In planar four-position kinematics, the centers of circles containing four positions of a point in a moving rigid body form the center point curve. This curve can be parameterized by analyzing a “compatibility linkage” obtained from a complex number formulation of the four-position problem. In this paper, we present another derivation of the center point curve using a special form of dual quaternions and the fact that it is identical to the pole curve. The defining properties of the pole curve lead to a parameterization by kinematic analysis of the opposite pole quadrilateral as a four-bar linkage. Thus the opposite pole quadrilateral becomes the compatibility linkage. This derivation generalizes to provide parameterizations for the center point cone of spherical kinematics and the central axis congruence of spatial kinematic theory.

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