On Characterization of Absolute Geometries

Abstract

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einführung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if ${\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}$. Denoting the foot of a on the line ${\overline{c, d}}$ with ${a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}$, the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if ${a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}$ or${a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}$ respectively. Let LS be the set of all L-S quadrangles and LS _r , LS _h or LS _e the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS =  LS _r or LS =  LS _h or LS =  LS _e . To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” ${o:=\overline{a,c}\cap\overline{b,d}}$, and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.