catenoid

A catenoid is the surface of revolution obtained by rotating a catenary about its axis. It is the simplest non-planar minimal surface, that is, a surface whose mean curvature is zero at every point. The catenoid has the remarkable property of being the minimum-area surface whose boundary consists of two parallel circles of equal radius, whose centres are aligned along a line perpendicular to the planes of those circles. This property is demonstrated by the soap-film experiment: when two coaxial circular rings are immersed in a soap solution, the film that forms between them takes the shape of a catenoid, provided that the distance between the rings does not exceed approximately 0.66 times their common diameter; beyond this limit, the film breaks. A catenoid can be parametrized using hyperbolic functions and is related to the helicoid by a continuous isometric deformation. It was studied by Euler as early as 1740, in the context of his foundational work on the calculus of variations.