Spring (math)

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A Spring

A left-handed and a right-handed spring.

For other meanings of the term, see Spring (disambiguation).

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.

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)

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

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