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ROTATING ROD SPIRAL
Curve studied by Johannes Bernoulli in 1742 (Opera,
T IV, p. 248).
Homemade name ; other names : spiral of the hyperbolic cosine, spiral of the hyperbolic sine. 
Polar equation: . 
The rotating rod spiral
is the trajectory of a massive ring sliding without friction on a horizontal
rod in rotation around a centre when the ring goes to infinity. See on
this
website a derivation of the polar equation above.
Case b = a: the equation can be written .  Case b = – a: the equation can be written .  Case b = – 2a. 

Note that the curve is similar to the curve in the first case if ab > 0 because, then, , and it is similar to the curve in the second case if ab < 0, because, then, ; in all the cases where a and b are not zero, the spiral therefore has a symmetry axis, like the Archimedean spiral; but the two infinite branches have asymptotes: the logarithmic spirals.
After this website, there are also cases where the ring oscillate on the rod ; the curve is so an epitrochoid.
These spirals are inverses of Poinsot
spirals.
See also the spiral
of the hyperbolic tangent.
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© Robert FERRÉOL 2022