Catenoid

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

A catenoid is one of several types of minimal surfaces. In mathematical terms, a minimal surface is a surface with a mean curvature of zero. Physical models of minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film. A minimal surface made by rotating a catenary once around the axis is called a catenoid. A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity is called a helicoid. A catenoid and a helicoid are representative minimal surfaces.

One can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal.