  We examine a computational geometric problem concerning the structure of
polymers. We model a polymer as a polygonal chain in three dimensions. Each
edge splits the polymer into two subchains, and a dihedral rotation rotates one
of these chains rigidly about this edge. The problem is to determine, given a
chain, an edge, and an angle of rotation, if the motion can be performed
without causing the chain to self-intersect. An Omega(n log n) lower bound on
the time complexity of this problem is known.
  We prove that preprocessing a chain of n edges and answering n dihedral
rotation queries is 3SUM-hard, giving strong evidence that solving n queries
requires Omega(n^2) time in the worst case. For dynamic queries, which also
modify the chain if the requested dihedral rotation is feasible, we show that
answering n queries is by itself 3SUM-hard, suggesting that sublinear query
time is impossible after any amount of preprocessing.
