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We give a constructive proof for the existence of a unique rational motion ofminimal degree in the dual quaternion model of Euclidean displacements with agiven rational parametric curve as trajectory. The minimal motion degree equalsthe trajectory's degree minus its circularity. Hence, it is lower than thedegree of a trivial curvilinear translation for circular curves.

The Rational Motion of Minimal Dual Quaternion Degree With Prescribed Trajectory