Systematic Escape Using Billiard Moves

We consider a robot that can move in a straight line, until it hits any wall of a given 2D rectangle room. If the robot hits a corner of the room then it will stop, otherwise the robot bounces off the wall using the laws of reflection and again move in another straight line. The room may contain an unbroken opening of unit length on a wall, which is unknown to the robot. If the robot reaches any point of the opening with a non-zero angle, then it escapes through the opening. The objective of the problem is to devise an algorithm for the robot which enables it to find if there is an opening on the perimeter of any given rectangle or detect that the room contains no opening. This work is a continuation of our previous publication [1], where we presented an algorithm that enables the robot to find the opening or correctly declare that there is none, when any two adjacent sides of the rectangle are integer and co-prime. In this work, we study the problem for rectangles of any lengths and we have proposed a strategy where the robot is guaranteed to find the opening or correctly declare that there is none. Additionally, We provide other interesting results related to our proposed algorithm.