You might find The Billiards SimulationThe Billiards Simulation useful. It seems to be a student project aimed at helping people visualize a chaotic dynamical system.
This is not an exact answer to your question. The program described at http://serendip.brynmawr.edu/chaos/doc.htmlhttps://serendipstudio.org/chaos/doc.html handles regular polygons, circles, ellipses, and "stadiums" (half-circles joined by straight lines), and not arbitrary plane polygons which you are looking for. You might want to contact the authors for additional information.