- #1

Joppy

MHB

- 284

- 22

I'm lost in at the first line were it is stated that a given collection of polygons need not be convex. How is this possible? I am trying to understand translation surfaces from the perspective of dynamical systems, specifically, billiard systems. In this setting we can 'unfold' the trajectory of a point particle. But surely this unfolding process only works for trajectories confined to convex regions?

I suspect my confusion comes from the fact that generating a translation surface from unfolding a billiard trajectory, and generating one given the definition from Wiki are different things. I also don't understand what is meant by $s_j = s_i + \vec{v}_i$. Are we saying that for every side in a plane of polygons, there exist some other side which lies in the same direction?

Thanks