circular helix (original) (raw)

The space curve traced out by the parameterization

𝜸⁢(t)=[a⁢cos⁡(t)a⁢sin⁡(t)b⁢t],t∈ℝ,a,b∈ℝ

is called a circular helix (plur. helices).

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.

Figure 1: A plot of a circular helix with a=b=1, and κ=τ=1/2.

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

d⁢𝜸d⁢t⋅𝐤=[-a⁢sin⁡ta⁢cos⁡tb]⁢[0 0 1]=b≡∥d⁢𝜸d⁢t∥⁢cos⁡φ=a2+b2⁢cos⁡φ.

Therefore,

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,

P=γ3⁢(t+2⁢π)-γ3⁢(t)=b⁢(t+2⁢π)-b⁢t=2⁢π⁢b,

and P is also a constant.