We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.
"A Natural Parameterization of the Roulettes of the Conics Generating the Delaunay Surfaces." J. Geom. Symmetry Phys. 33 27 - 45, 2014. https://doi.org/10.7546/jgsp-33-2014-27-45