Crumpling up a line: how to tell if a Line self intersects?

Question

I'm crumpling up lines by rotating them at random positions. That is I take a line, a random rotation angle, and a random pivot point, then every point beyond that gets rotated about the pivot by the angle.

rotator[points_, θ_] := 
 With[{r = RotationMatrix[θ], p1 = points[[1]]},
  (r . (# - p1) + p1) & /@ points
  ]
crumple[points_, θ_] := 
 Module[{head, tail, t = RandomReal[{-θ/2, θ/2}]},
  {head, tail} = 
   TakeDrop[points, RandomInteger[{2, Length[points] - 1}]];
  Join[head, rotator[tail, t]]
  ]
points = {#, 0} & /@ Subdivide[2, 100];
result = NestList[crumple[#, 0.8] &, points, 500];
Manipulate[
 Graphics[{Thick, Line[result[[i]]]}, Axes -> True, 
  PlotRange -> {{-1, 1}, {-1, 1}}, AspectRatio -> 1], {i, 1, 
  Length[result], 1}]

The trouble is the line will eventually self intersect given enough crumpling. So I'd like to be able to test if a Line self intersects so I can avoid any random rotations that lead to invalid configurations.