Spring (mathematics)

In geometry, a spring is a surface of revolution in the shape of a helix with thickness, generated by revolving a circle about the path of a helix. The torus is a special case of the spring obtained when the helix is crushed to a circle.

Definition

A spring wrapped around the z-axis can be defined parametrically by::$x(u,\; v)\; =\; left(R\; +\; rcos\{v\}\; ight)cos\{u\},$:$y(u,\; v)\; =\; left(R\; +\; rcos\{v\}\; ight)sin\{u\},$:$z(u,\; v)\; =\; rsin\{v\}+\{Pcdot\; u\; over\; pi\},$

where :$u\; in\; [0,\; 2npi]\; left(n\; in\; mathbb\{R\}\; ight),$:$v\; in\; [0,\; 2pi]\; ,$:$R\; ,$ 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)

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

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

The interior volume of the spiral is given by

:$V\; =\; 2pi^2\; n\; R\; r^2\; =\; left(\; pi\; r^2\; ight)\; left(\; 2pi\; n\; R\; ight).\; ,$

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 [cite web |url=http://mathworld.wolfram.com/Helix.html | title=http://mathworld.wolfram.com/Helix.html] 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 spanned by the normal and binormal of the curve. This would be closer to the shape of a physical spring. The mathematics involved would however be much more involved.

References

ee also

*spiral
*helix