Points in the plane, lines in space

Abstract

The 1911 Grünwald -Blaschke mapping is reviewed from the point of view of a particular Clifford algebra. This is a mapping between the group of proper Euclidean displacements of the plane and an open set in 3-dimensional real projective space. The image of the set of group elements which displace an arbitrary point to another fixed point is a line in the projective space. In this way, a correspondence is established between point-pairs in the plane and lines in 3-dimensional projective space. The space of lines in 3 dimensions is an object of classical study usually called the Klein quadric. The action of the group of planar rigidbody displacements on the Klein quadric is different from the usually considered action of the spatial group. The quadratic invariants with respect to this representation are found and interpretations in terms of point-pairs are given. Some subspaces of lines, including line complexes and congruences, are investigated and their interpretation as sets of point-pairs in the plane are given.