The space $\mathfrak R_{\mathrm c}(P)$ of *convex* realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (this result is usually attributed to Steinitz and Legendre). See below for more detailed definitions. I wonder how much of this is due to convexity, sphericity and low dimension respectively. Here I ask specifically what is known if we drop convexity:

Question:What is known about the space $\mathfrak R(P)$ ofgeneral(i.e. not necessarily convex) realizations? Has it been studied in the literature?

Typical questions that are interesting to me: is $\mathfrak R(P)$ connected/contractible after removing degenerate realizations? Since $\mathfrak R(P)$ has the structure of a (real) affine varierty, what are its irreducible components? etc

**Definitions and remarks**

By "polyhedron" I mean a 3-connected planar graph. Note that vertices, edges, and in particular, faces are well-defined. A "realization" thereof is then any embedding of the vertices into $\Bbb R^3$ so that vertices that belong to the same face lie in a common plane. In particular, this allows non-convex, self-intersecting and degenerate (collapsed into a plane, line or point) realizations. Some useful examples to think of are the Great icosahedron and the Bricard octahedron.

I am aware that I am asking about the realization space of a special type of matroids. While matroid realization spaces can be arbitrarily bad, I hope that my setting (3-dimensional, spherical) allows to say more.

notwant to impose any constraints beyond coplanarity. I am however more interested in realization spaces of polyhedra that are not simple/simplicial. $\endgroup$8more comments