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POINSOT SPIRAL

spirale bornée avec omega = 1/3spirale à asymptote avec omega = 1/3

Curve determined by Cotes in 1722.
Louis Poinsot (1777 - 1859): French mathematician.
Other names, along with the epispirals: Cotes spiral.

The Poinsot spiral is the curve with
 
Polar equation: with .

When
    -  we get a logarithmic spiral
    -  we get a bounded spiral similar to .
    -  we get a spiral with asymptote similar to .
 
The bounded Poinsot spiral is the orthogonal projection on the equatorial plane of the rhumb line of the sphere 
the bounded Poinsot spiral is the view from above of a rhumb line

The Poinsot spiral with an asymptote is the projection of the curve traced on a hyperboloid of two sheets

The Poinsot spirals are solutions to the problem consisting in determining the trajectories in space of a massive point subject to a force centred on O proportional to (this force is, according to the Binet formula, proportional to which is equal, here, to , with ); the other solutions are the epispirals with, as an intermediary case, the hyperbolic spiral, see this link.
 The trajectories of a marble thrown in a tube and in uniform rotation are the inverses of the bounded Poinsot spirals (Jean Bernoulli, Opera, T IV, p 248).
 
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© Robert FERRÉOL 2017