Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to point $p_{i+1}$. (Note that if every $M_i$ passed through the origin rather than $q_i \in C$, all $p_i$ would lie on $C$.)

In the example below, $p_1$ on $C$ reflects to $p_2$ near the origin, $p_2$ reflects out to $p_3$ near $(1,1.5)$, etc.

^{ Green connects $p_1,\ldots,p_5$. Dark lines are mirrors $M_i$. }

I expected that repeating this process would produce some smooth distribution around the origin, as the points reflect further and further out, when the mirrors effectively pass through the origin at that expanded scale. But I find not infrequently bands of density, as depicted below.

^{ Each distribution includes $n=5000$ reflected points. The unit circle is red. Approx. radii: $107,67,47,148$. }

Extending $n$ to larger values seems to continue banding effects. Here is the 3rd example above, the most uniform of those four, extended. Its radius increases from $47$ to $225$:

^{ Continuation of 3rd example above, to $n=25000$. Approx. radius $225$. }

. What explains this radial clustering/banding behavior?Q