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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