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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:

<math>x(u, v) = \left(R + r\cos{v}\right)\cos{u}</math>,

<math>y(u, v) = \left(R + r\cos{v}\right)\sin{u}</math>,

<math>z(u, v) = r\sin{v}+{P\cdot u \over \pi}</math>

where

<math>u \in [0,\ 2n\pi]\ \left(n \in \mathbb{R}\right)</math>,

<math>v \in [0,\ 2\pi]</math>,

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

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

The interior volume of the spiral is given by

<math>V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \,</math>

[edit] See also

• spiral
• helix