Kinematic mappings of plane affinities

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. 1. General kinematic spaces Generalizing the classical ideas, Karzel gave the following definitions (see [3]): Let ( P , ~ ) b e a linear space and ( P , . ) be a group. Denote for a ~ b by a,b the line through a,b C P and for T C P by (T ,~ (T ) ) the trace geometry in (P,(¢). ( P , ~ , . ) is called an incidence group i f x --* ax is a collineation for all a E P and is called a two-sided incidence group i f x --+ axb is a collineation for all a,b c P. A kinematic space is a two-sided incidence group in which 1,a is a subgroup for all a E P \ {1}. Now I call an incidence group (P,f#, • ) a general kinematic space (see [2]) i f fiarthermore the mapping x ~ x -1 is a collineation. (P, c~,. ) is a general kinematic space i f and only i f it is a two-sided incidence group in which any line G E f# with [G[/>3 and 1 C G is a subgroup. A construction of Marchi and Zizioli [4] can be applied to find new examples of such spaces: Let (P, ~ , + ) be a general kinematic space, U a subgroup of Aut(P, ~, +) I The preparation of this paper was sponsored by the NATO Scientific Affairs Division grant CRG 900103 0012-365X/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0012-365X(94)00375-0 122 H. HotjelDiscrete Mathematics 155 (1996) 121-125 and V a set o f subgroups of U such that the following condition holds: (K) Let V, V1, V2 E V, ~, fl E U \ { id }. Then (a) a, fl E V ==~ Fix a = Fix fl, (b) ~V0~ -1 E V, (c) Z l -¢V2 ~ r ' l n v 2 { i d } . Then we have (1)(a) I f ifu is the set of all cosets of subgroups of V together with the set of all 2-sets of U which are not contained in one of these eosets then (U, ifu, .) is a two-sided incidence group. (b) In the semidirect product (M, .) = P × U with the multiplication (a,~)(b, fl) = (a+a(b),af l) , let gr~l := {G × {id}; if ~ G ~ 0} U { { ( a a ( a ) , a ) ; ~ E V};a E P, V E V} and let ifM be the set of cosets of these subgroups of ~ M together with the set of all 2-sets of M which are not included in one of these cosets. Then (M, ifM, ") is a general kinematic space [2] which we will call the connected space and denote by (P, if; U, V). (c) For the subgroup Mo := {0} x U, (Mo, ifM(Mo)) is a space which is isomorphic to (U, i fu) . (d) (P, if; U, V) is a kinematic space i f and only i f u V = U and for all • E U\{id}, P = {x ~(x); x ~ P}.