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I want to calculate the parametric equations of the surface of a tape helically wound around a helix. The axis of the tape is basically a slinky https://mathworld.wolfram.com/Slinky.html.

I tried different methodologies (I started with tangent, normal and binormal vectors) without success; what I got is here Fig1 .

The surface I want to paramterize is the following one (the blue one): Fig 3

I appreciate any help you can provide. Thanks!

Update.

Using this equation:

=(+cos())cos(+) =(+cos())sin(+) =ℎ(+)+sin()

This is what I got: Fig 2

Giac
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2 Answers2

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The correct way to generalize the given parametrization is $$\begin{align} x(t,u) &= (R + a \cos (\omega t + u)) \cos t \\ y(t,u) &= (R + a \cos (\omega t + u)) \sin t \\ z(t,u) &= h t + a \sin (\omega t + u). \end{align}$$ $R$ controls the turning radius of the primary helical structure; $a$ is the radius of the winding of the surface around the primary helix--i.e., it is a "secondary" helical radius. Then $\omega$ is the frequency of windings for each full rotation around the primary helix, and $h$ is the vertical period for each full rotation. The parameter $t$ corresponds to the total angular rotations around the primary helix, an $u$ is a parameter that controls the width of the ribbon.

The following plot corresponds to the choices $$(R, a, \omega, h) = (8, 1.5, 15, 2),$$ for the parameter ranges $t \in [0, 4\pi]$ and $u \in [-1,1]$:

enter image description here

The drawback to this parametrization is that the orientation of the ribbon as it winds around the primary helix is not perpendicular to the tangent vector of the helix. You can see this in certain areas of the plot: the ribbon is sort of "warped." In order to correct this, you would need a much more complicated parametrization of a helical tube. For example,

$$\begin{align} x(t,u) &= R \cos t - a \cos t \sin (\omega t + u) + \frac {h a \sin t \sin (\omega t + u)}{\sqrt{R^2 + h^2}} \\ y(t,u) &= R \sin t - a \sin t \cos (\omega t + u) + \frac{h a \cos t \sin (\omega t + u)}{\sqrt{R^2 + h^2}} \\ z(t,u) &= h t + \frac{R a \sin (\omega t + u)}{\sqrt{R^2 + h^2}} \end{align}$$ for the same choices, yields a more pleasing result: enter image description here

heropup
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Starting from a parametrization of a 1D slinky $$ \begin{aligned}&x=(R+a\cos(\omega t))\cos(t)\\ &y=(R+a\cos(\omega t))\sin(t)\\ &x=ht+a\sin(\omega t) \end{aligned}$$ you can modify it to get a parametrization of your curve in the variables $(s,t)$: $$ \begin{aligned}&x=(R+a\cos(\omega t))\cos(t+s)\\ &y=(R+a\cos(\omega t))\sin(t+s)\\ &x=h(t+s)+a\sin(\omega t) \end{aligned},$$ where $t$ ranges in some interval $I$ (it could be $\mathbb R$) and $s\in(\varepsilon/2,\epsilon/2)$, where $\varepsilon$ is roughly the width of the band.

Edit: this one is probably closer to what you want, as it morally makes the width of the tape independent on the distance from the main symmetry axis:

$$ \begin{aligned}&x=(R+a\cos(\omega t))\cos\left(t+\frac{s}{R+a\cos(\omega t)}\right)\\ &y=(R+a\cos(\omega t))\sin\left(t+\frac{s}{R+a\cos(\omega t)}\right)\\ &x=h\left(t+ \frac{s}{R+a\cos(\omega t)}\right)+a\sin(\omega t) \end{aligned}.$$

Lorenzo Pompili
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  • Hi Lorenzo, thank you very much for your answer. However, this equation was one of the first ones I tried and it doesn't work. Am I missing something? Another point, why is roughly the width of the tape and not precisely the width? Also, shouldn’t s vary from – /(2a) to /(2a)? Thanks! – Giac Mar 08 '23 at 10:34
  • I updated the question so you can see what I got. Thanks again! – Giac Mar 08 '23 at 10:41
  • The width is not precisely $\varepsilon$ basically because the parametrization is not orthogonal. – Lorenzo Pompili Mar 08 '23 at 13:10
  • What is the problem of figure 2? I think I don’t get precisely what is the parametrization you want to obtain, can you explain it with more details? – Lorenzo Pompili Mar 08 '23 at 13:11
  • Hi Lorenzo, thanks again for your reply. The parametrisation I want is the following one: imagine having a solid helix with radius a. The simple helix equation (with radius R and constant h) describes the solid helix. Now, I wrap a tape around this solid helix. The ratio between the two twist pitches is equal to w. This is the surface that I would like to parametrise. I updated the question with a third picture that can make it clear. – Giac Mar 08 '23 at 15:35
  • I see almost no difference between figure 1 and 3. Maybe you mean that the width of the tape has to remain constant. In figure 1 the tape gets thinner when facing the axis around which the helix turns. Is this the problem with the parametrization I provided to you? – Lorenzo Pompili Mar 08 '23 at 21:03
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    Let me make a remark: ANY parametrization you make is never gonna be as wiping a tape around a solid helix, because the curvature of the tape is zero, while the helix has some zones where the curvature is positive and others while the curvature is negative (have you ever tried wiping some tape around some curved thing and noticed that it is impossible to make that perfect?). The tape HAS to be able to stretch. So you need to specify in which sense it stretches: it either has to get thinner when closer to the main axis, or it has to bend in some strange way in order to preserve its width, or … – Lorenzo Pompili Mar 08 '23 at 21:05
  • I wrote one parametrization that morally preserves the width of the tape – Lorenzo Pompili Mar 08 '23 at 21:17