next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
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 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).
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017