# Catenoid

A catenoid
A catenoid obtained from the rotation of a catenary

A catenoid is a surface in 3-dimensional Euclidean space arising by rotating a catenary curve about its directrix (see animation on the right). Not counting the plane, it is the first non-trivial minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744.[1]

Early work on the subject was published also by Jean Baptiste Meusnier.[2] There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[3]

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 circles 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

Animation showing the deformation of a helicoid into a catenoid.

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.

## Architecture

Further information: Igloo § Engineering
design of building igloos spiral

The Inuit learned to develop their igloo structures by implementing a catenary of revolution shape which offers optimal balance between height and diameter of the structure without risk of collapsing under the weight of compacted snow.[4] This is slightly different from what is typically called a catenoid in that the catenary is rotated about its center, forming a surface with the topology of a bowl rather than that of a cylinder.

## References

1. ^ Eulero, Leonhardo (1952) [1744]. Carathëodory, ed. Methodus inveniendi lineas curvas : maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti [A method for finding the property of maxima or minima of curves and, as the sense of receiving, or Solution of isoperimetric problems in the broadest sense] (in Latin). 24 (MDCCXLIV ed.). Bernae: Orell Füssli Turici. ISBN 9783764314248. Archived from the original on archivedate. Retrieved 1 August 2015. Check date values in: |archive-date= (help)
2. ^ 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.
3. ^ Catenoid at MathWorld
4. ^ Handy, Richard L. (Dec 1973). "The Igloo and the Natural Bridge as Ultimate Structures" (PDF). Arctic. Arctic Institute of North America. 26 (4): 276–277. doi:10.14430/arctic2926.