Abstract

A mathematical billiard system is composed of a planar or multidimensional surface and a moving object whose trajectory is defined by its initial position, speed vector, and some reflection law. This study considered a novel reflection law for billiard systems of regular n-gons in which an object, starting on one of the sides and moving with any given slope, reflects towards the interior with a prescribed constant angle. We mapped the position of such an object as a function of the starting point and the slope in the case of a regular triangle and square. We proved that the object’s path converges to an inner rectangle for all initial conditions and all slopes on a square billiard. In a triangular billiard, the path converges to an inner equilateral triangle for slopes less than or equal to the root of three. The path converges for set of degenerate points for each slope greater than the root of three. We have also constructed a numerical model for the object’s trajectory and determined equations yielding the speed of the object’s convergence to a stable path within a square. We have performed relevant simulations and discussed the resulting data in the context of the proposed model. The analysis of these systems can be used to develop independent control mechanisms for ground robots in delivery missions working within contested environments. It can also be implemented to create simpler and smaller microchips in the form of quantum billiards, offering an innovative way to encode information.