PlanetMath: circular helix

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

circular helix

(Definition)

The space curve traced out by the parameterization

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

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.

**Figure:** A plot of a circular helix with , and $\kappa = \tau = 1/2$ .
$\includegraphics{helix2.eps}$

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)

$\displaystyle \frac{d\boldsymbol{\gamma}}{dt}\cdot\mathbf{k}=\begin{bmatrix}-a\... ...frac{d\boldsymbol{\gamma}}{dt}\bigg\rVert\cos\varphi=\sqrt{a^2+b^2}\cos\varphi.$

Therefore,

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

as was to be shown.

There is also another parameter, the so-called pitch of the helix 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

is also a constant.

Anyone with an account can edit this entry. Please help improve it!

"circular helix" is owned by rspuzio. [ full author list (6) | owner history (1) ]

(view preamble)

Also defines:

circular helices

Log in to rate this entry.
(view current ratings)

Cross-references: consecutive, separation, parameter, parallel, unit vector, position vector, axis, tangent, angle, point, property, curve, Cartesian coordinates, torsion, curvature, Frenet frame, space curve
There are 3 references to this entry.

This is version 10 of circular helix, born on 2003-01-26, modified 2008-06-11.
Object id is 3928, canonical name is Helix.
Accessed 5174 times total.

Classification:

AMS MSC:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

No messages.

Interact