Spring (mathematics)

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

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

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

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

where

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

${\displaystyle v\in [0,\ 2\pi ),}$

${\displaystyle R\,}$ is the distance from the center of the tube to the center of the helix,

${\displaystyle r\,}$ is the radius of the tube,

${\displaystyle P\,}$ 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),

${\displaystyle n\,}$ is the number of rounds in a spring.

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

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

The interior volume of the spiral is given by

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

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 ${\displaystyle P\,}$ 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.

• Spiral
• Helix