Cycloid Evolute

x[t_] := Sin[t] + t;
y[t_] := Cos[t];
tmin = -5; tmax = 5;
n = 41;
notzero[x_] := If[x == 0, 0.001, x];
normals = N@Table[{y'[t], -x'[t]}, {t, tmin, tmax, (tmax - tmin)/n}];
ps = N@Table[{x[t], y[t]}, {t, tmin, tmax, (tmax - tmin)/n}];
ts = N@Table[t, {t, tmin, tmax, (tmax - tmin)/n}];
X[t_] := x[t] -
   y'[t] ( x'[t]^2 + y'[t]^2)/notzero[x'[t] y''[t] - x''[t] y'[t]];
Y[t_] := y[t] +
   x'[t] ( x'[t]^2 + y'[t]^2)/notzero[x'[t] y''[t] - x''[t] y'[t]];
frame[\[Tau]_] :=
  Show[
   Graphics[{

     }, PlotRange -> {{-4, 4}, {-4, 2}}, ImageSize -> {300, 300}],
   Reverse@Table[
     ParametricPlot[
      ps[[i]] + t normals[[i]]/notzero@Norm[normals[[i]]], {t, -100,
       100},
      PlotStyle ->
       If[i == \[Tau] || i == \[Tau] - Length@ts,
        Directive[Thickness[0.01], RGBColor[0.4, 0.4, 0.9]],
        Directive[Thick, Opacity[0.4], RGBColor[0.1, 0.1, 0.1]]]
      ], {i, If[\[Tau] > Length[ts], \[Tau] - Length[ts], 1],
      If[\[Tau] > Length[ts], Length[ts], \[Tau]]}],
   ParametricPlot[{{x[t], y[t]}, {X[t], Y[t]}}, {t, tmin, tmax},
    PlotStyle -> {Directive[Thick, RGBColor[0.9, 0.2, 0.3]],
      Directive[Thick, RGBColor[1.0, 0.1, 0.2]]}],
   Graphics[{PointSize[0.05], RGBColor[0.2, 0.2, 0.6],
     Point[ps[[Mod[\[Tau], Length@ts, 1]]]],
     \[Theta] = ts[[Mod[\[Tau], Length@ts, 1]]];
     Point[{Sin[\[Theta] + Pi] + \[Theta], Cos[\[Theta] + Pi] - 2}],
     Thick, Black,
     Line[{{-4, -1}, {4, -1}}],
     Circle[{\[Theta], 0}, 1],
     Table[
      Line[{{Sin[\[Theta] + dt], Cos[\[Theta] + dt]} + {\[Theta],
          0}, {Sin[\[Theta] + dt + Pi],
          Cos[\[Theta] + dt + Pi]} + {\[Theta], 0}}]
      , {dt, 0, Pi, Pi/2}],

     Line[{{-4, -3}, {4, -3}}],
     Circle[{\[Theta], -2}, 1],
     Table[
      Line[{{Sin[\[Theta] + dt],
          Cos[\[Theta] + dt]} + {\[Theta], -2}, {Sin[\[Theta] + dt +
            Pi], Cos[\[Theta] + dt + Pi]} + {\[Theta], -2}}]
      , {dt, 0, Pi, Pi/2}]
     }]
   ];
Manipulate[frame[\[Tau]], {\[Tau], 1, 2 Length[ts], 1}]