A Helix is parameterized as $\langle R \cos(t), R \sin(t), \alpha t\rangle$ and one can visualize it as "wrapping" around a cylinder of radius R. I would like to accomplish the same thing but wrapping around a torus(or one can think of bending the cylinder into a torus).
$\langle R_1 \cos(t) + R_2 \cos(\beta t), R_1 \sin(t), R_2 \sin(\beta t)\rangle$ is sort of a solution but is not quite right as "sides" of the enclosed hypothetical torus are flat.
A seemingly better result is $\langle(R_2 + \cos(t)) \cos(\beta t), \sin(t), (R_2 + \cos(t))\sin(\beta t)\rangle$ and it looks almost right but it seems there might need to be a special relationship between $\beta$ and $R_2$ because some turns seem to get out of whack slightly.