Fragments of Euclidean and hyperbolic geometry (Q2731727)

By fragments of Euclidian and hyperbolic geometry the author understands geometries which one obtains by weakening the assumptions defining the Euclidean or hyperbolic geometry. For several of these fragments the author presents axiomatizations which are expressed by means of quantifier-free axioms in first-order languages.NEWLINENEWLINENEWLINEThe first part deals with fragments of Euclidean geometry. In II1 Tarski's axiom system \(L(B,\equiv)\) is presented which is based on a set of ``points'' and structured by a betweenness relation \(B\) and congruence relation \(\equiv\) (collinearity ``\(L\)'' is derived from \(B\)). The first 10 resp. all 11 axioms characterize geometries \(E_2\) resp. \(E_2'\) whose models can be represented as Cartesian planes over an ordered Pythagorean resp. Euclidean field \(F\). Then in II2 the author gives the ``simplest'' \(L(B,\equiv)\) axiom systems for \(E_2\) and \(E_2'\) and (in II3) an independent axiom system for \(E_2\) in which metric properties are separated from order properties.NEWLINENEWLINENEWLINEThen there are considered axiom systems defining geometries \(BD_2, D_2 \) and \(D_2'\) such that their models are Cartesian planes over ordered fields, fields \(F\) of char\((F) \not= 2\) with \(F^{(2)} \not= F\) and fields with \(-1 \notin F^{(2)}\). For the affine plane a simple axiom system \(L(\|)\) based on a quaternary (parallelity) relation \(ab\|cd\) is given in II4 which is then augmented by axioms based on a ternary operation (describing reflection in lines) characterizing the geometries \(D_2\). It follows a universal axiom system \(\Delta\) for translation planes using besides a ternary collinarity relation a ternary operation \(P(abc) \) describing the image of \(c\) under the translation which maps \(a\) into \(b\) (II5).NEWLINENEWLINENEWLINEThen the axioms of Pappus (C1) and Desargues (C2) are expressed and discussed, what means ``to apply three times (C1)'' in order to prove \(\Delta \vee (C1) \Longrightarrow (C2)\). In this manner are treated moreover in Chap. II equiaffine geometry, André's central translation structures, metric-Euclidean and rectangular planes and in Chap. III (absolute geometry) and IV (hyperbolic geometry) geometries where neither the Euclidean parallel postulate nor a Euclidean metric is required, like metric planes, elementary hyperbolic geometry (in the sense of Menger), Klingenbergs hyperbolic planes (the point set can be represented by \(\{(x,y) \in K^2 \mid x^2 + y^2 \leq 1\}\) where \((K,+,.,\leq)\) is an arbitrary ordered field), Bachmann's ``Treffgeradenebenen'' and Baer's ``generalized hyperbolic planes'' \(L(\in ,\perp ,\alpha)\) where \(\alpha\) is a ternary relation on the set of lines describing the ends. Finally ``general affine geometry'' in the sense of Szczerba (it can be represented by open convex subsets of an affine plane over an ordered skewfield) is discussed.NEWLINENEWLINENEWLINEThere is also a larger survey of the literature dealing with such geometric structures.