A visualization of the 'unfolding' method in dynamical systems, mapping 2D billiard trajectories onto a 3D torus.
The Inspiration
While reading a paper on dynamical systems (Menini, Possieri, Tornambè), I encountered the concept of “unfolding” a billiard table. Instead of calculating complex reflections at every wall, you can “mirror” the world and treat the ball’s path as a straight line.
Visual Gallery
The simulation demonstrates two distinct regimes:
- Periodic Motion: The ball retraces its path perfectly, creating a closed loop.
- Dense Motion: The trajectory never repeats, eventually painting the entire surface of the torus.
The Math
To visualize this, I mapped the flat 2D billiard coordinates $(x,y)$ onto a 3D torus using the parametric embedding:
\begin{cases}
X = (R + r\cos\omega)\cos\varphi \\
Y = (R + r\cos\omega)\sin\varphi \\
Z = r\sin\omega
\end{cases}
This allows us to see how a “straight line” in the unfolded plane wraps beautifully around the 3D donut shape.