The dual Euler parameters of a rigid spatial motion are used to define a curve on a dual unit hypersphere. The dual velocity, curvature, and torsion of this curve form a set of instantaneous parameters related to the instantaneous invariants of the motion. In this paper these new parameters are used to reformulate the kinematic theory of line trajectories. The distribution parameter, Disteli formulas, and the inflection congruence are examined in detail.

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