How many reflections before absorption on a perfect rectangular mirror?

[lugita15]

I have a mathematical conjecture. It has to do with physics, but I call it a mathematical conjecture because the cases I which I generalized into a conjecture were done purely mathematically, with no actual physical experimentation.

Consider a perfect rectangular mirror which obeys the law of reflection exactly and which has side lengths x and y which are coprime integers (meaning they have no common factors except 1). Light is emitted from the topleft corner of the rectangle at a 45 degree angle below the horizontal. Obviously, it will hit the mirror at various points. If the light is absorbed when it hits one of the corners of the rectangle, how many points on the rectangle will the light hit before it is absorbed, as a function of x and y? Alternatively, we can formulate the problem in terms of a pool table, assuming the pool ball and the pool holes are both infinitely small, and the pool ball makes elastic collisions with the sides of the table and friction never slows the ball down.

After considering many values of x and y, I've reached the conjecture that the answer is f(x,y)=x+y-1, but I have thus far not been able to rigorously prove or disprove this conjecture.

Any help would be greatly appreciated.
Thank You in Advance.

 

[Hurkyl]

Draw a straight line segment from (0, 0) to (xy, -xy). How many points on the interior of this line segment have its first coordinate divisible by x or its second coordinate divisible by y?

Exercise: explain why this problem should give the same answer as your problem. (Try drawing pictures)



In general, you want to use the least common multiple of x and y; this let's you solve the problem when they are not coprime.

 

[ObsessiveMathsFreak]

I've seen "How many reflections" problems like this one in talks before, but generally the question is phrased as, "How many locations and directions give rise to only finitely many reflections". The the regions which give only finitely many reflections, turned out to be fractal. Moreover they were usually disjoint, and as you added more mirrors, the genus of the regions started increasing. It got very horrible, but fascinating.

The restrictions you have posed seem to make this problem more tractable however. You could try posing it as a graph problem, with the points on the edges of the rectangle as nodes, and the edges being the 45 degree ray paths coming out of them.