Weird Wheels & Roads

Wheel[road_, x_, xmax_] :=

  NDSolve[{\[Theta]'[x] == -1/func, \[Theta][0] == -Pi/
      2}, \[Theta], {x, 0, xmax}];
PolygonRoad[x_, sides_] :=
  Module[{\[Alpha] = ArcSinh[Tan[Pi/sides]]},
   -Cosh[If[x >= 0,
      NestWhile[# - 2 \[Alpha] &, x, Abs[#] >= \[Alpha] &],
      NestWhile[# + 2 \[Alpha] &, x, Abs[#] >= \[Alpha] &]]]
   ];
frame[xt_] := Module[{soln},
   func = PolygonRoad[x + xt, 6];
   (*func=Cos[x+xt]-Sqrt[17];*)
   xmax = 3 Pi;
   soln = Wheel[func, x, xmax];
   Show[
    ParametricPlot[-func First[{Cos[\[Theta][x]],
         Sin[\[Theta][x]]} /. soln], {x, 0, xmax}, Axes -> None,
     PlotRange -> {{-xmax/3, xmax/3}, {-2, 2}}],
    Plot[func, {x, -xmax/3, xmax/3}, Axes -> None]
    ]];
Manipulate[frame[xt], {xt, 0, Pi/2}]