Plane sets invisible in finitely many directions

TL;DR

The paper constructs plane sets that are invisible in finitely many directions, containing a circle and approximating it within any small margin, by union of infinitely many smooth boundary domains.

Contribution

It introduces a method to create plane sets invisible in specified directions that contain a given circle and approximate it arbitrarily closely.

Findings

Existence of such invisible sets for given directions

Construction of sets containing the circle within any epsilon neighborhood

Sets are unions of infinitely many domains with smooth boundaries

Abstract

We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with $\pi$, for any $\varepsilon > 0$ there exists a set invisible in the chosen directions that contains the circle and is contained in its $\varepsilon$-neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary.