Kinematic mappings of plane affinities

Abstract

In 1911 W. Blaschke and J. Grnwald described the group B of proper motions of the euclidean plane E in the following way: Let ( P, G ) be the real three-dimensional projective space, let E ⊂ P be an isomorphic image of E , and let U ∈ G such that E ⌣ U is the projective closure of E in P. Then there is a bijection κ : B → P′ := PβU called the kinematic mapping and an injective mapping E × E → G ; ( u, v) → [ u, v] called the kinematic line mapping such that [ u, v] := { β ∈ P′; β( u) = v} where the operation is defined by conjugation. A principle of transference is valid by which statements on group operations of ( B , E ) correspond with statements on incidence in the trace geometry of P′. Following Rath (1988) I will show that a similar concept holds for the group of affinities of the real plane where (P, G) is part of and spans the six-dimensional real projective space.