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# 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 by Leonhard Euler in 1744.

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

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

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

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

for $(u,v) \in (-\pi, \pi] \times (-\infty, \infty)$, with deformation parameter $-\pi < \theta \le \pi$.