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

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.

It was Pierre Varignon who studied the curve as first, in 1704. Later Johann Bernoulli and Roger Cotes worked on the curve.

Properties

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

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

where

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

and

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

The tangential angle of the hyperbolic curve is

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

Other spirals

External links

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