catenoid

The catenoid is a surface of revolution obtained by rotating the catenary about its axis. It is a minimal surface, meaning its mean curvature vanishes everywhere, and it is the simplest nonplanar minimal surface. The standard form can be written as x^2 + y^2 = a^2 cosh^2(z/a) for a > 0, or by the parameterization X(u,v) = (a cosh(v/a) cos u, a cosh(v/a) sin u, v), with u in [0, 2π) and v in (-∞, ∞).

The catenoid has axial symmetry about the z-axis, negative Gaussian curvature, and is non-compact, extending infinitely along the axis. It is the surface obtained by rotating the catenary y = a cosh(x/a) around the z-axis.

Historically, the surface was described by Leonhard Euler in 1744 and is named for its relation to the catenary, the curve y = a cosh(x/a). It is notable as the first nontrivial minimal surface discovered and is often cited as a classic example in the study of minimal surfaces and Plateau's problem.

In physical models, a soap film spanning two parallel circular rings can form a catenoid when the rings are sufficiently close; if the separation is too large, no minimal surface bridges the rings and the film collapses into two disks.