Point-reflections in metric plane
type: Item
instance of: scholarly article
title: Point-reflections in metric plane
zbMATH Open document ID: 1124.51009
published in: Beiträge zur Algebra und Geometrie
publication date: 2007-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/231346
full work available at URL: 6.html
review text: Starting from the group theoretical axiom-system for absolute planes (= metric planes) introduced by Arnold Schmidt and simplified by \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. Auflage, Springer-Verlag, Berlin (1973; Zbl 0254.50001)], one has the following classifications: A metric plane is called elliptic if there are three line-reflections whose product is the identity, otherwise non-elliptic and singular (or metric Euclidean) if there are three noncollinear points such that the product of their point-reflections is again a point-reflection, otherwise ordinary. The author presents for non-elliptic metric planes axiom-systems \(\mathfrak B\) and \( \Sigma \) firstly in terms of line-reflections and secondly in terms of two ternary geometric operarions, \( F\) and \(\pi \) which can be interpreted as follows: if \( a,b,c \) are three points with \(a \neq b \) then \(F(a,b,c) \) is the foot of the perpendicular from \(c\) to the line \( \overline{a,b} \) and if \( a,b,c, \) are collinear, \( \pi(a,b,c) \) denotes the point \(d\) such that the point-reflection \( \widetilde d \) is equal the product of the point-reflections \( \widetilde a \circ \widetilde b \circ \widetilde c \). In [Bull. Pol. Acad. Sci., Math. 51, No. 1, 49--57 (2003; Zbl 1036.03051)], the author proved that \(\Sigma \) is an axiom-system for non-elliptic metric planes. Then he turns to his main intention, the axiomatization of non-elliptic ordinary planes in terms of point-reflections (cf. Sect. 3). Here he can use the fact that three points \(a,b,c \) are collinear if and only if the product of their point-reflections \( \widetilde a \circ \widetilde b \circ \widetilde c \) is involutory. The illustrative background for his axiom-system are the properties of motions which are products of point-reflections. So there is given the individual constant 1 (for the identity) and a binary operation \( \circ \) where \( a \circ b \) stands for the composition of the motions …
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 5152738
author: Victor V. Pambuccian
reviewed by: Helmut Karzel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5152738