Equiform kinematics and the geometry of line elements (Q860108)

The authors study the manifold of line elements of the 3-dimensional Euclidean space \(E_3\). In order to describe this manifold they use generalized Plücker coordinates. This yields a point model on a 5-dimensional quadratic cone in a 6-dimensional projective space \(P_6\). Then the authors consider linear mappings of this manifold and define linear complexes of line elements. They show that the infinitesimal velocity vector field of a one-parametric equiform motion of \(E_3\) belongs to a linear complex of line elements. This is a generalisation of the corresponding result for motions within the displacement group of \(E_3\) and associated linear line complexes. Additionally, further properties of linear submanifolds of line elements (concurrency, complanarity, intersections,\dots) are worked out in detail.