's center is also visible. Note that the self-intersection curves are subsets of the planes y = 0 and x = 0. The surface above is a special case of the more general Enneper's surface of degree n (second equations below). These surfaces tend to be even more complicated and difficult to visualize. Below is an animation for the case n = 2 of Enneper's surface of degree n, as radius increases. Note how the self-intersections become more complicated as the surface grows.

Definition:

Enneper's minimal surface is parameterized by:

The more general Enneper's surface of degree n is parameterized in polar coordinates by:

Note: The exponent on r is 1+2n.

Properties:

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

Mean curvature of the surface.
Surface colored by Mean curvature.