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Concept
Enneper surface
Summary
In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by:
It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory.
The Weierstrass–Enneper parameterization is very simple, , and the real parametric form can easily be calculated from it. The surface is conjugate 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 where
Its coefficients satisfy the implicit degree-6 polynomial equation
The Jacobian, Gaussian curvature and mean curvature are
The total curvature is . Osserman proved that a complete minimal surface in with total curvature is either the catenoid or the Enneper surface.
Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.
It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization for integer k>1. It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in for n up to 7.
See also
for higher order algebraic Enneper surfaces.
Official source
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Catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler. Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa.
Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.