POINSOT SPIRAL  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).