This paper uses line geometry to find an elegant solution to the kinematic registration problem involving reconstruction of a spatial displacement from data on three homologous points at two finitely separated positions of a rigid body. The bisecting linear line complex of two position theory in kinematics is used in combination with recent results from computational line geometry to present an elegant computational geometric method for the solution of this old problem. The results have applications in robotics, manufacturing, and biomedical imaging. The paper considers when minimal, over-determined, and perturbed sets of point data are given.

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