POINSOT SPIRAL
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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  |
  |
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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