Revolution of the sinusoid

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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