Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers.
A starting point is chosen at random within the box at (x,y), such that 0 < x < m, and 0 < y < n.
From the starting point, create a vector with integer components. Once the vector hits a side of the box, it will reflect off the side, similar to the billiards problem, but not necessarily at a 45 degree angle.
Provided that the vector will continue reflecting in the box for a very long, but finite time (say, 10,000 units), for a given vector, I want to determine
1) whether the vector will eventually hit a corner.
2) whether the vector will eventually hit the starting point.