Your images of rotating figures are quite intriguing and they suggest a number of questions, both two- and three-dimensional. Can you describe the exact shape of the umbra and the penumbra of the figures you have generated? It seems that you are always going to get a symmetric convex polygon, except for a pair of branches of a hyperbola in the middle. Do you agree?
In three-space, I suspect that the form you are looking at from the outside is a collection of cones, frustra of cones, and one portion of a hyperboloid of one sheet?
Do you yet have any information on the projections into three-space of a hypercube rotating in the fourth dimension? That promises to be quite interesting, and it should bring up analogues of the hyperboloids in the next dimension. We might want to employ some coordinates in the description of these things, since you are doubtless using coordinates already when you are setting up your pictures.