Catenoid

Surface of revolution of a catenary

animation of a catenary sweeping out the shape of a catenoid as it rotates about a central point — A catenoid obtained from the rotation of a catenary

In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution).^[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: ${\begin{aligned}x&=c\cosh {\frac {v}{c}}\cos u\\y&=c\cosh {\frac {v}{c}}\sin u\\z&=v\end{aligned}}$ 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

Continuous animation showing a helicoid deforming into a catenoid and back to a helicoid — 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 ${\begin{aligned}x(u,v)&=\cos \theta \,\sinh v\,\sin u+\sin \theta \,\cosh v\,\cos u\\y(u,v)&=-\cos \theta \,\sinh v\,\cos u+\sin \theta \,\cosh v\,\sin u\\z(u,v)&=u\cos \theta +v\sin \theta \end{aligned}}$ 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.

References

^ Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Springer Science & Business Media. p. 141. ISBN 9783642116988.
^ ^a ^b ^c 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b 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. Bibcode:2006PNAS..10311106C. doi:10.1073/pnas.0510379103. PMC 1544050. PMID 16847265.
^ 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.