Catenoid

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three-dimensional diagram of a catenoid
A catenoid
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

A catenoid is a type of surface in topology, 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.

The dome shape of Inuit igloos can be derived from rotation of a catenary about its central axis.

Geometry[edit]

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][non-primary source needed]

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

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[edit]

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

for , with deformation parameter ,

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

Architecture[edit]

Further information: Igloo § Engineering
cutaway diagram showing snow blocks in an igloo placed in an ascending spiral
spiral sequence of snow blocks in igloo construction

The Inuit learned to pattern the structure of their igloos, or snow houses, after a shape with a catenary arch cross-section, which offers an optimal balance between height and diameter, avoiding the risk of collapsing under the weight of compacted snow.[6] This differs from what is normally called a catenoid in that the catenary is rotated about its central axis, forming a surface with the topology of a bowl rather than that of a cylinder.

References[edit]

  1. ^ Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Springer Science & Business Media. p. 141. ISBN 9783642116988. 
  2. ^ a b c 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. ^ 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. 
  5. ^ "Catenoid". Wolfram MathWorld. Retrieved 15 January 2017. 
  6. ^ Handy, Richard L. (Dec 1973). "The Igloo and the Natural Bridge as Ultimate Structures" (PDF). Arctic. Arctic Institute of North America. 26 (4): 276–281. doi:10.14430/arctic2926. The Eskimo snow igloo is not a hemisphere as frequently depicted, but a catenoid of revolution with an optimum height-to-diameter ratio. This shape eliminates ring tension and shell moments and therefore prevents failure by caving or bulging. 

Further reading[edit]

External links[edit]