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# circular helix

The space curve traced out by the parameterization

$\boldsymbol{\gamma}(t)=\left[\begin{array}[]{c}a\cos(t)\\ a\sin(t)\\ bt\end{array}\right],\quad t\in\mathbb{R},\;a,b\in\mathbb{R}$ |

is called a *circular helix* (plur. helices).

Its Frenet frame is:

$\displaystyle\mathbf{T}$ | $\displaystyle=\frac{1}{\sqrt{a^{2}+b^{2}}}\begin{bmatrix}-a\sin t\\ \hphantom{-}a\cos t\\ b\end{bmatrix}\,,$ | ||

$\displaystyle\mathbf{N}$ | $\displaystyle=\begin{bmatrix}-\cos t\\ -\sin t\\ 0\end{bmatrix}\,,$ | ||

$\displaystyle\mathbf{B}$ | $\displaystyle=\frac{1}{\sqrt{a^{2}+b^{2}}}\begin{bmatrix}\hphantom{-}b\sin t\\ -b\cos t\\ a\end{bmatrix}\,.$ |

Its curvature and torsion are the following constants:

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

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

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

$\displaystyle\frac{d\boldsymbol{\gamma}}{dt}\cdot\mathbf{k}=\begin{bmatrix}-a% \sin t\\ \hphantom{-}a\cos t\\ b\end{bmatrix}\begin{bmatrix}0\;0\;1\end{bmatrix}=b\equiv\bigg\lVert\frac{d% \boldsymbol{\gamma}}{dt}\bigg\rVert\cos\varphi=\sqrt{a^{2}+b^{2}}\cos\varphi.$ |

Therefore,

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

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,

$\displaystyle P=\gamma_{3}(t+2\pi)-\gamma_{3}(t)=b(t+2\pi)-bt=2\pi b\,,$ |

and $P$ is also a constant.

## Mathematics Subject Classification

53A04*no label found*

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