For people interested in treating (epi/hypo)trochoids as line envelopes, there are nice papers on this subject by Simoson and Quenell, among others. Simoson's treatment, in particular, suggests modifying the original code by John to a 3D version, given below:

base = 360; max = 359;
plist = {Sin[(2.*Pi*#/base)], Cos[2.*Pi*#/base]} & /@ Range[0, base - 1];
colors = Table[ColorData["BalancedHue", x], {x, 0, 1, 1./base}];

zlist[p_, m_] := Table[{Append[p[[i + 1]], 0],
                        Append[p[[Mod[IntegerPart[m*i], base] + 1]], 1]},
                       {i, base - 1}]

Manipulate[Graphics3D[{{Directive[EdgeForm[Directive[Black, Opacity[1/2]]], 
                                  FaceForm[]], Cylinder[{{0, 0, 0}, {0, 0, 1}}]},
                       {Directive[AbsoluteThickness[1/2], colors[[IntegerPart[m]]]], 
                        Line[zlist[plist, m]]}}, Boxed -> False],
           {m, 1.1, base - 0.1, 0.1}, SaveDefinitions -> True]

I would follow John's advice and run the code on the desktop, my jaw dropped!
Thank you @John Shonder for sharing :)

Where is that tie-dyed shirt I used to love?

Nice Job!