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REVOLUTION OF THE SINUSOID
Surface studied in 2012 by G. Claeser and P. Calvache.
Homemade name, I'm am open to suggestions if you have a better name... 
Cylindrical and Cartesian equations:
and .
Cartesian parametrizations: 

1) as a surface of revolution of the sinusoid around Oz: .  
2) as a translation surface: , locus of the middles of the segments lines joining the two symmetric circular helices: . 

3) as the surface of revolution of one of the previous helices: 

The revolution of the sinusoid is the surface of revolution obtained by rotation of a sinusoid around its axis of translation.
But it is remarkable that surface is also obtained by translation of a circular helix on a symmetric helix with respect to the axis (compare to the right helicoid which is obtained by translation of a helix on itself).
Therefore, it is also obtained by rotation of a circular helix around a generatrix of the cylinder on which it is traced.
The section of this surface by a cylinder tangent to the axis and passing by the vertices is therefore composed of two symmetric circular helices.
Remark (Lapalissade): these helices are the... helices of this surface of revolution.... 

Obviously, there exist other curves obtained by revolution of a sinusoid, for example: . 

Do not mistake for the onduloid, and compare to the egg box.
See on this
page a polyhedron with diamond faces that approximates this surface.
The Gherkin in London, was more or less built on this model.
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© Robert FERRÉOL
2017