Details

The parametric equations of the nodoid are

with where and are parameters, with mean curvature . The different types are parallel surfaces. In type "H", the signs of the first sine and cosine in the expressions for and are reversed and in type "K", these terms are dropped.

References

[1] C. Delaunay, "Sur la Surface de Révolution Dont la Courbure Moyenne Est Constante," Journal de Mathématiques Pures et Appliquées, 1(6), 1841 pp. 309–320.

[2] "Delaunay Surfaces." Geometrie Werkstatt. (Jun 25, 2014) www.math.uni-tuebingen.de/user/nick/gallery/Delaunay.html.

[3] I. M. Mladenov, "Delaunay Surfaces Revisited," Comptes Rendus de l'Academie Bulgare des Sciences, 55, 2002 pp. 5–19. articles.adsabs.harvard.edu//full/2002CRABS..55e..19M/E000023.000.html.

[4] W. Rossman, "The First Bifurcation Point for Delaunay Nodoids," Experimental Mathematics, 14(3), 2005 pp. 331–342. www.emis.de/journals/EM/expmath/volumes/14/14.3/Rossman.pdf.

[5] E. Bendito, M. J. Bowick, A. Medina, and Z. Yao, "Crystalline Particle Packings on Constant Mean Curvature (Delaunay) Surfaces", Physical Review E, 88, 2013 pp. 012405. journals.aps.org/pre/abstract/10.1103/PhysRevE.88.012405.