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Catenoid

[kat-n-oid]
A catenoid is a three-dimensional shape made by rotating a catenary curve around the $x$ axis. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Meusnier. There are only two surfaces of revolution which are also minimal surfaces: the plane and the catenoid.

A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.

One can bend a catenoid into the shape of a helicoid without stretching. In other words, one can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal. A parametrization of such a deformation is given by the system

$x\left(u,v\right) = cos theta ,sinh v ,sin u + sin theta ,cosh v ,cos u$

$y\left(u,v\right) = -cos theta ,sinh v ,cos u + sin theta ,cosh v ,sin u$

$z\left(u,v\right) = u cos theta + v sin theta ,$

for $\left(u,v\right) in \left(-pi, pi\right] times \left(-infty, infty\right)$, with deformation parameter $-pi < theta le pi$,

where $theta = pi$ corresponds to a right-handed helicoid, $theta = pm pi / 2$ corresponds to a catenoid, $theta = pm pi$ corresponds to a left-handed helicoid,

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