A catenoid is a three-dimensional shape made by rotating a catenary curve around the $x$ axis.

A catenoid is one of several types of minimal surfaces.

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.

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

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

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

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

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

Minimal surfaces

Caténoïde | Катеноид