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[[Image:Catenoid.svg|thumb|right|A catenoid]]
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A '''catenoid''' is a [[surface]] in 3-dimensional Euclidean space arising by rotating a [[catenary]] [[curve]] about its directrix. Not counting the plane, it is the first [[minimal surface]] to be discovered. It was found and proved to be minimal by [[Leonhard Euler]] in 1744.<ref>L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, in: Opera omnia I, 24</ref> Early work on the subject was published also by [[Jean Baptiste Meusnier]].<ref>Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785</ref> There are only two [[minimal surfaces of revolution]] ([[surfaces of revolution]] which are also minimal surfaces): the [[plane (geometry)|plane]] and the catenoid.<ref>[http://mathworld.wolfram.com/Catenoid.html Catenoid at MathWorld]</ref>
 
The catenoid may be defined by the following parametric equations:
 
:<math>x=c \cosh \frac{v}{c} \cos u</math>
 
:<math>y=c \cosh \frac{v}{c} \sin u</math>
 
:<math>z=v</math>
 
:where u and v are real parameters and c is a non-zero real constant.
 
In cylindrical coordinates:
 
:<math>\rho =c \cosh \frac{z}{c}</math>
 
:Where c is a real constant.
 
A physical model of a catenoid can be formed by dipping two [[circle]]s 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|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 the shape of a [[helicoid]] without stretching. In other words, one can make a [[continuous function|continuous]] and [[Isometry|isometric]] deformation of a catenoid to a [[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>x(u,v) = \cos \theta \,\sinh v \,\sin u + \sin \theta \,\cosh v \,\cos u</math>
 
:<math>y(u,v) = -\cos \theta \,\sinh v \,\cos u + \sin \theta \,\cosh v \,\sin u</math>
 
:<math>z(u,v) = u \cos \theta + v \sin \theta \,</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==
<references/>
 
==External links==
*{{springer|title=Catenoid|id=p/c020800}}
*[http://www.mathcurve.com/surfaces/catenoid/catenoid.shtml "Caténoïde" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
*[http://www.princeton.edu/~rvdb/WebGL/catenoid.html Animated 3D WebGL model of a catenoid]
 
[[Category:Minimal surfaces]]
 
[[de:Minimalfläche#Die Katenoiden]]

Latest revision as of 21:46, 21 July 2014

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