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.



Finding particular closed paths in geometric plane regions
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 $2^n$.*


