Consider this polygon as the setting for a dynamical billiard:
When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the plane! I want to make it the boundary of a flat Riemannian surface homeomorphic to a disk (just like the area enclosed by an ordinary simple polygon). The billiard map on this surface will "see" only one "flap" at a time:
The vertex in the middle of the digram has an interior angle $< -180^\circ$, or equivalently an exterior angle $> 360^\circ$, hence the title question.
- What would a geometer call this surface and the polygon that bounds it? Is it even a "polygon" if we're not embedding it in the plane?
- Would it be clearer to call it a Riemann surface? I only care about geodesics and reflection angles. (Assume that my complex-analysis-fu is weak.)
- What should the unusual vertex be called: a "cusp", a "branch point", an angle $> 360^\circ$, or...?
- What would make the diagrams clearer?
I just want to refer to this billiard table in an offhand example, so I don't want to spend too much time describing it. On the other hand, I don't want it to be confused with a self-intersecting polygon in the plane, and ideally, I don't want to sound like a crazy person. :)