Spring (mathematics)

A Spring

A left-handed and a right-handed spring.

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]

## Contents

* 1 Definition
* 2 Other definitions
* 3 References
* 4 See also

## Definition

A spring wrapped around the z-axis can be defined parametrically by:

[math]\displaystyle{ x(u, v) = \left(R + r\cos{v}\right)\cos{u}, }[/math]
[math]\displaystyle{ y(u, v) = \left(R + r\cos{v}\right)\sin{u}, }[/math]
[math]\displaystyle{ z(u, v) = r\sin{v}+{P\cdot u \over \pi}, }[/math]

where

[math]\displaystyle{ u \in [0,\ 2n\pi)\ \left(n \in \mathbb{R}\right), }[/math]
[math]\displaystyle{ v \in [0,\ 2\pi), }[/math]
[math]\displaystyle{ R \, }[/math] is the distance from the center of the tube to the center of the helix,
[math]\displaystyle{ r \, }[/math] is the radius of the tube,
[math]\displaystyle{ P \, }[/math] is the speed of the movement along the z axis (in a right-handed Cartesian coordinate system, positive values create right-handed springs, whereas negative values create left-handed springs),
[math]\displaystyle{ n \, }[/math] is the number of rounds in a spring.

The implicit function in Cartesian coordinates for a spring wrapped around the z-axis, with [math]\displaystyle{ n }[/math] = 1 is

[math]\displaystyle{ \left(R - \sqrt{x^2 + y^2}\right)^2 + \left(z + {P \arctan(x/y) \over \pi}\right)^2 = r^2. }[/math]

The interior volume of the spiral is given by

[math]\displaystyle{ V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \, }[/math]

## Other definitions

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[1] increases (ratio of the speed [math]\displaystyle{ P \, }[/math] and the incline of the tube).

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.

The torus can be viewed as a special case of the spring obtained when the helix degenerates to a circle.

## References

1. ↑ "Helix". http://mathworld.wolfram.com/Helix.html.

## See also

* Spiral
* Helix

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