Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles,
and is non-self-intersecting; 
also known as a [rectilinear polygon](https://en.wikipedia.org/wiki/Rectilinear_polygon).
Treat every edge of $P$ as a perfect mirror, reflecting lightrays at equal angles.

Let $x$ and $y$ be points in $P$ that can be connected via a lightray path.
Assume rays die at vertices, so
such a path avoids vertices (except possibly at its endpoints if $x$ or $y$ is a vertex).

I am interested in shortest reflection paths between $x$ and $y$, where path length is measured
by the number $k$ of reflections.
For example, below the upper path from $x$ (green) to $y$ (red) reflects $k=19$ times,
while the lower path reflects $k=16$ times.
($\theta$ is the start angle w.r.t. the horizontal.)

[![LightRaysCorridor][1]][1]

I have two questions:

> ***Q1***.
Is there an upperbound on the number of reflections in a shortest path from
$x$ to $y$, expressed as a function of the minimum distance between parallel
edges of $P$ that can see one another vertically or horizontally?

> ***Q2***.
What is an algorithm for computing the shortest path between given $x$ and $y$?

It is known that for a given $x$, there can only be a finite number of dark points
in $P$ (discussed in [an earlier MO question](https://mathoverflow.net/q/379423/6094)).
Assume $y$ is not one of those dark points, so that there exists a path from $x$ to $y$.

In ***Q1*** I'm trying to capture the width of corridors, but perhaps the minimum
distance between parallel edges is the wrong quantity.
Concerning ***Q2***, there has been work on reflections in polygons,
but not *shortest* reflection paths, as far as I know.
It could be that the results in this paper lead to an algorithm:

> Aronov, Boris, Alan R. Davis, Tamal K. Dey, Sudebkumar Prasant Pal, and D. Chithra Prasad. "Visibility with multiple reflections." *Discrete & Computational Geometry* 20 (1998): 61-78.

I've simplified my question to orthogonal polygons, but my real target is
simple polygons with angles rational multiples of $\pi$.

  [1]: https://i.sstatic.net/U8iE9.jpg