Hello everyone.
I've been looking for a solution to a problem that only seems to get more difficult the more I search or an answer.
It starts out with finding a solution to write the equation for a helix torus, which is not too difficult, but then it starts to get confusing when I ask for a double helix, but as a 'twisted ribbon'.
It would be similar to a DNA ribbon but solid, like if you were to take a strip of metal and twisted it at both ends.
I can show a figure of that, it would be the figure under "Helix B"...
The ribbon would have a curve that follows the axis of a torus.
Finding an equation thus far would be a great help.
My problem however does go into further detail from here, the 'ribbon' that I have in mind has a more exotic shape if I were to show a cross section of it and would eventually have to be converted into another expression; but I wouldn't expect anyone to want to go into further detail.
Would it possible to have someone get me started and then figure out where I need to go from here?
what this does is that a line segment from (0,0,-1/2) to (0,0,1/2), parameterized by $s$, translate that out to the perimeter of a circle radius R, and rotate that segment about the line tangent to the circle, as you noted, $2 \omega$ times in the entire circle. $\theta=1$ does a single circle. $\omega$ can be varied as you like. $\theta$ should probably remain 1. You can vary $R$ and you can vary the limits on $s$ to get different sized toroid ribbons.
it's in Mathematica. I posted the whole sheet. What you see is what there is.
I tried running it at wolframalpha.com and it doesn't seem to want to deal with it. It should be pretty straightforward to do this in whatever the equivalent of POVRAY is these days. (it's been a while since i toyed with rendering...)