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 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[edit]

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[edit]

Asymptote[edit]

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[edit]

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[edit]

The tangential angle of the hyperbolic curve is

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

References[edit]

  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 .

External links[edit]