Hyperbolic spiral

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A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. It has the polar equation rθ = a, and is the inverse to the Archimedean spiral.

Hyperbolic spiral for a=2
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Hyperbolic spiral for a=2

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.


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