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:
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Infinitesimal Area:The infinitesimal area of a patch on the surface is given by |
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Gaussian Curvature:
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Mean Curvature:
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