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.
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.)