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I need to write some code to draw a 2D helix. Not a spiral, but more like the 2D projection of a normal helix. This would be like the representation of the gluon particle in a Feynman Diagram. This is shown in the image.

enter image description here

What would the equation of this shape look like?

Youssef Moawad
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2 Answers2

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You can indeed use the projection of a 3D helix.

The parametric equations of an horizontal helix can be written

$$\begin{cases}x=r\cos t,\\y=r\sin t,\\z=at.\end{cases}$$

Then rotating by an horizontal angle $\theta$ and dropping the $z$ coordinate,

$$\begin{cases}x'=r\cos t\cos\theta-at\sin\theta,\\y'=r\sin t\end{cases}.$$

These are cycloids.

enter image description here

  • They are not pure, but scaled (prolate) cycloids... projections of a helix in $(x−y) $ plane rotated about $y $ axis through angle $θ$. – Narasimham Jul 01 '17 at 16:27
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Feynman

As a start,

$$ (x,y)= ( a t - k \sin t , a - k\, \cos t) ,(0<t< 10 \pi) $$

a = 1 ; k = 1.8;
ParametricPlot[{ a t - k Sin[t]  , a - k  Cos[t] + .08 t},  { t, 
   0, 10 Pi} , AspectRatio -> Automatic, GridLines -> Automatic]

in Mathematica code. If you don't want the ramp up, remove the last $.08 t$. Can vary $k/a$ ratio to influence the curling, and also the $t$ limits to lengthen the helix or shorten it...

Narasimham
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