For what measures of ∠a are there infinitely many intersections?

Question

Suppose you have an xy coordinate plane with two circles both with a radius of one, centered at (-2,2) and (2,2). You have a line segment with one endpoint at (0,0) and forms an angle (∠a) with the x-axis. The line segment continues until it intersects one of the circles and then “bounces off” -that is it forms a new line segment with an endpoint at the intersection, forming an equivalent angle with the tangent of the circle at that point to the angle formed by the original segment and that tangent (as if the segments show the path of light and the circles are mirrors). The new segment extends until it intersects a circle and “bounces off” to form a new segment and so on. For what measure of ∠a are there infinitely many intersections? (The light never stops bouncing).

Answer 1

I wrote a Mathematica routine to compute the number of bounces as a function of angle $\theta$, assuming the light beam first strikes the circle on the right, at point $(2-\cos\theta,2-\sin\theta)$.

Using machine precision I found a maximum of $27$ bounces for some values in the interval

$$ 0.40923372682221769905317<\theta< 0.40923372682221769905375 \quad\text{(in radians).}$$

The limiting value leading to infinitely many bounces must then lie within this interval.

I could improve this result (at the expense of longer execution time). Notice that angle $\angle a$ defined in the question can be obtained from $$ \angle a=\arctan{2-\sin\theta\over2-\cos\theta}. $$