Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts centered on the (pinless) origin, and shoots off at angle $\theta$
w.r.t. the horizontal.
It reflects from pins in the natural manner:
<br /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
<img src="http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg" alt="PinBall" />
<br />
What happens?
More specifically,

> <b>Q</b>.
For a given $r$, what is the measure of the set of angles $\theta$
for which the pinball remains a finite distance from the origin
forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to
Polya's Orchard problem (e.g., explored in the MO
question,
<a href="http://mathoverflow.net/questions/62922/">"Efficient visibility blocker
s in Polya’s orchard problem"</a>),
but the reflections produce complex interactions.
It is also similar to Pach's enchanted forest problem, mentioned
in the MO question,
<a href="http://mathoverflow.net/questions/38307/trapped-rays-bouncing-between-two-convex-bodies/38314#38314">"Trapped rays bouncing between two convex bodies"</a>, 
but it seems simpler than that unsolved problem, due to the lattice regularity.
Has it been considered before, in some guise?
If so, pointers would be appreciated.  Thanks!