circular helix

The space curve traced out by the parameterization


is called a circular helix (plur. helices).

Its Frenet frame is:

𝐓 =1a2+b2⁒[-a⁒sin⁑ta⁒cos⁑tb],
𝐍 =[-cos⁑t-sin⁑t0],
𝐁 =1a2+b2⁒[b⁒sin⁑t-b⁒cos⁑ta].

Its curvaturePlanetmathPlanetmath and torsion are the following constants:


A circular helix can be conceived of as a space curve with constant, non-zero curvature, and constant, non-zero torsion. Indeed, one can show that if a space curve satisfies the above constraints, then there exists a system of Cartesian coordinatesMathworldPlanetmath in which the curve has a parameterization of the form shown above.

FigureΒ 1: A plot of a circular helix with a=b=1, and ΞΊ=Ο„=1/2.

An important property of the circular helix is that for any point of it, the angle Ο† between its tangent and the helix axis is constant. Indeed, if we consider the position vector of that arbitrary point, we have (where 𝐀 is the unit vector parallelMathworldPlanetmathPlanetmath to helix axis)

d⁒𝜸d⁒t⋅𝐀=[-a⁒sin⁑ta⁒cos⁑tb]⁒[0 0 1]=b≑βˆ₯d⁒𝜸d⁒tβˆ₯⁒cos⁑φ=a2+b2⁒cos⁑φ.



as was to be shown.

There is also another parameter, the so-called pitch of the helix P which is the separationMathworldPlanetmathPlanetmath between two consecutive turns. (It is mostly used in the manufacture of screws.) Thus,


and P is also a constant.

Title circular helix
Canonical name CircularHelix
Date of creation 2013-03-22 13:23:25
Last modified on 2013-03-22 13:23:25
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Definition
Classification msc 53A04
Related topic SpaceCurve
Related topic RightHandedSystemOfVectors
Defines circular helices