Rolling shapes

(*define r of theta here:*)
superellipse[\[Theta]_, m_,
   n_] := (Cos[m \[Theta] /4]^(2 n) + Sin[m \[Theta] /4]^(
    2 n))^(-1/(2 n));
r[\[Theta]_] := 1/(1 + 0.9 Cos[\[Theta]]);
r[\[Theta]_] := superellipse[\[Theta], 4, 10];
y[\[Theta]_] := r[\[Theta]] Sin[\[Theta]];
x[\[Theta]_] := r[\[Theta]] Cos[\[Theta]];
angle[\[Theta]_] := ArcTan[x'[\[Theta]], y'[\[Theta]]];
soln = NDSolve[{xt'[t] == Sqrt[r'[t]^2 + r[t]^2], xt[0] == 0},
   xt[t], {t, 0, 4 Pi}];
X[tt_] := First[xt[t] /. soln /. {t -> tt}];
point[t_, tt_] := {X[t], 0} +
   RotationMatrix[-angle[t]].(r[tt] {Cos[tt], Sin[tt]} -
      r[t] {Cos[t], Sin[t]});
Manipulate[
 Show[Graphics[{PointSize[Medium], Point[point[\[Tau], \[Alpha]]]},
   PlotRange -> {{X[0] - 3, X[4 Pi] + 3}, {-1, 3}},
   ImageSize -> {500}],
  ParametricPlot[point[\[Tau], tt], {tt, 0, 2 Pi},
   PlotStyle -> Darker@Red],
  ParametricPlot[point[tt, \[Alpha]], {tt, 0, \[Tau]},
   PlotStyle -> Red]], {\[Tau], 10^-3, 4 Pi}, {\[Alpha], 0, 2 Pi}]