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| [[Image:Spring2.png|frame|A Spring]]
| | Hello and welcome. My title is Irwin and I totally dig that name. My day job is a librarian. Her family lives in Minnesota. Body building is what my family members and I appreciate.<br><br>Here is my site [http://Xrambo.com/blog/192034 http://Xrambo.com/blog/192034] |
| [[Image:Helix.png|frame|A left-handed and a right-handed spring.]]
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| In [[geometry]], a '''spring''' is a surface in the shape of a coiled tube, generated by sweeping a [[circle]] about the path of a [[helix]].{{Citation needed|date=May 2009}}
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| ==Definition==
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| A spring wrapped around the z-[[Coordinate axis|axis]] can be defined parametrically by:
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| :<math>x(u, v) = \left(R + r\cos{v}\right)\cos{u}, </math>
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| :<math>y(u, v) = \left(R + r\cos{v}\right)\sin{u}, </math>
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| :<math>z(u, v) = r\sin{v}+{P\cdot u \over \pi}, </math>
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| where
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| :<math>u \in [0,\ 2n\pi)\ \left(n \in \mathbb{R}\right),</math>
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| :<math>v \in [0,\ 2\pi),</math>
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| :<math>R \,</math> is the distance from the center of the tube to the center of the [[helix]],
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| :<math>r \,</math> is the radius of the tube,
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| :<math>P \,</math> is the speed of the movement along the z axis (in a [[Right-hand rule|right-handed]] [[Cartesian coordinate system]], positive values create right-handed springs, whereas negative values create left-handed springs),
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| :<math>n \,</math> is the number of rounds in circle.
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| The [[implicit function]] in Cartesian coordinates for a spring wrapped around the z-[[Coordinate axis|axis]], with <math>n</math> = 1 is
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| :<math>\left(R - \sqrt{x^2 + y^2}\right)^2 + \left(z + {P \arctan(x/y) \over \pi}\right)^2 = r^2.</math>
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| The interior [[volume]] of the spiral is given by
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| :<math>V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \,</math>
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| ==Other definitions==
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| Note that the previous definition uses a vertical circular cross section. This is not entirely accurate as the tube becomes increasingly distorted as the Torsion<ref>{{cite web |url=http://mathworld.wolfram.com/Helix.html | title=http://mathworld.wolfram.com/Helix.html}}</ref> increases (ratio of the speed <math>P \,</math> and the incline of the tube).
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| An alternative would be to have a circular cross section in the plane perpendicular to the helix curve. This would be closer to the shape of a physical spring. The mathematics would be much more complicated.
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| The [[torus]] can be viewed as a special case of the spring obtained when the helix degenerates to a circle.
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| ==References==
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| {{reflist}}
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| ==See also==
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| *[[spiral]]
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| *[[helix]]
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| [[Category:Surfaces]]
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Hello and welcome. My title is Irwin and I totally dig that name. My day job is a librarian. Her family lives in Minnesota. Body building is what my family members and I appreciate.
Here is my site http://Xrambo.com/blog/192034