This paper presents a method for kinematic registration in three dimensions using a classical technique from two-dimensional kinematics, namely the Reuleaux method. In three dimensions the kinematic registration problem involves reconstruction of a spatial displacement from data on a minimum of three homologous points at two finitely separated positions of a rigid body. When more than the minimum number of homologous points are specified or when errors in specification of these points are considered, the problem becomes an over determined approximation problem. A computational geometric method is presented, resulting in a linear solution of the over determined system. The results have applications in robotics, manufacturing, and biomedical imaging. The paper considers the kinematic registration when minimal, over-determined, infinitesimal, and perturbed sets of homologous point data are given.