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:

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 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

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.

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

↑ 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)
↑ 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 at MathWorld
↑ 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.

External links

Hazewinkel, Michiel, ed. (2001), "Catenoid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Catenoid - WebGL model

Minimal surfaces

Associate family Bour's Catalan's Catenoid Chen–Gackstatter Costa's Enneper Gyroid Helicoid Henneberg K-noid Lidinoid Neovius Richmond Riemann's Saddle tower Scherk Schwarz Triply periodic

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