  We present two methods to interpolate between two given rigid body
displacements. Both are based on linear interpolation in the ambient space of
well-known curved point models for the group of rigid body displacements. The
resulting motions are either vertical Darboux motions or cubic circular
motions. Both are rational of low degree and lie in the cylinder group defined
by the two input poses. We unveil the essential parameters in the construction
of these motions and discuss some of their properties.
