Nodoid

A nodoid is a surface of revolution with constant mean curvature (CMC). Specifically, it is a surface that is topologically a cylinder but with a periodically bulging and constricted profile, resembling a string of beads. Nodoids are part of a one-parameter family of CMC surfaces of revolution discovered by Delaunay in 1841, along with the cylinder, the sphere, the catenoid, and the unduloid.

The nodoid's profile curve, when rotated around an axis, generates the surface. The axis of rotation is a line of symmetry for the profile curve. Unlike the catenoid, which is formed by rotating a catenary, the profile curve of a nodoid is described by the trace of the focus of an ellipse rolling along a line.

Nodoids can be visualized as deformations of a cylinder, with the amount of bulge and constriction being determined by a parameter. As this parameter approaches a certain limit, the nodoid approaches a chain of tangent spheres. In the other limit, the nodoid approaches a cylinder. The unduloid, another member of Delaunay's family, has a more uniform wavy profile compared to the nodoid's more pronounced constrictions.

The constant mean curvature property of the nodoid means that at every point on the surface, the average of the two principal curvatures is the same. This property is significant in various areas of mathematics and physics, including differential geometry, capillarity, and the study of minimal surfaces. Nodoids and the other Delaunay surfaces represent important examples of surfaces minimizing area subject to constraints on enclosed volume or surface tension.