The Catenoid

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Introduction:

The surface of revolution of a catenary, called a catenoid, has the property that its mean curvature is everywhere zero; we say that it is a minimal surface. Although the catenoid looks substantially like the hyperboloid, there are substantial differences in their values away from the plane z = 0, as well as in their properties. The catenoid also has the fascinating property that it can be deformed into a helicoid in such a way that every surface along the way is a minimal surface which is locally isometric to the helicoid. The animation at right shows this deformation.

Definition:

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.