This is a question implicitly raised by Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?.

Suppose $X$ is a closed subset of $\mathbb R^n$.

What can you say about

The shadow hull of $X$: the intersection of complements of lines disjoint from $X$, and

The sight hull: the intersection of complements of rays disjoint from $X$?

Do these sets have standard mathematical or computer graphics / computational geometry names? Are there other good characterizations? How hard are they to compute (say for a polyhedron, so that complexity is fairly well-defined)?

Note: two sets with the same shadow hull cast the same shadows (in light cast by any parallel beam). Two sets with the same sight hull have the same visible surface. Sight hull $\subset$ shadow hull $\subset$ convex hull.

**Addendum** While we're at it: one could also define the "Knife Hull" to be the intersection of complements of half-planes disjoint from the set. In other words, it's the best outer approximation of a small shape that you can carve using a long wide straight-edged knife. This makes it sight hull $\subset$ shadow hull $\subset$ knife hull $\subset$ convex hull.

All of these are invariant under affine transformations.

drip hull, the enclosing shape that cannot hold water in any orientation. Maybe this needs a separate posting... $\endgroup$ – Joseph O'Rourke Sep 18 '10 at 19:38