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| {{unreferenced|date=September 2012}}
| | Hello and welcome. My name is Ling. The thing I adore most bottle tops collecting and now I have time to take on new things. Her family members life in Delaware but she requirements to transfer simply because of her family members. Bookkeeping is what he does.<br><br>Feel free to surf to my site ... [http://Totoagc.com/module_page_g13/232511 car warranty] |
| [[image:hyperspiral.svg|thumb|200px|right|Hyperbolic spiral for a=2]]
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| A '''hyperbolic spiral''' is a [[Transcendental function|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]].
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| It has the [[coordinates (elementary mathematics)#Polar coordinates|polar]] equation:
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| : <math>r=\frac{a}{\theta}</math>
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| 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:
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| :<math>x = r \cos \theta, \qquad y = r \sin \theta,</math>
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| leads to the following parametric representation in [[Cartesian coordinate system|Cartesian coordinates]]:
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| :<math>x = a {\cos t \over t}, \qquad y = a {\sin t \over t},</math>
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| where the [[Parameter#Mathematical|parameter]] ''t'' is an equivalent of the polar coordinate θ.
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| The spiral has an [[asymptote]] at ''y'' = ''a'': for ''t'' approaching zero the [[ordinate]] approaches ''a'', while the [[abscissa]] grows to infinity:
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| :<math>\lim_{t\to 0}x = a\lim_{t\to 0}{\cos t \over t}=\infty,</math>
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| :<math>\lim_{t\to 0}y = a\lim_{t\to 0}{\sin t \over t}=a\cdot 1=a.</math>
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| It was [[Pierre Varignon]] who studied the curve as first, in 1704. Later [[Johann Bernoulli]] and [[Roger Cotes]] worked on the curve.
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| ==Properties==
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| Using the representation of the hyperbolic spiral in polar coordinates, the curvature can be found by
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| :<math>\kappa = {r^2 + 2r_\theta^2 - r r_{\theta \theta} \over (r^2+r^2_\theta)^{3/2}}</math>
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| where
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| :<math>r_\theta = {d r \over d \theta} = {-a \over \theta^2} </math>
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| and
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| :<math>r_{\theta \theta} = {d^2 r \over d \theta^2} = {2 a \over \theta^3}.</math>
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| Then the curvature at <math>\theta</math> reduces to
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| :<math>\kappa(\theta) = {\theta^4 \over a (\theta^2 + 1)^{3/2}}.</math>
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| The curvature tends to infinity as <math>\theta</math> tends to infinity. For values of <math>\theta</math> 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.
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| The tangential angle of the hyperbolic curve is
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| <math>\phi(\theta) = -\tan^{-1} \theta. </math>
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| ==Other spirals==
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| * [[Archimedean spiral]].
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| ==External links==
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Hyperbolic_spiral Online exploration using JSXGraph (JavaScript)]
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| [[Category:Spirals]]
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| {{Geometry-stub}}
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| [[pt:Espiral logarítmica]]
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Hello and welcome. My name is Ling. The thing I adore most bottle tops collecting and now I have time to take on new things. Her family members life in Delaware but she requirements to transfer simply because of her family members. Bookkeeping is what he does.
Feel free to surf to my site ... car warranty