curve[\[Theta]_] := {0.7 Sin[\[Theta]], Cos[\[Theta]]};
\[Theta]2[curve_, \[Theta]_, l_] := Module[{solns, \[Theta]2},
   solns = \[Theta]t /. (Quiet@
       NSolve[Norm[curve[\[Theta]t] - curve[\[Theta]]] == 
         l, \[Theta]t]);
   \[Theta]2 = SortBy[solns, Mod[\[Theta] - #, 2 Pi] &][[1]]
   ];
l = 1.3;
n = 300;
\[Theta]s = Table[N@\[Theta], {\[Theta], 0, 2 Pi, 2 Pi/n}];
\[Theta]2s = 
  Table[\[Theta]2[curve, \[Theta], l], {\[Theta], \[Theta]s}];
m = 20;
js = Flatten@Table[If[j != 0.5, j, {}], {j, 1/m, 1 - 1/m, 1/m}];
AppendTo[js, 0.5];
Manipulate[
 Module[{p, q, lower, upper},
  p = curve[\[Theta]s[[Mod[i, n, 1]]]];
  q = curve[\[Theta]2s[[Mod[i, n, 1]]]];
  If[i < n,
   lower = 1;
   upper = i;
   ,
   lower = i - n;
   upper = n;
   ];
  Show[
   Graphics[{
     {Thick, Black, Line@(curve /@ \[Theta]s)},
     Table[
      {
       If[j == 1/2, Directive[Thick, Black], 
        ColorData["DarkRainbow"][j]],
       Line@(j curve /@ (\[Theta]s[[lower ;; upper]]) + (1 - 
             j) curve /@ (\[Theta]2s[[lower ;; upper]]))},
      {j, js}],
     {
      PointSize[Medium],
      {Darker@Red, PointSize[Large], Point[{p, q}]},
      {
       Table[
        {Darker@ColorData["DarkRainbow"][j], Point[{j p + (1 - j) q}]},
        {j, js}]
       }
      },
     Line[{p, q}]
     }, ImageSize -> 200]
   ]
  ],
 {i, 1, 2 n, 1}]