Let $P$ be an opaque polyhedron.
Assuming parallel projection, let's define a viewpoint to be a point on the unit sphere around the center of $P$.
Let's say that two viewpoints $v_1$ and $v_2$ are topological equivalent, if
- we see the same set $F$ of faces from $v_1$ and $v_2$
- there is a path $p$ of viewpoints between $v_1$ and $v_2$ such that we see $F$ from all viewpoints on $p$ (i.e. there is no viewpoint on $p$ such that we see any other face not contained in $F$ and no face from $F$ will disappear)
We can prove that if $P$ is convex, any two viewpoints which satisfy (1.) are topological equivalent, i.e. they satisfy (2.) too.
Does anyone know a simple example of a non-convex $P$ such that there are two viewpoints satisfying (1.) but not (2.)?