View source for Catenoid

{{Short description|Surface of revolution of a catenary}}
[[Image:Catenoid.svg|thumb|right|alt=three-dimensional diagram of a catenoid|A catenoid]]
[[Image:Catenoid.gif|thumb|right|alt=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 (mathematics)|surface]], arising by rotating a [[catenary]] curve about an axis (a [[surface of revolution]]).<ref>{{cite book|last1=Dierkes|first1=Ulrich|last2=Hildebrandt|first2=Stefan|last3=Sauvigny|first3=Friedrich|title=Minimal Surfaces|date=2010|publisher=[[Springer Science & Business Media]]|isbn=9783642116988|page=141|url=https://books.google.com/books?id=9YhBOg6vO-EC&pg=PA141|language=en}}</ref> It is a [[minimal surface]], meaning that it occupies the least area when bounded by a closed space.<ref name=Gullberg>{{cite book|last1=Gullberg|first1=Jan|title=Mathematics: From the Birth of Numbers|date=1997|publisher=[[W. W. Norton & Company]]|isbn=9780393040029|page=[https://archive.org/details/mathematicsfromb1997gull/page/538 538]|url=https://archive.org/details/mathematicsfromb1997gull|url-access=registration|language=en}}</ref> 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.<ref name=Gullberg/> 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 (topology)|surface]] in 3-dimensional Euclidean space to be discovered apart from the [[plane (geometry)|plane]]. The catenoid is obtained by rotating a catenary about its [[Directrix (conic section)|directrix]].<ref name=Gullberg/> It was found and proved to be minimal by [[Leonhard Euler]] in 1744.<ref>{{cite book|last1=Helveticae|first1=Euler, Leonhard |editor= Carathëodory Constantin |title=Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti |date=1952 |orig-year=reprint of 1744 edition |publisher=Springer Science & Business Media  |isbn=3-76431-424-9 |language=Latin |url=https://books.google.com/books?id=zNDdVFZalSAC}}</ref><ref name=Colding06>{{cite journal|last1=Colding|first1=T. H.|last2=Minicozzi|first2=W. P.|title=Shapes of embedded minimal surfaces|journal=Proceedings of the National Academy of Sciences|date=17 July 2006|volume=103|issue=30|pages=11106–11111|doi=10.1073/pnas.0510379103|pmc=1544050|pmid=16847265|bibcode=2006PNAS..10311106C|doi-access=free}}</ref>

Early work on the subject was published also by [[Jean Baptiste Meusnier]].<ref name=salvert>{{cite book|url=https://archive.org/details/mmoiresurlathor00salvgoog|format=PDF|last1=Meusnier|first1=J. B|title=Mémoire sur la courbure des surfaces|trans-title=Memory on the curvature of surfaces.|date=1881|publisher=F. Hayez, Imprimeur De L'Acdemie Royale De Belgique|location=Bruxelles|language=French|isbn=9781147341744|pages=477–510}}</ref><ref name=Colding06/>{{rp|11106}} There are only two [[minimal surfaces of revolution]] ([[surfaces of revolution]] which are also minimal surfaces): the [[plane (geometry)|plane]] and the catenoid.<ref>{{cite web|title=Catenoid|url=http://mathworld.wolfram.com/Catenoid.html|website=Wolfram MathWorld|accessdate=15 January 2017|language=en}}</ref>

The catenoid may be defined by the following parametric equations:
<math display=block>\begin{align}
x &= c \cosh \frac{v}{c} \cos u \\
y &= c \cosh \frac{v}{c} \sin u \\
z &= v
\end{align}</math>
where <math>u \in [-\pi, \pi)</math> and <math>v \in \mathbb{R}</math> and <math>c</math> is a non-zero real constant.

In cylindrical coordinates:
<math display=block>\rho =c \cosh \frac{z}{c},</math>
where <math>c</math> is a real constant.

A physical model of a catenoid can be formed by dipping two [[circle|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 ==

[[Image:helicatenoid.gif|thumb|right|256px|alt=Continuous animation showing a right-handed helicoid deforming into a catenoid, a left-handed helicoid, and back again|Deformation of a right-handed [[helicoid]] into a left-handed one and back again via 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 function|continuous]] and [[Isometry|isometric]] deformation of a catenoid to a portion of the [[helicoid]] such that every member of the deformation family is [[Minimal surface|minimal]] (having a [[mean curvature]] of zero).  A [[Parametric equation|parametrization]] of such a deformation is given by the system
<math display=block>\begin{align}
x(u,v) &= \sin \theta \,\cosh v \,\cos u + \cos \theta \,\sinh v \,\sin u \\
y(u,v) &= \sin \theta \,\cosh v \,\sin u - \cos \theta \,\sinh v \,\cos u \\
z(u,v) &=  v \sin \theta + u \cos \theta
\end{align}</math>
for <math>(u,v) \in (-\pi, \pi] \times (-\infty, \infty)</math>, with deformation parameter <math>-\pi < \theta \le \pi</math>, where:
* <math>\theta = \pi</math> corresponds to a right-handed helicoid, 
* <math>\theta = \pm \pi / 2</math> corresponds to a catenoid, and
* <math>\theta = 0</math> corresponds to a left-handed helicoid.

== References ==
{{reflist}}

== Further reading ==
* {{cite book |last1=Krivoshapko |first1=Sergey |last2=Ivanov |first2=V. N. |title=Encyclopedia of Analytical Surfaces |date=2015 |publisher=Springer |isbn=9783319117737 |chapter=Minimal Surfaces |chapter-url=https://books.google.com/books?id=cXTdBgAAQBAJ&pg=PA427 |language=en}}

== External links ==
* {{springer|title=Catenoid|id=p/c020800}}
* [http://www.princeton.edu/~rvdb/WebGL/catenoid.html Catenoid – WebGL model]
* [http://posner.library.cmu.edu/Posner/books/book.cgi?call=517.4_E88M_1744 Euler's text describing the catenoid] at Carnegie Mellon University
* [https://www.youtube.com/watch?v=31Om4VrSzb8 Calculating the surface area of a Catenoid]
* [https://mathworld.wolfram.com/MinimalSurfaceofRevolution.html Minimal Surface of Revolution]

{{Minimal surfaces}}

[[Category:Minimal surfaces]]

[[de:Minimalfläche#Das Katenoid]]