# 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:

- where and and is a non-zero real constant.

In cylindrical coordinates:

- where 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

- for , with deformation parameter ,

where corresponds to a right-handed helicoid, corresponds to a catenoid, and corresponds to a left-handed helicoid.

## References

- Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010).
*Minimal Surfaces*. Springer Science & Business Media. p. 141. ISBN 9783642116988. - Gullberg, Jan (1997).
*Mathematics: From the Birth of Numbers*. W. W. Norton & Company. p. 538. ISBN 9780393040029. - 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. - 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. - 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. - "Catenoid".
*Wolfram MathWorld*. Retrieved 15 January 2017.

## Further reading

- Krivoshapko, Sergey; Ivanov, V. N. (2015). "Minimal Surfaces".
*Encyclopedia of Analytical Surfaces*. Springer. ISBN 9783319117737.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Catenoid",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Catenoid - WebGL model
- Euler's text describing the catenoid at Carnegie Mellon University