Calculate if a line will pass through a given point?

Question

I'm not sure if this should be in math or physics so I'm asking in both.

Given a laser beam in closed reflective surface where point $(x,y)$ is the source of a laser beam which hits a reflective surface at $(x_1,y_1)$, how can I(if possible) tell if the beam will hit point $(x_2,y_2)$ at some point??

where the values of $x,y,x_1,y_1$ are given.

note: The beam could $(x_2,y_2)$ after multiple reflections.

Answer 1

1. I think that this is generally an open question, even for convex boundaries in 2D.

2. There are some cases where you have to be quite careful to define exactly what you mean. For instance, consider a deep V-shaped groove, with a laser shining into it from above. The laser bounces back and forth from wall to wall, and (if you set up the angles right), all odd-numbered segments of the path are parallel; all even-numbered segments are parallel. The "bounce points" approach the vertex of the V-shape. In this case, do you say that they reach the vertex, or not? It might take infinitely many bounces to do so, and when the path does "reach" the vertex, there's no well-defined "previous bounce".

Answer 2

The light ray can hit every point of present in the reflecting room.

Ponder over this construction :

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Edit 1 - A possible construction for a specific case

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Edit 2 - Multiple Reflections

Let there be 2 reflections from different surfaces. Then the following construction gives us the path of the light ray.

Let the last reflection be from the bottom surface:

And let the first reflection be from the left surface:

Now, draw the rays