We provide necessary and sufficient conditions for all projective transformations of the projectivized dual quaternion model of rigid body displacements that are induced by coordinate changes in moving and/or fixed frame. These transformations fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. Moreover, we fully characterize the constraint varieties of dyads with revolute and prismatic joints in the dual quaternion model. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a “null quadrilateral.” If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterize such dyads. We provide a complete characterization by introducing a new invariant, the “Study fibre projectivity,” and we present examples that demonstrate its potential to explain hitherto not sufficiently well-understood phenomena.