Spring (mathematics)
In geometry, a spring is a surface in the shape of a coiled tube, generated by sweeping a circle about the path of a helix.[citation needed]
Definition
A spring wrapped around the z-axis can be defined parametrically by:
- [math]\displaystyle{ x(u, v) = \left(R + r\cos{v}\right)\cos{u}, }[/math]
- [math]\displaystyle{ y(u, v) = \left(R + r\cos{v}\right)\sin{u}, }[/math]
- [math]\displaystyle{ z(u, v) = r\sin{v}+{P\cdot u \over \pi}, }[/math]
where
- [math]\displaystyle{ u \in [0,\ 2n\pi)\ \left(n \in \mathbb{R}\right), }[/math]
- [math]\displaystyle{ v \in [0,\ 2\pi), }[/math]
- [math]\displaystyle{ R \, }[/math] is the distance from the center of the tube to the center of the helix,
- [math]\displaystyle{ r \, }[/math] is the radius of the tube,
- [math]\displaystyle{ P \, }[/math] 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),
- [math]\displaystyle{ n \, }[/math] is the number of rounds in a spring.
The implicit function in Cartesian coordinates for a spring wrapped around the z-axis, with [math]\displaystyle{ n }[/math] = 1 is
- [math]\displaystyle{ \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]\displaystyle{ V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \, }[/math]
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[1] increases (ratio of the speed [math]\displaystyle{ P \, }[/math] 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.
References
See also
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