Geodesics in Rotating Frame

n = 6;
\[Omega] = 1/2;
p1s = Table[{-Sqrt[1 - y^2], y}, {y, -1, 1, 2/n}][[2 ;; -2]];
p1s2 = Table[{x, -Sqrt[1 - x^2]}, {x, -1, 1, 2/n}][[2 ;; -2]];
Manipulate[
 Show[
  Graphics[{
    Circle[{0, 0}, 1],
    Table[
     Line[{{Sin[\[Theta] + \[Omega] t],
        Cos[\[Theta] + \[Omega] t]}, {Sin[\[Theta] + \[Omega] t + Pi],
         Cos[\[Theta] + \[Omega] t + Pi]}}],
     {\[Theta], 0, Pi, Pi/5}],
    Table[
     Block[{
       pt =
        p1 Max[0, (1 - (t/(2 Pi))/(-p1[[1]]))] + {-p1[[1]],
           p1[[2]]} Min[1, (t/(2 Pi))/(-p1[[1]])]
       },
      {Disk[pt, 0.05],
       Opacity[0.5], Darker[Red], Line[{p1, pt}]
       }]
     , {p1, p1s}],
    Table[
     Block[{
       pt =
        p1 Max[0, (1 - (t/(2 Pi))/(-p1[[2]]))] + {p1[[
            1]], -p1[[2]]} Min[1, (t/(2 Pi))/(-p1[[2]])]
       },
      {Disk[pt, 0.05],
       Opacity[0.5], Darker[Red], Line[{p1, pt}]
       }]
     , {p1, p1s2}]
    }, PlotRange -> 1.1],
  Table[
   ParametricPlot[
    RotationMatrix[\[Omega] (\[Tau] - t)].(
      p1 Max[0, (1 - (\[Tau]/(2 Pi))/(-p1[[1]]))] + {-p1[[1]],
         p1[[2]]} Min[1, (\[Tau]/(2 Pi))/(-p1[[1]])]
      )
    , {\[Tau], -0.001, t}, PlotStyle -> Thick]
   , {p1, p1s}
   ],
  Table[
   ParametricPlot[
    RotationMatrix[\[Omega] (\[Tau] - t)].(
      p1 Max[
         0, (1 - (\[Tau]/(2 Pi))/(-p1[[2]]))] + {p1[[
          1]], -p1[[2]]} Min[1, (\[Tau]/(2 Pi))/(-p1[[2]])]
      )
    , {\[Tau], -0.001, t}, PlotStyle -> Thick]
   , {p1, p1s2}
   ]
  ],
 {t, 0, 2 Pi}
 ]