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HELIX OF THE ONE-SHEETED HYPERBOLOID OF REVOLUTION
Curve studied by Blaschke in 1908 [Mh.
Math. Phys. 19, p. 194]
See also [Loria 3d] p. 160. |
Cartesian parametrization: When Polar equation of the projection on xOy: |
Here, we consider the helix
of the one-sheeted 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 |
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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 one-sheeted hyperboloid, but the following curve, that does not have a name:
Intersection between the helicoid Cartesian parametrization: |
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Lift on
the hyperboloid of the spiral |
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View of the construction of these curves (by Robert March). |
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Buffon's gloriette in the Jardin des Plantes in Paris...
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![]() ...made by Alain Esculier. |
Picture: Remy Couderc
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© Robert FERRÉOL 2018