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A three-dimensional Laguerre geometry and its visualization. (English) Zbl 1073.51003

Weiß, Gunter (ed.), DSG CK 2003. Dresden Symposium Geometry: Constructive and kinematic/konstruktiv und kinematisch. Zum Gedenken an/in commemoration of Rudolf Bereis (1903–1966), Dresden, Germany, February 27–March 1, 2003. Proceedings. Dresden: Technische Universität Dresden (ISBN 3-86005-394-9/pbk). 122-141 (2003).
The authors discuss the Laguerre geometry associated with the real algebra \(L:={\mathbb R} [x]/ (x^3)\). The point set of this geometry is \({\mathcal P}(L):= \{ L(u,v) \mid u,v\in L\), \(Lu+Lv=L\}\). A point of the subset \({\mathcal A}:= \{ L(z,1)\mid z\in L\}\) is called a proper point. A proper point \(L(z,1)\) can be identified with \(z\in L\). Using this identification, H.-J. Samaga observed that the proper part \({\mathcal A}\cap C\) of a chain \(C\) is an affine line or a parabola or a cubic in the affine space given by the 3-dimensional real vector space \(L\).
For the entire collection see [Zbl 1048.51001].

MSC:

51B15 Laguerre geometries
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
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