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HELIX OF THE ONESHEETED HYPERBOLOID OF REVOLUTION
Curve studied by Blaschke in 1908 [Mh.
Math. Phys. 19, p. 194]
See also [Loria 3d] p. 160. 
Cartesian parametrization:
traced on the hyperboloid ,
with .
When , , we get the line x = a, by = az. Polar equation of the projection on xOy: . 
Here, we consider the helix
of the onesheeted hyperboloid
of revolution with vertical axis, a curve with constant slope
with respect to a horizontal plane.
View of the 3 kinds of helices:
 in red for a slope < b/a, the helix goes to infinity  in blue, the line with slope b/a.  in green for a slope > b/a; the curve has a form of bound (its projection on xOy cannot cross the circle with radius ). 
Be careful, if the cylindrical helix is indeed the intersection between a right helicoid and a cylinder, this method does not yield the helix of the onesheeted hyperboloid, but the following curve, that does not have a name:
Intersection between the helicoid
and the hyperboloid .
Cartesian parametrization:; the figure opposite was traced thanks to the parametrization for which the coordinate lines are the curves under examination here. 

Lift on the hyperboloid of the spiral , which is asymptotic to the Archimedean spiral . 

View of the construction of these curves (by Robert March). 

Buffon's gloriette in the Jardin des Plantes in Paris...

...made by Alain Esculier. 
Picture: Remy Couderc
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© Robert FERRÉOL 2018