Details
| Original language | English |
|---|---|
| Pages (from-to) | 121-125 |
| Number of pages | 5 |
| Journal | Discrete mathematics |
| Volume | 155 |
| Issue number | 1-3 |
| Publication status | Published - 1 Aug 1996 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Discrete mathematics, Vol. 155, No. 1-3, 01.08.1996, p. 121-125.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Kinematic mappings of plane affinities
AU - Hotje, Herbert
PY - 1996/8/1
Y1 - 1996/8/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0042126635&partnerID=8YFLogxK
U2 - 10.1016/0012-365X(94)00375-S
DO - 10.1016/0012-365X(94)00375-S
M3 - Article
AN - SCOPUS:0042126635
VL - 155
SP - 121
EP - 125
JO - Discrete mathematics
JF - Discrete mathematics
SN - 0012-365X
IS - 1-3
ER -