Kinematic mappings of plane affinities

✍ Scribed by Herbert Hotje

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 229 KB
• Volume: 155
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00375-s

✦ Synopsis

In 1911 W. Blaschke and J. Grnwald described the group ~ of proper motions of the euclidean plane d' in the following way: Let (P, fg)be the real three-dimensional projective space, let ~ C P be an isomorphic image of g, and let U ~ f# such that ~ tA U is the projective closure of in P. Then there is a bijection x : ~ ~ P' := P \ U called the kinematic mapping and an injective mapping ~ × ~ ~ ~; (u, v) ~ [u, v] called the kinematic line mapping such that [u, v] := {fl C P'; fl(u) = v} where the operation is defined by conjugation. A principle of transference is valid by which statements on group operations of (~, ¢) 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,~) is part of and spans the six-dimensional real projective space.