# Hyperbolic spiral

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 opposite 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 pol equation:

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

It begins at an infinite distance from the pole in the centre (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:

$x = r \cos \theta, \qquad y = r \sin \theta,$

leads to the following parametric representation in Cartesian coordinates:

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

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:

$\lim_{t\to 0}x = a\lim_{t\to 0}{\cos t \over t}=\infty,$
$\lim_{t\to 0}y = a\lim_{t\to 0}{\sin t \over t}=a\cdot 1=a.$

### Curvature

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

$\kappa = {r^2 + 2r_\theta^2 - r r_{\theta \theta} \over (r^2+r^2_\theta)^{3/2}}$

where

$r_\theta = {d r \over d \theta} = {-a \over \theta^2}$

and

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

Then the curvature at $\theta$ reduces to

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

The curvature tends to infinity as $\theta$ tends to infinity. For values of $\theta$ 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 tangential angle of the hyperbolic curve is

$\phi(\theta) = -\tan^{-1} \theta.$

## 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.