Consider non-degenerate simple planar polygons. Intuitively, we can classify them as follows:
[
]
For $n=6$ I think these are all the classes but I haven't checked super carefully:
(Second image fixed, thanks to Jaap Scherphuis's comment.)
These classes make sense intuitively, but is there any clean classification of them, along the lines of "these are the simple non degenerate planar $n$-gons, taken up to maps $f :\mathbb{R}^2 \to \mathbb{R}^2$ with property $P$"?
We can't take, say, P="affine," because not every convex polygon is affine regular.
I do know of one way to tell these polygons apart, which is to take a clockwise parametrization of the boundary and check whether we make a left or right turn at each vertex. For instance the classes for $n=4$ are RRRR and RRRL, respectively.
But that's not the kind of criterion I'm looking for. I want something that's stated in terms of regions of the plane, or maybe in terms of Jordan curves, not something that refers to polygons directly.
Edit: in the comments, Mark S. asks for further motivation. In the $RRR\ldots R$ case, the polygon is convex, and we can define convexity for arbitrary subsets of a real vector space. The classification of polygons described here basically measures the way in which a polygon fails to be convex. As such, it makes sense to ask if this has a definition that goes through in the same general context as the definition of convexity does.
I think whatever appropriate notion here is should be, for polygons, equivalent to classifying by the sequence of convex and non-convex vertices, but if I were shown a criterion that's obviously the "right" notion but doesn't exactly match this, day for large $n$, I'd be perfectly happy with it.
The inspiration was a question about dissecting a polygon into convex polygons. (E.g.: describe all distinct ways to dissect a rectangle into eight convex pentagons.) I suspect, but have not proven, that this is determined by the class (in the above sense) of the polygon being dissected.
You might consider the classification by means of a "convex polygon tree", which is recursively defined as follows:
the CPT of a convex polygon is a single node labelled with the number of vertices;
the CPT of a concave polygon is a parent node labelled with the number of vertices of the convex hull of the polygon, and having as siblings the CPT of the differences between the convex hull and the polygon.
Whether the order of the siblings or the number of edges between them matters or not is your choice.