  We study an isomorphism between the group of rigid body displacements and the
group of dual quaternions modulo the dual number multiplicative group from the
viewpoint of differential geometry in a projective space over the dual numbers.
Some seemingly weird phenomena in this space have lucid kinematic
interpretations. An example is the existence of non-straight curves with a
continuum of osculating tangents which correspond to motions in a cylinder
group with osculating vertical Darboux motions. We also suggest geometrically
meaningful ways to select osculating conics of a curve in this projective space
and illustrate their corresponding motions. Furthermore, we investigate
factorizability of these special motions and use the obtained results for the
construction of overconstrained linkages.
