Point-reflections in metric plane (Q880223)

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 \(a\) and \(b\). A point or point-reflection \( a \) is expressed by \( P(a) : \Leftrightarrow a \neq 1 \wedge a \circ a = 1 \). Further abreviations are ``\( L(abc) : \Leftrightarrow (a \circ b) \circ c = (c \circ b) \circ a\)'' -- if \( a,b,c \) are points then \(L(abc)\) expresses that the points are collinear, ``\( \sigma(ab):= (a \circ b) \circ a \)'' -- \( \sigma(ab) \) is the point obtained by reflecting \(b\) in \(a\). His first axioms (B1, B2, B3 and P1) ensure that for three collinear points \(a,b,c\) the product \( (c \circ b) \circ a\) is again a point collinear with \(a,b \). The main difficulties he had to resolve are to express ``orthogonality'', the ``foot'' and the definition of ``line-reflections''. For this purpose the author needs thirteen further axioms (P2--P14) and finally for the complete characterization of non-elliptic ordinary planes an axiom which claims that each (motion) \(a\) can be represented as a product of finite many points \(p_i\) hence \((a = p_1 \circ (\dots\circ p_n)\dots)\) (cf. P15). In the last section, the author gives an axiom-system for the group generated by the point-reflections of a metric-Euclidean plane.