Generators for the motion group of metric vector spaces

Abstract

The purpose of this paper is to generalize the following statement: Given two points a and b in the Euclidean plane, every proper motion of the plane can be written as a product of rotations which fix either a or b. While similar statements have been proved for various plane geometries (see [1, p. 164], [4] and [5]), we consider metric vector spaces of arbitrary dimension. For the results see Theorems 1, 2, 3 and Propositions 5, 6, 7.Let V be a vector space over a field K and let q: V--+ K be a quadraticfor any asV and VI: ={vEVIf(v,x) = 0 for every x~V}. For every aeV we have a translation r,: V ~ V; x ~ x + a. If q(a) ¢ 0, we also have a reflection era:For a(sV ± the product r-ba~zb is a reflection in the affine hyperplane a ± + b. The product of two reflections fixing a common point c~V is a rotation fixing c. We set T: = {%la~V}, S: = {a, lasV and q(a)v L 0}, and $2: = {aaablaa, abeS}. The group (Sw T) generated by Sw T is the motion group, and M: = (S2w T) is the proper motion group of the metric vector space (V, q), which for char K = 2 may coincide with the motion group. Given a, bEV, we investigate the following two questions:(i) Is M generated by the rotations fixin 9 a or b? (ii) Is M generated by the rotations fixing b and the translation ra-b?Without loss of generality we may assume b = 0. In order to avoid trivial cases, we assume dim V >~ 1. M operates faithfully on V. The stabilizer of a s Vis denoted by Ma. Hence Mo = (S 2) and Ma = Tal(S 2 )~'al and our questions are the following: (i) ls M generated by M o ~ M~? (ii) Is M generated by M o ~ {r~}? REMARK. Since (MouM,) ~ (Mou {%}), (ii) has a positive answer if (i) has one. NOTATIONS. B(a): = {ap l p~ ( SZ ) } F(a): = {v e VN V ± I q(v) = q(a)} (B(a)): the additive group generated by B(a) Geometriae Dedicata 22 (1987) 22~233.

Projective Metric Geometry and Clifford Algebras

“…We refer to [15,16,39] for proofs and to [4,[9][10][11][12]14,29,37], [43, p. 18], [46], [50, 1.6.3], [53, pp. 156-159] for further details, generalisations and additional references.…”

“…, comprises precisely those p ∈ Cl i (V , Q) which satisfy pq = (−1) ∂p∂q qp for all homogeneous q ∈ Cl(V , Q); see [31, (3.5.2)] or [40, p. 152]. By (14), for all p ∈ Lip × (V , Q) and all vectors x ∈ V , we have…”

Projective metric geometry and Clifford algebras

“…We refer to [15], [16], [39] for proofs and to [4], [9], [10], [11], [12], [14], [29], [37], [43, p. 18], [46], [50, 1.6.3], [53, pp. 156-159] for further details, generalisations and additional references.…”

“…, comprises precisely those p ∈ Cl i (V, Q) which satisfy pq = (−1) ∂ p∂q q p for all homogeneous q ∈ Cl(V, Q); see [31, (3.5.2)] or [40, p. 152]. By (14), for all p ∈ Lip × (V, Q) and all vectors x ∈ V, we have…”