Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all silhouettes of a convex figure —unordered, that is to say, you know only the silhouettes, not the viewing angles associated with each one—is that always sufficient to uniquely reconstruct the shape?
Or are there convex shapes that are not related via any rigid transformation that have indistinguishable silhouette profiles?
I have been working on this problem, and I'm stumped. So far, I've tried generating asymmetric (chiral) shapes, and attempting to answer the question in two dimensions instead of three.
Here's one attempt to formalize these terms: if $P$ is a (bounded) convex subset of $\mathbb{R}^3$, let $u\in S^2$ be a viewing direction. The silhouette of $P$ in the $u$ direction is the set of all vectors $v\perp u$ such that the line through $v+u$ and parallel to $u$ intersects the object.
Two convex subsets $P, Q\in\mathbb{R}^3$ have the same silhouette profile if there is a rigid transformation $f:\mathbb{R}^3\rightarrow \mathbb{R}^3$ such that the set of all silhouettes of $f(P)$ is equal to the set of all silhouettes of $Q$.
The question is whether there are convex subsets $P,Q\subseteq \mathbb{R}^3$ that are not the same shape (not related by a rigid transformation), but have the same silhouette profile.
(These definitions carry over to other dimensions with $\mathbb{R}^3$ replaced with $\mathbb{R}^n$, and $S^2$ replaced with $S^{n-1}$.)
Edit: In order to entirely erase the viewing direction from the silhouette, I should quotient these silhouettes by the relation that considers silhouettes equivalent if they can be transformed into one another by a rigid transformation.
