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. :)
Do not call it a Riemann surface, since this notion is too weak for your purposes.
You can call it "a flat surface with conical singularities" or just a "flat surface".
You can call the point with total angle $>2\pi$ a "singular point" or "cone point".
Surfaces like this do appear routinely in the theory of billiards, see e.g. "Rational billiards and flat structures" by Masur and Tabachnikov, In Handbook Dynamical Systems, Elsevier, http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf