Line orbits on a circle

After a little break time, I decided that a good way to start the year is by posting something that I was thinking on few days ago.

At the beginning of the Winter break, one of my friends back home post me a question about determining the foci a hyperbola just with compass and straightedge constructions. Thinking a little about it, I was trying to find some basic property of the foci of a hyperbola, and it came to my mind that they must have the same property no matter what kind of conic we are looking at, since all conics are equivalent under the $PSL(2,\mathbb{R})$ group.

So I tried to find some kind of property that the foci of all conics share, and one of them can be regarded as some kind of reflection property.

If you have a set of lines that passes through one of the foci of a conic and you take the reflection of these lines on the conic, the resulting set of lines passes through the other focus.

In the case of a circle, the two foci coincide in the center of the circle, so the condition holds, but this made me think of analyzing the orbit of a line in a circle. By this, what I mean is the following: start with a line that intersects the circle, then at the intersection points, reflect the line through the circle and keep with this process. Call the resulting intersection points on the circle, the orbit of the line. Now, a very natural question would be for which kind of lines does this orbit is finite? has a limit point? is dense in the circumference?

At the beginning, the answer seems really straightforward, one could say that the lines that lead to finite orbits are the ones that belong to sides of a regular polygon, which means that the lines leading to finite orbits are the ones whose distance to the center of the circle equals the apothem of some regular polygon inscribed in the circle.

However, this condition is quite weak, since there are more such lines. To look at these, we can take a different approach. One easy way is to look this is by taking the arc length instead. If we take the length of the circumference to be 1, then if the arc lengths of the pieces into which is divided the circle by the line are rationals, we have that the orbit is finite. Moreover, if the arc length is $q/p$, the orbit has length $p$.

This is equivalent to studying the circle regarded as $\mathbb{R}/\mathbb{Z}$, and here the orbit for a value $x$ is the set $\{[nx]: n\in\mathbb{Z}\}$ from where we have that it is finite iff $x\in\mathbb{Q}$ and it is dense in the circle otherwise.

Another interesting question would be to find sufficient conditions for finite orbits on a general conic, and this might be studied using the group structure of the circle $S^1$ and the projective properties of $PSL(2,\mathbb{R})$.