# Extensive Definition

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(u,v) = \cos \theta \,\sinh v \,\sin u + \sin
\theta \,\cosh v \,\cos u

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

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

for (u,v) \in (-\pi, \pi] \times (-\infty,
\infty), with deformation parameter -\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,

## References

catenoid in French: Caténoïde

catenoid in Italian: Catenoide

catenoid in Polish: Katenoida

catenoid in Russian: Катеноид

catenoid in Chinese: 懸鏈曲面