Crossing the road

hmax = 2;
Cost[h_, y_, \[Lambda]_] :=
 With[{p1 = {-1, 0}, p2 = {-0.8, y}, p3 = {0.8, h - y}, p4 = {1, h}},
  Norm[p2 - p1] + Norm[p3 - p2] + Norm[p4 - p3] +
   \[Lambda] Norm[p3 - p2]
  ];
Manipulate[
 Block[{yo, n},
  yo = yt /. (Minimize[Cost[h, yt, \[Lambda]], yt][[2]]);
  With[{p1 = {-1, 0}, p2 = {-0.8, y}, p3 = {0.8, h - y}, p4 = {1, h},
    p2o = {-0.8, yo}, p3o = {0.8, h - yo}},
   n = Sin[VectorAngle[p2o - p1, {-1, 0}]]/
     Sin[VectorAngle[p3o - p2o, {-1, 0}]];
   Graphics[{
     GrayLevel[0.7],
     Rectangle[{-1, -0.5}, {1, hmax + 0.5}],
     GrayLevel[0.5],
     Rectangle[{-0.8, -0.5}, {0.8, hmax + 0.5}],
     PointSize[Large], Blue, Point[p1], Red, Point[p4],
     (*Black,Point[p2],Point[p3],Line[{p1,p2,p3,p4}],*)
     Dashed,
     Purple, Point[p2o], Point[p3o], Line[{p1, p2o, p3o, p4}]
     }]
   ]
  ],
 {{\[Lambda], 1}, 0, 5}
 ]