This paper develops relationships between the instantaneous invariants of a motion and the local shape of the trajectories generated during the motion. We consider the point trajectories generated by planar and spherical motions and the line trajectories generated by spatial motion. Those points or lines which generate special trajectories are located on (and define) so-called boundary loci in the moving body. These boundary loci define regions, within the body, for which all the points or lines generate similarly shaped trajectories. The shapes of these boundaries depend directly upon the invariants of the motion. It is shown how to qualitatively determine the fundamental trajectory shapes, analyze the effect of the invariants on the boundary loci, and how to combine these results to visualize the motion trajectories to the third order.

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