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I know the parametric equation for a $3D$ helix is:

$x = R \cos t$

$y = R \sin t$

$z = h t$

Can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix around helix" / Slinky:

$x = [R + a \cos(\omega t)] \cos t$

$y = [R + a \cos(\omega t)] \sin t$

$z = h t + a \sin(\omega t)$.

helix around helix

I don't understand what are the variables '$a$' and '$\omega$' supposed to represent. I assume '$h$' is the height?

How could I expand those equations for an $n$-number of helices? (say this is $2$ helices, what would change in the formula for $3$ helices?)

I would like to be able to generate such structures in Solid Works. I can get the simple $3D$ helix from the equations, but if I try with the "Slinky" one it only makes strange (yet beautiful) shapes.

Any help? I know this is way over my head, but the coil-made of a coil-made of a coil structure has obsessed me for quite some time now (I would like to sculpt it in real-life, but first model it in $3D$)

Thanks!

Glorfindel
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cannotcompute
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  • $\omega $ oscilates rapidly and gives the slinky effect, a is the amplitude , reflected in a rather small radius for the coil of the slinky , R is the widest loop of the helix and h is the pitch of the larger helix. Thanks, fun to draw! – Alan Sep 25 '14 at 00:31
  • Could you explain what value should the Omega (w) have in the equation? It's not just a number, it has to be some kind of longer function (the oscillation, like a sinewave function or something?)? Sorry, my math terminology is lacking. – cannotcompute Sep 25 '14 at 05:00
  • I got a nice result using: $ R = 3$ , $a = .4$ , $ h = .2$ , $\omega = 40$ – Alan Sep 25 '14 at 09:48
  • Ok, I figured out how to plot this in GnuPlot!Thanks Alan! – cannotcompute Sep 26 '14 at 00:50

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