Internal distance constraints in polymers often have a characteristic offset distance between the constrained monomers. This paper introduces a new kind of such constraint, the displacement constraint, which has particularly simple properties. The displacement constraint has both a characteristic offset distance and a characteristic offset direction. A simple, exact expression for the free energy of an ideal polymer with displacement constraints is derived. This expression is used to obtain bounds on the free energy of an ideal polymer with the more usual length constraints, which have a characteristic offset length but no characteristic direction. The form of these results suggest that the offset length can make an important contribution to the free energy if the offset distance is not negligible in comparison to the chemical distance between constrained monomers.