Finding particular closed paths in geometric plane regions

Question

Let $X_m$ denote a set of $m\geq 3$ lines in $\mathbb{R}^2$ that are not all parallel. Consider the problem of determining a closed path of $kn$ points in $X_m$ $k, n \in \mathbb{Z}^+$, such that the "forward orbit" of the path has angles of incidence with the lines in $X_m$ which are strictly contained in a given set $\{\theta_1, \theta_2, ..., \theta_n\}$, and further each angle must occur in a particular order when traversing the sequence of points.

In other words, if $p_1, p_2, ..., p_n, p_{n+1}$ (with $p_{n+1} = p_1$) are a set of such points in $X_m$, join points $p_i, p_{i+1}$ with a line segment, so that that line segment makes angle $\theta_i$ with line $L_j \subset X_m$, where $p_{i+1} \in L_j$. The question then is whether such a closed path can be found for a given $X_m$, set of incidence angles $\{\theta_1, \theta_2, ..., \theta_n\}$, and ordering at which the angles must appear.

This appears to be similar to a weakening of the problem of finding periodic billiard paths in polygons, however it still appears to be hard, and I am wondering if anyone is aware of a solution, or an analogous problem that is discussed in literature.

Answer 1

The answer, given in this paper: https://arxiv.org/abs/2112.02207, turns out to be "yes", there always exist such closed curves. The theorem stating the answer to this question is given in the introduction to the linked paper, and I restate it here for reference:

Theorem: For any space $X_m$ with labeled lines, let $\theta_1, \theta_2,...,\theta_n$, $n\geq m \geq 3$, be any sequence of acute or right angles, and let $L_{a_1}, L_{a_2}, ..., L_{a_n}$ be a sequence of line labels such that no two consecutive labels are the same, including $L_{a_n}$ and $L_{a_1}$, and each of the $m$ possible labels occur at least once in the sequence. Then there exists a closed curve $\Gamma$ over $X_m$ that admits an incidence angle sequence $\theta_1, \theta_2, ..., \theta_n$ with respect to the line sequence $L_{a_1}, L_{a_2}, ..., L_{a_n}$ when traversed in a fixed direction. Moreover, there are either uncountably many such curves, or no more than $2^n$.