Making Tracks

The underlying curve here is the intersection of the quartic surface $(1-x^2)^2+(1-y^2)^2=(1-z^2)^2$ with the unit sphere. This surface comes up in an interesting way in a project I'm currently working and which I'll describe in more detail at some point; for now, here are ten Gaussian translates.

Code:

DynamicModule[{cols, rands, y},
 SeedRandom[40];
 cols = RGBColor /@ {"#B36458", "#353E55"};
 rands = Table[RandomVariate[NormalDistribution[0, .05], 3], {10}];
 Manipulate[
  Graphics3D[{cols[[1]], Thickness[.005], Opacity[.5], 
    Table[y = Haversine[Clip[? s, {0, ?}]];
     {2 (-1)^m Norm[rands[[m]]] + # & /@ cols[[1]], 
      Line[Table[{Sqrt[(1 - y^2)/(1 + y^2)], y, y Sqrt[(1 - y^2)/(1 + y^2)]}*{i, j, k} + rands[[m]], 
         {s, Max[0, t - 1], Min[t, 1], 1/50}]]}, {i, {-1, 1}}, {j, {-1, 1}}, {k, {-1, 1}}, {m, 1, 10}]}, 
       Lighting -> "Neutral", PlotRange -> 1.2, Boxed -> False, Axes -> None, 
   ViewAngle -> ?/8, ImageSize -> 540, Background -> cols[[2]]], {t, 0, 2}]
 ]