# Hyperbolic spiral

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Hyperbolic spiral for a = 2

A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral.[1] A hyperbolic spiral is the inverse of an Archimedean spiral[2] and is a type of Cotes' spiral.

Pierre Varignon first studied the curve in 1704.[2] Later Johann Bernoulli and Roger Cotes worked on the curve as well.

## Equation

The hyperbolic spiral has the polar equation:

${\displaystyle r={\frac {a}{\theta }}}$

It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is infinite. Applying the transformation from the polar coordinate system:

${\displaystyle x=r\cos \theta ,\qquad y=r\sin \theta ,}$

leads to the following parametric representation in Cartesian coordinates:

${\displaystyle x=a{\frac {\cos t}{t}},\qquad y=a{\frac {\sin t}{t}},}$

where the parameter t is an equivalent of the polar coordinate θ.

## Properties

### Asymptote

The spiral has an asymptote at y = a: for t approaching zero the ordinate approaches a, while the abscissa grows to infinity:

${\displaystyle \lim _{t\to 0}x=a\lim _{t\to 0}{\frac {\cos t}{t}}=\infty ,}$
${\displaystyle \lim _{t\to 0}y=a\lim _{t\to 0}{\frac {\sin t}{t}}=a\cdot 1=a.}$

### Curvature

Using the representation of the hyperbolic spiral in polar coordinates, the curvature can be found by

${\displaystyle \kappa ={\frac {r^{2}+2r_{\theta }^{2}-rr_{\theta \theta }}{(r^{2}+r_{\theta }^{2})^{3/2}}}}$

where

${\displaystyle r_{\theta }={\frac {dr}{d\theta }}={\frac {-a}{\theta ^{2}}}}$

and

${\displaystyle r_{\theta \theta }={\frac {d^{2}r}{d\theta ^{2}}}={\frac {2a}{\theta ^{3}}}.}$

Then the curvature at ${\displaystyle \theta }$ reduces to

${\displaystyle \kappa (\theta )={\frac {\theta ^{4}}{a(\theta ^{2}+1)^{3/2}}}.}$

The curvature tends to infinity as ${\displaystyle \theta }$ tends to infinity. For values of ${\displaystyle \theta }$ between 0 and 1, the curvature increases exponentially, and for values greater than 1, the curvature increases at an approximately linear rate with respect to the angle.

### Tangents

The polar tangential angle of the hyperbolic spiral is

${\displaystyle \psi (\theta )=-\arctan \theta .}$

## References

1. ^ Bowser, Edward Albert (1880), An Elementary Treatise on Analytic Geometry: Embracing Plane Geometry and an Introduction to Geometry of Three Dimensions (4th ed.), D. Van Nostrand, p. 232.
2. ^ a b Lawrence, J. Dennis (2013), A Catalog of Special Plane Curves, Dover Books on Mathematics, Courier Dover Publications, p. 186, ISBN 9780486167664.