- Catenoid
A catenoid is a three-
dimension alshape made by rotating acatenary curve around the x axis. Not counting the plane, it is the firstminimal surface to be discovered. It was found and proved to be minimal byLeonhard Euler in 1744 [L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, in: Opera omnia I, 24] . Early work on the subject was published also by Meusnier [Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785] . There are only two surfaces of revolution which are also minimal surfaces: the plane and the catenoid [ [http://mathworld.wolfram.com/Catenoid.html Catenoid at MathWorld] ] .A physical model of a catenoid can be formed by dipping two
circle s into a soap solution and slowly drawing the circles apart.[
thumb|right|256px|Animation_showing_the_deformation_of_a_helicoid_into_a_catenoid._Generated_with_Mac_OS_X_
Grapher .]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 ahelicoid such that every member of the deformation family is minimal. A parametrization of such a deformation is given by the systemx(u,v) = cos heta ,sinh v ,sin u + sin heta ,cosh v ,cos u
y(u,v) = -cos heta ,sinh v ,cos u + sin heta ,cosh v ,sin u
z(u,v) = u cos heta + v sin heta ,
for u,v) in (-pi, pi] imes (-infty, infty), with deformation parameter pi < heta le pi,
where heta = pi corresponds to a right-handed helicoid, heta = pm pi / 2 corresponds to a catenoid, heta = pm pi corresponds to a left-handed helicoid,
References
Wikimedia Foundation. 2010.