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:

x(u,v)=(R+rcos⁡v)cos⁡u,
y(u,v)=(R+rcos⁡v)sin⁡u,
z(u,v)=rsin⁡v+P⋅uπ,

where

u∈[0, 2nπ) (n∈ℝ),
v∈[0, 2π),
R is the distance from the center of the tube to the center of the helix,
r is the radius of the tube,
P 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),
n is the number of rounds in a spring.

The implicit function in Cartesian coordinates for a spring wrapped around the z-axis, with n = 1 is

(R−x2+y2)2+(z+Parctan⁡(x/y)π)2=r2.

The interior volume of the spiral is given by

V=2π2nRr2=(πr2)(2πnR).

## 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 P 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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