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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), 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.

Other spirals

Archimedean spiral.

External links

Online exploration using JSXGraph (JavaScript)

Mathematics Encyclopedia

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