Some background: I'm trying to simulate biological cells as polygons in 2D. Real biological cells have an internal cytoskeleton which enables them to conform to a variety of shapes, many non-convex. I'm trying to find a control scheme to control the shape of the cell that prevents cells from forming complex polygons that self intersect while still providing as many shapes they can assume as possible.
At first I was thinking that the polygon could have edges of uniform length, and the cell controller could specify targets for the internal angles of the polygon, but that doesn't prevent self intersection.
My current idea is that the polygon vertices are specified in polar coordinates $(\theta, r)$ from some origin (the cell nucleus). As long as the $r$ term is positive and the $\theta$ terms are strictly increasing I think this guarantees a simple polygon. But this prevents "c" shaped polygons, which I'd like to have if at all possible.
Is the above class of polar polygons named something? And are there any other classes of polygons that would work here? It's important that I can guarantee the simple-ness of the polygon without much computational effort (O(n) on the number of vertices/edges), and that it allows the largest range of shapes possible.