TITLE: Put P inside Q! polygons/polyhedra
QUESTION [5 upvotes]: We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P and Q are convex and the input is the list of vertices, edges and faces of P,Q.

REPLY [3 votes]: There is quite a bit of literature specifically on polygon containment,
much of deriving from robotics applications. The paper below solves the containment problem,
allowing translations and rotations, with a polynomial-time algorithm (roughly $O(n^6)$).

Avnaim, Francis, and Jean Daniel Boissonnat. "Polygon placement under translation and rotation." Informatique theorique et applications. 23(1). 1989. p.5-28.

Here is a snapshot from their introduction. The two polygons,
named $E$ and $I$, have $n$ and $m$ edges respectively. This Intro reviews
related, prior work:

 
 
 


Their algorithm extends to handle the case of polyhedra in $\mathbb{R}^3$,
but for translation only (no rotation).