Parameterizations of Delaunay Surfaces from Scratch

Clementina D. Mladenova and Ivaïlo M. Mladenov ()
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Clementina D. Mladenova: Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria
Ivaïlo M. Mladenov: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria

Mathematics, 2024, vol. 12, issue 10, 1-16

Abstract: Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form.

Keywords: Delaunay surfaces; surfaces of revolution; constant mean curvature surfaces; elliptic functions and integrals; roulettes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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