Crossing the road- smooth version

h = 3.5;
Needs["VariationalMethods`"];
Unset[x];
frame[dip_] := Module[{eqn, soln},
   c[x_] :=
    1 - dip ((1 + Tanh[5 (x + 0.7)])/2 - (1 + Tanh[5 (x - 0.7)])/2 +
        0.1 Exp[-10 x^2]);
   eqn = EulerEquations[Sqrt[1 + y'[x]^2]/c[x], y[x], x];
   soln =
    First[NDSolve[{eqn, y[-1] == 0, y[1] == h}, y[x], {x, -1, 1}]];
   Show[
    Graphics[{
      Table[
       {Lighter@ColorData["LightTemperatureMap"][1 - c[x]],
        Rectangle[{x, -0.1}, {x + 0.1, h + 0.1}]}
       , {x, -1.1, 1.1, 0.05}],
      {RGBColor[0.62, 0.15, 0.2], Disk[{-1, 0}, 0.1],
       Disk[{1, h}, 0.1]}
      }
     , PlotRange -> {{-1.1, 1.1}, {-0.1, h + 0.1}},
     AspectRatio -> h/2],

    Plot[{Evaluate[y[x] /. soln], Evaluate[y[x] /. soln]}, {x, -1, 1},
      PlotStyle -> {Directive[Thickness[0.05],
        RGBColor[0.62, 0.15, 0.2]], Directive[White]}]
    ]
   ];
frame[0.3]