Spring (mathematics)
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a spring is a surface in the shape of a coiled tube, generated by sweeping a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
about the path of a
helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
.


Definition

A spring wrapped around the z-
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
can be defined parametrically by: :x(u, v) = \left(R + r\cos\right)\cos, :y(u, v) = \left(R + r\cos\right)\sin, :z(u, v) = r\sin+, where :u \in
helix_ A_helix_()_is_a_shape_like_a__corkscrew_or__spiral_staircase.__It_is_a_type_of__smooth_space_curve_with_tangent_lines_at_a_constant_angle_to_a_fixed_axis._Helices_are_important_in_biology,_as_the__DNA_molecule_is_formed_as__two_intertwined_helic_...
,_ :r_\,_is_the_radius_of_the_tube, :P_\,_is_the_speed_of_the_movement_along_the_z_axis_(in_a_Right-hand_rule.html" "title=",\ 2n\pi)\ \left(n \in \mathbb\right), :v \in [0,\ 2\pi), :R \, is the distance from the center of the tube to the center of the
helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
, :r \, is the radius of the tube, :P \, is the speed of the movement along the z axis (in a Right-hand rule">right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
Cartesian coordinate system, positive values create right-handed springs, whereas negative values create left-handed springs), :n \, is the number of rounds in a spring. The implicit function in Cartesian coordinates for a spring wrapped around the z-
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
, with n = 1 is :\left(R - \sqrt\right)^2 + \left(z + \right)^2 = r^2. The interior
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of the spiral is given by :V = 2\pi^2 n R r^2 = \left( \pi r^2 \right) \left( 2\pi n R \right). \,


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 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 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 In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
can be viewed as a special case of the spring obtained when the helix degenerates to a circle.


References

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See also

* Spiral *
Helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
Surfaces