Drawn Caustic

f[x_] := If[Abs[x] < 1, -Sqrt[1 - x^2], 0];
(*f[x_]:=x^2;*)
n = 100;
T = 300;
lines = Table[line[{-1 + 2 i/n, 1}, Pi, T, 0.1, f, 0.01, 3],
   {i, n - 1}];
T = Length[lines];

Clear[line];
line[p0_, \[Theta]0_, n_, dx_, f_, \[Sigma]_: 0.01, r_: 5] :=
  Module[{p = p0, p1, \[Theta] = \[Theta]0, out, x},
   out = {p};
   Do[
    \[Theta] += RandomVariate[NormalDistribution[0, \[Sigma]]];
    p1 = p;
    p += dx {Sin[\[Theta]], Cos[\[Theta]]};
    x = (p[[1]] + p1[[1]])/2;

    If[And[f[p1[[1]]] < p1[[2]], f[p[[1]]] > p[[2]]],
     With[{\[Beta] = ArcTan[f'[x]]},
      \[Theta] = Pi - \[Theta] - 2 \[Beta];
      ];
     p = {x, f[x]};
     ];
    AppendTo[out, p];
    , {i, n}];
   MovingAverage[out, r]
   ];

frame[t_] := Rotate[(Show[
     Graphics[{
       Opacity[0.5], Thickness[0.003],
       Table[
        {ColorData["DarkRainbow"][i/n],
         Line[lines[[i]][[
           If[t <= T, 1, t - T] ;; If[t < T, t, T]]]]},
        {i, Length@lines}
        ]
       }, PlotRange -> {{-1, 1}, {-1.1, 1}}, Background -> White,
      ImageSize -> 250],
     Plot[f[x], {x, -1, 1}, PlotStyle -> Directive[Thick, White]]
     ]), Pi];
Manipulate[
 frame[t],
 {{t, T}, 0, 2 T + 1, 1}]