circular helix

The space curve traced out by the parameterization

$$𝜸(t)=\left[\begin{array}{c}\hfill a\cos (t)\hfill \\ \hfill a\sin (t)\hfill \\ \hfill bt\hfill \end{array}\right],t\in ℝ,a,b\in ℝ$$

is called a circular helix (plur. helices).

Its Frenet frame is:

$𝐓$$={\displaystyle \frac{1}{\sqrt{a^2+b^2}}}\left[\begin{array}{c}\hfill -a\sin t\hfill \\ \hfill a\cos t\hfill \\ \hfill b\hfill \end{array}\right],$

$𝐍$$=\left[\begin{array}{c}\hfill -\cos t\hfill \\ \hfill -\sin t\hfill \\ \hfill 0\hfill \end{array}\right],$

$𝐁$$={\displaystyle \frac{1}{\sqrt{a^2+b^2}}}\left[\begin{array}{c}\hfill b\sin t\hfill \\ \hfill -b\cos t\hfill \\ \hfill a\hfill \end{array}\right].$

Its curvature and torsion are the following constants:

$\kappa ={\displaystyle \frac{a}{a^2+b^2}},\tau ={\displaystyle \frac{b}{a^2+b^2}}.$

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

Figure 1: A plot of a circular helix with $a=b=1$, and $\kappa =\tau =1/2$.

An important property of the circular helix is that for any point of it, the angle $\phi$ 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 parallel to helix axis)

${\displaystyle \frac{d𝜸}{dt}}\cdot 𝐤=\left[\begin{array}{c}\hfill -a\sin t\hfill \\ \hfill a\cos t\hfill \\ \hfill b\hfill \end{array}\right]\left[\begin{array}{c}\hfill \mathrm{0\hspace{0.33em}0\hspace{0.33em}1}\hfill \end{array}\right]=b\equiv \parallel {\displaystyle \frac{d𝜸}{dt}}\parallel \cos \phi =\sqrt{a^2+b^2}\cos \phi .$

Therefore,

$\cos \phi ={\displaystyle \frac{b}{\sqrt{a^2+b^2}}}\text{constant},$

as was to be shown.

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

$P={\gamma }_3(t+2\pi )-{\gamma }_3(t)=b(t+2\pi )-bt=2\pi b,$

and $P$ is also a constant.