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 aroung the z-axis can be defined parametrically by:

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

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

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

where

$u\; \backslash in2n\backslash pi\backslash \; \backslash left(n\; \backslash in\; \backslash mathbb\{R\}\backslash right)$,

$v\; \backslash in2\backslash pi$,

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

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

The interior volume of the spiral is given by

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

See also

• spiral

Surfaces