Kinematic mappings of plane affinities

authored by

Herbert Hotje

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, script G sign)be the real three-dimensional projective space, let ℰ̄ ⊂ P be an isomorphic image of ℰ, and let U ∈ script G sign such that ℰ̄ ∪ U is the projective closure of ℰ̄ in P. Then there is a bijection κ : ℬ → P′ := P \U called the kinematic mapping and an injective mapping ℰ̄ × ℰ̄ → script G sign; (u, v) → [u, v] called the kinematic line mapping such that [u, v] := {β ∈ P′; β(u) = v} where the operation is denned 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, script G sign) is part of and spans the six-dimensional real projective space.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Discrete mathematics

Volume

155

Pages

121-125

No. of pages

5

ISSN

0012-365X

Publication date

01.08.1996

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Theoretical Computer Science, Discrete Mathematics and Combinatorics

Electronic version(s)

https://doi.org/10.1016/0012-365X(94)00375-S (Access: Open)