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.

Definition

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

$x(u,v)=(R+r\cos v)\cos u,$

$y(u,v)=(R+r\cos v)\sin u,$

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

where

$u\in [0,2n\pi )(n\in \mathbb{ℝ}),$

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

$R$ 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 a spring.

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

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

The interior volume of the spiral is given by

$V=2{\pi }^{2}nR{r}^2=\left(\pi r^2\right)\left(2\pi nR\right).$

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 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.

See also

• Spiral
• Helix