When , the solution is the hyperbolic spiral.

A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral.

Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus.

The hyperbolic spiral:

A hyperbolic spiral is the opposite of an Archimedean spiral and are a type of Cotes' spiral.

The first form corresponds to an epispiral, and the second to one of Poinsot's spirals; the third form corresponds to a hyperbolic spiral, also known as a reciprocal spiral, which is sometimes not counted as a Cotes's spiral.