A Delaunay surface is a surface of revolution in the family of constant mean curvature surfaces constructed from focal roulettes of conic sections, together with the elementary sphere and circular cylinder cases. The convention used here includes the catenoid, whose mean curvature is zero (Bendito et al. 2014).
In Delaunay's construction (Delaunay 1841), a conic section rolls without slipping along a line. The path
of a focus is revolved about that line.
An ellipse gives an unduloid,
a hyperbola gives a nodoid,
and a parabola gives a catenoid.
The following table compares the five nonplanar types. Here 
is the radius of a sphere
or circular cylinder, and 
is mean curvature. Its sign
depends on the choice of normal vector (Koiso and
Palmer 2008, Bendito et al. 2014).
| surface | rolling construction or limit |  | geometry |
| unduloid | noncircular ellipse | nonzero | periodic bulges and necks |
| nodoid | hyperbola | nonzero | self-intersecting continuation |
| catenoid | parabola | 0 | minimal surface |
| cylinder | circle, the circular limit of the ellipse |  | constant radius |
| sphere | limit as unduloid necks shrink to zero |  | round closed surface |
As the necks of an unduloid shrink to zero at fixed nonzero mean curvature, the limiting configuration is a chain of tangent spheres. An individual sphere is smooth, but the whole touching chain is not a regular surface. The plane is the other minimal surface of revolution, in addition to the catenoid, and is a degenerate zero-curvature case rather than one of the five nonplanar types.
The variational interpretation is stationarity of surface area under a prescribed volume, not a guarantee of global minimization (Koiso and Palmer 2008).
