  The trajectories of a rational motion given by a polynomial of degree n in
the dual quaternion model of rigid body displacements are generically of degree
2n. In this article we study those exceptional motions whose trajectory degree
is lower. An algebraic criterion for this drop of degree is existence of
certain right factors, a geometric criterion involves one of two families of
rulings on an invariant quadric. Our characterizations allow the systematic
construction of rational motions with exceptional degree reduction and explain
why the trajectory degrees of a rational motion and its inverse motion can be
different.
