Do these equations create a helix wrapped into a torus?

Question

I don't have any graphing software capable of plotting this, but it makes sense to me.

First a circle:

$$ x = r\cos\theta \\ y = r\sin\theta $$

Then a sine wave wrapped into a circle:

$$ x = (r+\cos\theta)\cos\theta\\ y = (r+\sin\theta)\sin\theta $$

And finally, add in the z-coordinate:

$$ z = \sin\theta $$

Is this correct? If not, what's wrong with it?

$$ x = (r+\cos\theta)\cos\theta = r\cos\theta + \cos^2\theta\\ y = (r+\sin\theta)\sin\theta = r\sin\theta + \sin^2\theta\\ z = \sin\theta $$

Answer 1

You need two radii to decribe a torus. Let's call them $a$ and $b$. Then the parametric equations of the torus are: $$x = (a + b\cos u)\cos v$$ $$y = (a + b\cos u)\sin v$$ $$z = b\sin u$$

Then, to get a helical curve, set $v = ku$, where $k << 1$.

Here's the result with $a=3$, $b=1$, $k=0.05$:

Answer 2

Here's the parametrization for a helix of $n$ winds wrapped around a torus of major radius $R$ and minor radius $r$: $$\begin{align*} x&=(R+r\cos(nt))\cos(t)\\ y&=(R+r\cos(nt))\sin(t)\\ z&=r\sin(nt) \end{align*}$$

Making this into a function in Mathematica,

HelixPlot[R_, r_, n_] := 
 Show[ParametricPlot3D[{(R + r Cos[t]) Cos[u], (R + r Cos[t]) Sin[u], 
    r Sin[t]}, {t, 0, 2 Pi}, {u, 0, 2 Pi}, PlotPoints -> 30, 
   Mesh -> None, PlotStyle -> Opacity[0.3]], 
  ParametricPlot3D[{(R + r Cos[n*t]) Cos[t], (R + r Cos[n*t]) Sin[t], 
    r Sin[n*t]}, {t, 0, 2 Pi}, PlotPoints -> 30, 
   PlotStyle -> {Thick, Black}]]

we can toy with different settings.

HelixPlot[6, 2, 5] produces

HelixPlot[6, 1, 10] produces

HelixPlot[6, 5, 20] produces