1

I'm trying to find the parametric equations for a helix around a helix around a circle (helix on helix on circle) That is: I would like to start with a circle, add a helix around it and a helix around the helix.(See video)

I'm ok even if the second helix is not perfectly orthogonal to the first helix as long as we can have a simpler parametrization. I'm ok also if the curve represents a helix around a helix around a helix.

I know the helix around a helix around an axis is quite easy but I was not able to find a solutions for this case. I'm interested in this parametric curve as a way to represent time and I would like to write a program to show data attached to that curve.

Edit: I already know the parametric equations of a helix around a torus:

$$x(t) = (R+ r\cos(nt)) \cos(t)$$

$$y(t) = (R+ r\cos(nt)) \sin(t)$$

$$z(t) = s t + r \sin(nt)$$

where $R$ is the radius of the torus

$r$ is the radius of the helix

$n$ is the winding number

$s$ vertical velocity ($0$ if we want a closed curve).

What I'm looking for is the next level helix on top of that.

Sebastiano
  • 7,649
Guzman Tierno
  • 13
  • I totally endorse Ted Shifrin's answer as the definitive method for parametrizing these curves. I link to three older threads simply to connect all four 1, 2, 3. – Jyrki Lahtonen Oct 15 '20 at 12:38

1 Answers1

2

HINT: You are looking at a curve spiraling at a constant rate about a torus. Start by finding parametric equations of a torus. Now make the two angles linear functions of $t$. It appears you want one to go much faster than the other, so that suggests how you should relate those linear functions.

EDIT: Based on your comment, you want a curve that spirals around a given space curve. The way to do that is to take an orthonormal basis for the normal plane to the curve at each point and go around a circle as you move along the curve. In particular, take the Frenet frame $T,N,B$ for the curve, parametrized say by $\alpha(s)$. Now consider $$\alpha(s)+\cos\theta(s)N(s)+\sin\theta(s)B(s)$$ where $\theta$ is a linear function of $s$. (If you don't know about Frenet frames, see this or my differential geometry text, linked in my profile. It's most convenient to work with arclength-parametrized curves $\alpha(s)$, but the chain rule will do the heavy lifting for you if they're not.)

Ted Shifrin
  • 115,160
  • I'm looking for a curve that spirals around the curve you describe (I already know the parametric equations for the curve you described). – Guzman Tierno Jun 03 '20 at 22:38
  • Your question really needs to be made clearer. So you want a curve spiraling around the curve in your picture. Helix is not going to be an accurate word, because it has technical mathematical meaning. But I will edit appropriately. – Ted Shifrin Jun 03 '20 at 22:42
  • I understand what you say about the Frenet frame T,N,B, what I was trying to ask is: is there a direct parametric representation of that curve? – Guzman Tierno Jun 03 '20 at 22:48
  • I just gave you a "direct parametrization." You have the spiraling curve on the torus — that's $\alpha$, and then you have to calculate its Frenet frame and do as I said. Is it a simple, elegant formula? Um, NO. This isn't too hard to do in Mathematica or other CAS, but not much fun to do by hand. – Ted Shifrin Jun 03 '20 at 22:59
  • OK, I understand, I hoped it was possible to calculate everything and get an explicit formula but you're right, your formula will work fine especially in a program. thx. I was hoping for a formula that gives x(t), y(t), z(t) using t ... I thought it was possibile to generalize in some way the mechanism and find a pattern. – Guzman Tierno Jun 03 '20 at 23:02
  • Of course, what I'm suggesting will give an explicit formula. I just will not work it out by hand. If you find a superior solution, you should post it. Otherwise, I think you need to do what I suggested and take my answer as the answer. :P – Ted Shifrin Jun 03 '20 at 23:12
  • Ok, got it! Does Wolfram Alpha (or other programs) have explicit commands to calculate Frenet vectors? – Guzman Tierno Jun 03 '20 at 23:15
  • Not that I know of. Having taught differential geometry dozens of times, I wrote my own Mathematica program to do so. And others. If you have access to Mathematica (as opposed to Wolfram Alpha), I'm happy to share a notebook with you. – Ted Shifrin Jun 03 '20 at 23:17
  • Thank you very much, I installed and used Mathematica when at the university but I don't have it anymore. I think I'll just write some code and forget about the explicit formula. Thank you. – Guzman Tierno Jun 03 '20 at 23:21
  • Just to let you know that I was able to use Frenet vectors and Rodrigues' Rotation Formula https://mathworld.wolfram.com/RodriguesRotationFormula.html to get this: http://www.guzman.it/3/toro.html that was part of something I'm working on – Guzman Tierno Jun 07 '20 at 00:59
  • That looks cool. I never knew Rodrigues' formula per se, but it's just an application of the change of basis theorem (which I know all too well). Well done! By the way, you should accept the answer at this point :P – Ted Shifrin Jun 07 '20 at 01:08