Invisibility via reflecting coating

We construct a subset A of the unit disc with the following properties. (i) The set A is the finite union of disjoint line segments. (ii) The shadow of A is arbitrarily close to the shadow of the unit disc in “most” directions. (iii) If the line segments are considered to be mirrors reflecting light according to the classical law of specular reflection then most light rays hitting the set emerge on the other side of the disc moving along a parallel line and shifted by an arbitrarily small amount. We also construct a set which reflects almost all light rays coming from one direction to another direction but its shadow is arbitrarily small in other directions, except for an arbitrarily small family of directions.