Curly chaps

Lines[l_] := Module[{curr, \[Theta] = 0, out = {}},
   curr = {{0, 0}, {1, 0}};
   AppendTo[out, curr];
   Do[
    \[Theta] = N@l[[i]];
    curr = Map[# + (curr[[2]] - curr[[1]]) &, curr];
    curr[[2]] = RotationTransform[\[Theta], curr[[1]]][curr[[2]]];
    AppendTo[out, curr];
    , {i, Length[l]}
    ];
   out
   ];
angles = Table[0.08 i, {i, 3635}];
lines = Lines[angles];
anglesaccum = Join[{0}, Accumulate[angles]];
frame[n_] :=

  Graphics[{{Opacity[0.8], ColorData["DarkRainbow"][0],
     Thickness[0.005], Line@lines[[;; n]]},
    If[n > 1, {
      Darker@Red, EdgeForm[], FaceForm[Opacity[0.4]], Thickness[0.01],
       Disk[lines[[n, 1]],
       0.5, {anglesaccum[[n]], Pi + anglesaccum[[n - 1]]}
       ]
      }, {}],
    Darker@Blue,
    If[n < 200, Map[Disk[First[#], 0.05] &, lines[[;; n]]], {}]},
   PlotRange -> All, ImageSize -> {300, 300}, Background -> White];
ts = Join[Range[35], 35 + Range[20]*2, 75 + Range[40]^2];
Manipulate[frame[ts[[i]]], {i, 1, Length@ts, 1}]