Parameterizations of Delaunay Surfaces from Scratch

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 fir...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-05, Vol.12 (10), p.1570
Main Authors: Mladenova, Clementina D., Mladenov, Ivaïlo M.
Format: Article
Language:English
Subjects:
Astronomers
constant mean curvature surfaces
Delaunay surfaces
Delaunay, Charles-Eugene
elliptic functions and integrals
Euclidean space
Geometry
Influence
Mathematical research
Parameters
roulettes
surfaces of revolution
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12101570