Catenoid

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 [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 circles into a soap solution and slowly drawing the circles apart.

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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 a helicoid such that every member of the deformation family is minimal. A parametrization of such a deformation is given by the system

$x(u,v)\; =\; cos\; heta\; ,sinh\; v\; ,sin\; u\; +\; sin\; heta\; ,cosh\; v\; ,cos\; u$

$y(u,v)\; =\; -cos\; heta\; ,sinh\; v\; ,cos\; u\; +\; sin\; heta\; ,cosh\; v\; ,sin\; u$

$z(u,v)\; =\; u\; cos\; heta\; +\; v\; sin\; heta\; ,$

for $(u,v)\; in\; (-pi,\; pi]\; imes\; (-infty,\; infty)$, with deformation parameter ,

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