n = 10;
dy = 10;
ss = Table[s, {s, 0.1, 2, 0.1}];
d[s_, t_,
   e_] := (s EllipticE[
     ArcTan[s Tan[2 Pi t]] + Floor[4 t] Pi - Floor[2 t] Pi, e]);
frame[t_] := (Show[{
     Table[
      Module[{l, e},
       e = 1 - 1/s^2;
       l = 4 s EllipticE[e];
       ParametricPlot[
        {
         {d[s, t, e] \[Theta]/(2 Pi), dy s},
         {Cos[\[Theta]] Sin[2 \[Pi] t] +
            s Cos[2 \[Pi] t] Sin[\[Theta]],
           Cos[2 \[Pi] t] Cos[\[Theta]] -
            s Sin[2 \[Pi] t] Sin[\[Theta]]} + {
           d[s, t, e] -
            ((s^2 - 1) Sin[2 \[Pi] t] Sign[Cos[2 Pi t]])/Sqrt[
            1 + s^2 Tan[2 \[Pi] t]^2],
           Abs[Cos[2 \[Pi] t]] Sqrt[1 + s^2 Tan[2 \[Pi] t]^2] + dy s}
         },
        {\[Theta], 0, 2 Pi}, Axes -> None,
        PlotStyle -> If[s == 1, Red]]
       ]
      , {s, ss}],
     Graphics[{
       Line[{{0, 0}, {0, 22}}],
       {
        Module[{ps},
         ps = Table[{d[s, t, 1 - 1/s^2], dy s}, {s, ss}];
         {Point[ps],
          Darker@Red, Thick, Line[ps]}
         ],
        Opacity[0.4],
        Table[{
          Line[{{0, dy s}, {7, dy s}}]}
         , {s, ss}]
        }}]
     }
    , PlotRange -> {{-2.1, 12}, {0, 24}}, ImageSize -> 300]
   );
Manipulate[
 frame[t],
 {t, 0, 1}]