Catenoid
From Wikipedia, the free encyclopedia
A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. 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.[1] Early work on the subject was published also by Meusnier.[2] There are only two surfaces of revolution which are also minimal surfaces: the plane and the catenoid.[3]
A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.
One can bend a catenoid into the shape of a helicoid without stretching. In other words, one can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal. A parametrization of such a deformation is given by the system
for ![(u,v) \in (-\pi, \pi] \times (-\infty, \infty)](http://upload.wikimedia.org/math/d/1/1/d11b64d7f07089a4a1d3fded937de015.png)
, with deformation parameter 
,
where θ = π corresponds to a right-handed helicoid, 
corresponds to a catenoid, θ = 0 corresponds to a left-handed helicoid,
[edit] References
- ^ L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, in: Opera omnia I, 24
- ^ Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785
- ^ Catenoid at MathWorld
