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 = u_{0}, v = v_{0},
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.


