Arithmetic Billiard Paths Revisited: Escaping from a Rectangular Room

In this work, we consider a problem where a robot can move in a straight line inside a 2D rectangular room with integer lengths until it hits any part of the wall of the room. If the robot hits any part of a wall other than the corners or any point of an opening, then the robot bounces off the wall and follows a new direction in another straight line following the laws of symmetric reflection. The robot needs to escape through an opening on the wall that has a minimum length of one unit. The robot can only escape through the opening if it reaches any point of the opening with a non-zero angle.We present an efficient algorithm for which the robot is guaranteed to find the opening if there is any or declare that there is none. We prove that the algorithm works if and only if the sides of the rectangle are co-prime. As a by-product of our main result, we also provide some interesting results related to the coverage of the interior of the rectangle when the robot follows similar algorithms to escape from the rectangular room.