Hyperbolic Spiral

Polar equation:
r = a/θ


Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point


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The hyperbolic spiral originated with Pierre Varignon in 1704. It was studied by Johann Bernoulli between 1710 and 1713 and it was also studied by Cotes in 1722.

The roulette of the pole of a hyperbolic spiral rolling on a straight line is a tractrix.

Pierre Varignon (1654-1722) was professor of mathematics at Collège Mazarin and later at Collège Royal. Led into mathematics by reading Euclid he also read Descartes' Géométrie and thereafter devoted himself to the mathematical sciences. He was one of the first French scholars to recognise the value of the calculus. His chief contributions were to mechanics.

Taking the pole as the centre of inversion, the hyperbolic spiral r = a/θ inverts to the spiral of Archimedes r = aθ.


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JOC/EFR/BS January 1997

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http://www-history.mcs.st-andrews.ac.uk/Curves/Hyperbolic.html