## Billiard ball trajectories

Given a polygon, the problem of finding general closed billiard-ball trajectories seems to be difficult (unsolved to my knowledge). One can however try to find closed billiard ball trajectories in regular polygons and of the following kind:
The first segment AB, starting from an inner point A and ending at the first reflexion point B should be parallel to a side of the polygon.

Free movable in the figure above are:
The polygon and point A (switch to selection-tool (Ctrl+1) to catch and modify)
Movable-On-Contour is:
The point B (switch to select-on-contour-tool (Ctrl+2) to catch and modify)

The recipe for the construction of the trajectory:
1) Start from two points A inside the polygon, and B on its contour.
2) Draw the Tangent t(B) at B and the reflected A' of A on t.
3) Draw the line s = [A',B] and find its second intersection C with the polygon.
4) Take A = B and B = C and repeat steps 2+3 to find the next point D ...
5) Take A = C and B = D and repeat steps 2+3 to find the next point E ...
.......
repeat N times
.......

In the preceding construction the tangents to the polygon and the reflected points are hidden.
The precedure lends itself for programming.

Further study: M. Berger , Geometry I, Springer 1987, ch. 9.4, p. 214
For a much more elegant example look at: BilliardTrajectories.html .