Salient loci existing for special properties of the nth-order motion state are investigated. Generalized analytical expressions are given for the loci as functions of the motion parameters in a rectangular coordinate system. Metric geometry is used to show that the well known Bresse circles are the proper circles to a pencil of circles. Circles similar to the Bresse circles are shown to exist for the nth-order motion state with an associated coaxial pencil of circles. A detailed treatment for special properties of velocity, acceleration, and jerk is presented.

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