Define the shadow of a convex polyhedron $P$ in direction $u$ to be the orthogonal projection of $P$ onto a plane whose normal is $u$. The shadow is a convex $k$-gon. I am wondering to what degree $P$ is determined by knowing the value of $k$ for all the distinct shadows. Let me try to make precise what I mean by "the combinatorics of the shadows."
On a unit sphere $S$, mark all the directions parallel to a face
of $P$. Equivalently, intersect $S$ with planes through its
center parallel to each face of $P$.
Here is the result for a regular tetrahedron:
Each cell $c$ of this arrangement of great circles corresponds to
a distinct shadow in the sense that, for all $u \in c$,
the shadow in direction $u$ is a $k$-gon, with $k$ fixed.
In this example, the shadows for a cell are either triangles or quadrilaterals,
3-gons or 4-gons.
Now form a graph, the shadow graph $G_S$, the dual graph of
the arrangement: each node is a cell, with an edge connecting
cells that share a positive-length arc.
Label each node with the integer $k \ge 3$ if that cell's
shadows are $k$-gons.
For the tetrahedron example, $G_S$ has 14 nodes,
labeled as shown below:
Now imagine you are given the labeled $G_S$ as input. I would like to
know to what extent $P$ is determined.
(1) Does this arrangement and/or its dual graph have a name in the literature? I've seen the term Gaussian sphere in the early computer vision literature, but that term is not prevalent. I feel I may be missing a key search phrase.
(2) Does $G_S$ determine $V$, $E$, and $F$, the number of vertices, edges, and faces of $P$, i.e., the $f$-vector?
(3) Given $G_S$, can you find some representative $P$ that realizes those combinatorics?
(4) More generally, what is the class of $P$ that realizes a given $G_S$?
(4a) A very specific version of this question is: Are all the $P$ that yield the $G_S$ illustrated above tetrahedra?
(5) The question generalizes to polytopes in $\mathbb{R}^d$, which is the root of my question.
Perhaps some of these questions are easier under genericity assumptions, e.g., no two faces are parallel. Edit. For example, under the assumption that no three faces are parallel to a line (which implies no two faces are parallel [thanks to Giovanni Viglietta for the correction]), $G_S$ has $m=F^2-F+2$ nodes for $F$ faces, so $F=\frac{1}{2}(1+\sqrt{ 4m -7})$, which partially solves (2) above.
I'd appreciate pointers to relevant literature!
