Spring (mathematics)

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.

Definition

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

$x(u, v) = \left(R + r\cos{v}\right)\cos{u},$

$y(u, v) = \left(R + r\cos{v}\right)\sin{u},$

$z(u, v) = r\sin{v}+{P\cdot u \over \pi},$

where

$u \in [0,\ 2n\pi)\ \left(n \in \mathbb{R}\right),$

$v \in [0,\ 2\pi),$

$R \,$ 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 circle.

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

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

The interior volume of the spiral is given by

$V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \,$

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. "http://mathworld.wolfram.com/Helix.html". External link in |title= (help)

See also

• spiral
• helix