This article discusses systematically the characterization of instantaneous point-line motions, and the higher-order relationship between a point-line motion and the associated rigid body motions. The transformation of a point-line between two positions is depicted as a pure translation along the point-line followed by a screw displacement about their common normal and expressed with a unit dual quaternion referred to as the point-line displacement operator. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. Such a treatment leads to a consistent expression or unified treatment for the transformation of lines, point-lines, and rigid bodies. The relationships between point-line motions and rigid body motions are addressed in detail up to the third order.

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