Enneper's Minimal Surface
Date: 10/22/2001 at 22:49:59 From: Robert Vermillion Subject: Enneper Minimal Surface What is an Enneper Minimal Surface? Everywhere that I've seen talks about what they look like, but so far I have not located any plans for generating an EMS. Why is it minimal as opposed to maximum? What quantity of the surface is minimal or how can one recognize a given curve as an EMS? Could you assist? Thank you kindly, sir.
Date: 10/23/2001 at 00:18:32 From: Doctor Pete Subject: Re: Enneper Minimal Surface Hi, Enneper's minimal surface is defined in 3-dimensional Cartesian coordinates by the equations x = u - (u^3)/3 + uv^2 y = v - (v^3)/3 + vu^2 z = u^2 - v^2, for real u, v. That is, the point (u,v) is mapped to (x(u,v),y(u,v),z(u,v)), which is a point on Enneper's minimal surface. While this surface has self-intersections in 3-space, it is minimal because it has the property of minimizing surface area. This notion can be made more precise by saying that the surface satisfies a set of differential equations, but another way of describing minimal surfaces is that their mean curvatures at every point is zero. To understand the concept of mean curvature, we begin by describing the curvatures of a surface at a point. Consider a vector normal to the surface at the given point; this vector is perpendicular to the tangent plane at that point. If you look at the 2-d curve formed by the intersection of any plane containing the normal vector, it has a curvature at the given point. Depending on the particular plane you choose, the curvature may be larger or smaller; the principal curvatures are the maximum and minimum curvatures thus obtainable. The mean curvature is the arithmetic mean (average) of the two principal curvatures. Geometrically speaking, the surface is "saddle-shaped" at that point. If, at every point on the surface, the mean curvature is zero, then the surface is minimal. Minimal surfaces are sometimes called "soap-bubble films," in the sense that these surfaces minimize their surface area for a given boundary. If you imagine a loop of wire shaped into a circle, then the minimal surface that spans the wire is a disk, which is a subset of the plane, which is a minimal surface. If you take two such loops, parallel to each other, and dip them into a soap solution, you may obtain a catenoid, which is another example of a minimal surface. Other embedded minimal surfaces include helicoids, Scherk surfaces, and more recently the Costa-Hoffman-Meeks family of minimal surfaces. If you're interested in minimal surfaces, I recommend a textbook on differential geometry. A background in multivariable calculus would be essential in order to discuss relevant concepts which include curvature of a surface at a point, the Gauss map and global curvature. - Doctor Pete, The Math Forum http://mathforum.org/dr.math/
Date: 10/23/2001 at 02:17:42 From: Robert Vermillion Subject: Re: Enneper Minimal Surface Thanks for shedding informative light upon the hitherto dark subject of the Enneper Minimal Surface. Your explanation was pleasantly concise, erudite, comprehensible, and satisfyingly explanatory. When I have another such question, I'll come to the same place. :) Robert Vermillion
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