Do these equations create a helix wrapped into a torus? 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?
$$
x = (r+\cos\theta)\cos\theta = r\cos\theta + \cos^2\theta\\
y = (r+\sin\theta)\sin\theta = r\sin\theta + \sin^2\theta\\
z = \sin\theta
$$
 A: 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

A: 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 $a=3$, $b=1$, $k=0.05$:

