Hypotrochoid

r = 1/3;
c = "SandyTerrain";
map[point_, t_] :=
  RotationMatrix[-t].(RotationMatrix[t/r].point + {0, 1 - r});
Manipulate[Show[Graphics[{
    Circle[{0, 0}, 1],
    Darker@Darker@Red,
    Circle[(1 - r) {Sin[t], Cos[t]}, r],
    Line[{map[{0, -1}, t], map[{0, 1}, t]}], PointSize[Medium],
    Table[{ColorData[c][(y + 1)/2], Point[map[{0, y}, t]]}, {y, -1, 1,
       1/6}]
    }, PlotRange -> {{-2, 2}, {-2, 2}}],
  Table[ParametricPlot[map[{0, y}, \[Tau]], {\[Tau], 0, t},
    PlotStyle -> ColorData[c][(y + 1)/2]], {y, -1, 1, 1/6}]
  ],
 {t, 10^-3, 2 Pi}]