Rolling a square in a square

RollSub[p_, dir_, \[Theta]_] := Module[{sq, rp},
   sq = {p, p + dir, p + dir + RotationMatrix[Pi/2].dir,
     p + RotationMatrix[Pi/2].dir};
   sq = Map[RotationMatrix[-\[Theta]].(# - (p + dir)) + (p + dir) &,
     sq];
   sq = Map[# &, sq]
   ];
Roll[p_, dir_, \[Theta]_] :=
  Module[{n = Floor[2 \[Theta]/Pi], sq0, sq},
   sq0 = RollSub[p, dir, 0];
   sq = Nest[RotateLeft[RollSub[#[[1]], dir, Pi/2]] &, sq0, n];
   RollSub[sq[[1]], dir, \[Theta] - n Pi/2]
   ];
RollAround[\[Theta]_] := Module[{p, dir, th, n, m},
  n = Mod[Floor[\[Theta]/Pi], 4] + 1;
  p = {{0, 0}, {3, 0}, {3, 3}, {0, 3}}[[n]];
  dir = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}[[n]];
  th = \[Theta] - Pi (n - 1);
  m = Mod[Floor[2 \[Theta]/Pi], 4] + 1;
  Nest[RotateRight, Roll[p, dir, th],
   1/4 (-3 - (-1)^m + 6 m) + 2 Floor[(n + 1)/2]]
  ];
Manipulate[
 Show[Graphics[{
    EdgeForm[Directive[Thick, RGBColor[0.1, 0.2, 0.1]]], FaceForm[],
    Rectangle[{0, 0}, {3, 3}],
    EdgeForm[Thick],
    Polygon[RollAround[Mod[\[Theta], 4 Pi]]]
    }, PlotRange -> {{-0.1, 3.1}, {-0.1, 3.1}}],

  ParametricPlot[
   RollAround[t], {t, If[\[Theta] < 4 Pi, 0, Mod[\[Theta], 4 Pi]],
    If[\[Theta] < 4 Pi, Max[\[Theta], 0.01], 4 Pi - 0.01]},
   PlotStyle -> RGBColor[0.450385`, 0.157961`, 0.217975`]]
  , ImageSize -> 320]
 , {\[Theta], 0, 8 Pi - 0.01}]