# Catenoid

A catenoid is a type of surface, arising by rotating a catenary curve about an axis.[1] It is a minimal surface, meaning that it occupies the least area when bounded by a closed space.[2] 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.[2] 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.[2] It was found and proved to be minimal by Leonhard Euler in 1744.[3][4]

Early work on the subject was published also by Jean Baptiste Meusnier.[5][4]:11106 There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[6]

The catenoid may be defined by the following parametric equations:

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

In cylindrical coordinates:

${\displaystyle \rho =c\cosh {\frac {z}{c}}}$
where ${\displaystyle c}$ 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

${\displaystyle x(u,v)=\cos \theta \,\sinh v\,\sin u+\sin \theta \,\cosh v\,\cos u}$
${\displaystyle y(u,v)=-\cos \theta \,\sinh v\,\cos u+\sin \theta \,\cosh v\,\sin u}$
${\displaystyle z(u,v)=u\cos \theta +v\sin \theta }$
for ${\displaystyle (u,v)\in (-\pi ,\pi ]\times (-\infty ,\infty )}$, with deformation parameter ${\displaystyle -\pi <\theta \leq \pi }$,

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

## References

1. Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Springer Science & Business Media. p. 141. ISBN 9783642116988.
2. Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W. W. Norton & Company. p. 538. ISBN 9780393040029.
3. Helveticae, Euler, Leonhard (1952) [reprint of 1744 edition]. Carathëodory Constantin (ed.). Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti (in Latin). Springer Science & Business Media. ISBN 3-76431-424-9.
4. Colding, T. H.; Minicozzi, W. P. (17 July 2006). "Shapes of embedded minimal surfaces". Proceedings of the National Academy of Sciences. 103 (30): 11106–11111. doi:10.1073/pnas.0510379103. PMC 1544050.
5. Meusnier, J. B (1881). Mémoire sur la courbure des surfaces [Memory on the curvature of surfaces.] (PDF) (in French). Bruxelles: F. Hayez, Imprimeur De L'Acdemie Royale De Belgique. pp. 477–510. ISBN 9781147341744.
6. "Catenoid". Wolfram MathWorld. Retrieved 15 January 2017.
• Krivoshapko, Sergey; Ivanov, V. N. (2015). "Minimal Surfaces". Encyclopedia of Analytical Surfaces. Springer. ISBN 9783319117737.