Catenoid

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A catenoid
A 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 Euler in 1744.

A physical model of a catenoid can be formed by dipping two circles into a soap solution, (popping any film in the center of the circle), and slowly drawing the circles apart.

Animation showing the deformation of a helicoid into a catenoid. Generated with Mac OS X Grapher.
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. An explicit parameterization 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.

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