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This spiral is a kind of Archimedean spiral.
 The hyperbolic
name reflects the relation between radius and angle being the same as the x,y relation
for the hyperbola. This relationship
leads also the name of the reciprocal
spiral.
The curve is a Cotes' spiral.
When a hyperbolic spiral is rolling over a line, then the path of the pole
forms a tractrix, being a roulette
of the spiral. Its inverse is the Archimides'
spiral.
It was Pierre Varignon who studied the curve as first, in 1704.
Later Johann Bernoulli (1710 - 1713) and Cotes (1722) worked on
the curve.
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