This paper investigates the construction of rational motions of minimal quaternionic degree that generate a given plane trajectory (a rational torse). It raises three main questions: under what conditions do such motions exist and what is the minimal degree required? When is the minimal motion unique? Extending earlier results [1], which proved that the minimal motion for any rational curve is unique and most often realized as a translation along the curve except for circular curves, where the degree is reduced and the motion takes a different form we show that such motions occur only when the Gauss map of the torse is rational. In that case, the minimal degree is generally half of the torse’s degree; however, if the Gauss map degree drops, the minimal degree increases and uniqueness is lost. The theoretical framework developed here has potential applications in robotics, computeraided design, and computational kinematics. An accompanying figure demonstrates two distinct motions that follow the same plane trajectory, with co-planar frames but different displacements.

References

[1] Z. Li, J. Schicho, H.-P. Schröcker: The rational motion of minimal dual quaternion degree with prescribed trajectory. Comput. Aided Geom. Design, Vol. 41, pp. 1–9, 2016.