Catenoid

A catenoid is a type of surface, arising by rotating a catenary curve about an axis. It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler.

Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa.

Geometry

The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. It was found and proved to be minimal by Leonhard Euler in 1744.

Early work on the subject was published also by Jean Baptiste Meusnier.: 11106  There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.

The catenoid may be defined by the following parametric equations:

$x=c\cosh {\frac {v}{c}}\cos u$
$y=c\cosh {\frac {v}{c}}\sin u$
$z=v$
where $u\in [-\pi ,\pi )$  and $v\in \mathbb {R}$  and $c$  is a non-zero real constant.

In cylindrical coordinates:

$\rho =c\cosh {\frac {z}{c}}$
where $c$  is a real constant.

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

The catenoid may be also defined approximately by the Stretched grid method as a facet 3D model.

Helicoid transformation

Because they are members of the same associate family of surfaces, one can bend a catenoid into a portion of a helicoid without stretching. In other words, one can make a (mostly) continuous and isometric deformation of a catenoid to a portion of the helicoid such that every member of the deformation family is minimal (having a mean curvature of zero). 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 $-\pi <\theta \leq \pi$ ,

where $\theta =\pi$  corresponds to a right-handed helicoid, $\theta =\pm \pi /2$  corresponds to a catenoid, and $\theta =0$  corresponds to a left-handed helicoid.