Dictionary - Definition of HYPERBOLIC SPIRAL

Definition: HYPERBOLIC SPIRAL


Part of Speech	Definition
Expression	1. (Math.), a spiral curve, the law of which is, that the distance from the pole to the generating point varies inversely as the angle swept over by the radius vector.[Websters].
Source: Webster's Revised Unabridged Dictionary (1913)		Top

Extended Definition: HYPERBOLIC SPIRAL

Hyperbolic spiral

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

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:

$\lim_{t\to 0}x = a\lim_{t\to 0}{\cos t \over t}=\infty,$

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

Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Hyperbolic spiral". Image Credit.
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Translations: HYPERBOLIC SPIRAL

Language	Translations (or nearest inflections or synonyms, in parentheses)
Italian	Spirale iperbolica (Hyperbolic spiral). Additional references: Italian, Italy, Croatia, Hyperbolic spiral. (volunteer & more translations)
Source: Eve, based on a combination of meta analysis and graph theory (for near and back translations).		Top