Manipulate[
 If[i == 1, 
  Q = ParametricPlot[{(R + r) Cos[t] - 
      d*Cos[((R + r)/r) t], (R + r) Sin[t] - 
      d*Sin[((R + r)/r) t]}, {t, 0, 80 Pi}, 
    ColorFunction -> (ColorData["Rainbow"][#3] &), 
    PlotStyle -> Thick]];
 If[i == 2, 
  Q = ParametricPlot[{(R - r) Cos[t] + 
      d*Cos[((R - r)/r) t], (R - r) Sin[t] - 
      d*Sin[((R - r)/r) t]}, {t, 0, 80 Pi}, 
    ColorFunction -> (ColorData["Rainbow"][#3] &), 
    PlotStyle -> Thick]]; Q,
 {{i, 1, "Curve:"}, {1 -> "Epitrochoid", 2 -> "Hypotrochoid"}}, {{R, 
   0}, -10, 10, Appearance -> "Open"}, {{r, 1}, -10, 10, 
  Appearance -> "Open"}, {{d, 0}, -10, 10, Appearance -> "Open"}]