Delaunay Curve

A Delaunay curve is the roulette traced by a focus of a conic section as the conic section rolls without slipping along a line. Revolution about the rolling line gives a Delaunay surface (Delaunay 1841, Bendito et al. 2014). The elliptic case is an undulary, also called an elliptic catenary, and the hyperbolic case a nodary, also called a hyperbolic catenary. The parabolic case is an ordinary catenary.

For a parabola, the resulting catenary has a parametrization

			(1)
			(2)

where . Its surface of revolution is a catenoid.