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 spherein 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!