Point-Line Distance Under Riemannian Metrics

A point-line is the combination of a directed line and an endpoint on the line. A pair of point-line positions corresponds to a point-line displacement, which is known to be associated with a set of rigid body displacements whose screw axes are distributed on a cylindroid. Different associated rigid body displacements generally correspond to different distances under Riemannian metrics on the manifold of $SE(3)$⁠. A unique measure of distance between a pair of point-line positions is desirable in engineering applications. In this paper, the distance between two point-line positions is investigated based on the left-invariant Riemannian metrics on the manifold of $SE(3)$⁠. The displacements are elaborated from the perspective of the soma space. The set of rigid body displacements associated with the point-line displacement is mapped to a one-dimensional great circle on the unit sphere in the space of four dual dimensions, on which the point with the minimum distance to the identity is indicated. It is shown that the minimum distance is achieved when an associated rigid body displacement has no rotational component about the point-line axis. The minimum distance, which has the inherited property of independence of inertial reference frames, is referred to as the point-line distance. A numerical example shows the application of point-line distance to a point-line path generation problem in mechanism synthesis.