Max Planck Institute for the Physics of Complex Systems: Visualisation of a 3D Billiard

Introduction

Billiards are very interesting dynamical systems because they exhibit a variety of dynamical behaviors. See also the Introduction to dynamical systems using billiards. The aim of this visualization is to visualize a special threedimensional billiard, introduced in the paper Topology in Chaotic Scattering, by D. Sweet, E. Ott and J. A. Yorke.

The Model

In this billiard four mirrored spheres are stacked like cannonballs. Three resting on the table and one is placed on top of the three in such a way, that the centers of the spheres form a regular tetrahedron (see first picture from left). There are four openings, each opening is located between three spheres (see second picture from the left). These four openings are covered with four transparent planes, with the colors blue, green red and black (see the right picture and the animation). The viewer looks inside through the transparent black opening. The viewing direction is visualized by the orange arrow in the animation. The picture, the viewer would see, is shown by the program. If the ray from the viewer's eye through the transparent black opening leaves the inner chamber to the left green transparent opening (maybe after many reflections), there would be a little green spot at this particular position.

images for explaining the physical model

animation for visualization of the 4 transparent faces

and the viewing direction (indicated by the orange arrow)

The Program

The Program has two purposes: allowing the user to zoom in to get an impression how this experiment looks like and helping the user to understand the fractal structure of the ray reflections. Since the program has been an application showing graphical data in high resolution long animations and videos would be very memory-consuming. Here are only two small examples: