Rolling Cardioid

Manipulate[
 Show[
  Graphics[{
    Thickness[Medium], PointSize[Medium],
    {GrayLevel[0.3], Disk[{0, 0}, 1]},
    {Darker@Red, Circle[{Sin[\[Theta]], -Cos[\[Theta]]}, 2]},
    Point[
     Table[{Sin[\[Theta]], -Cos[\[Theta]]} +
       2 {Sin[\[Phi] - \[Theta]/2],
         Cos[\[Phi] - \[Theta]/2]}, {\[Phi], 0, 2 Pi, 2 Pi/n}]
     ]
    },
   PlotRange -> 3, ImagePadding -> 10],
  ParametricPlot[
   Table[
    {Sin[\[Theta]t], -Cos[\[Theta]t]} +
     2 {Sin[\[Phi] - \[Theta]t/2], Cos[\[Phi] - \[Theta]t/2]},
    {\[Phi], 0, 2 Pi - 0.0001, 2 Pi/n}
    ]
   , {\[Theta]t, If[\[Theta] < 2 Pi, 0, \[Theta] - 2 Pi],
    If[\[Theta] < 2 Pi, \[Theta], 2 Pi]},
   PlotStyle -> Darker@Blue
   ]
  ], {\[Theta], 0.001, 4 Pi - 0.001}]