Universal hyperbolic geometry. II: A pictorial overview. (English) Zbl 1217.51009

This is a joint review of parts I [Geom. Dedicata 163, 215–274 (2013; Zbl 1277.51018)] and II of this series of papers.
In the Beltrami-Klein model of the hyperbolic plane, in which points belong to a certain domain \(D\) inside a projective plane over a Euclidean ordered field \(F\), where points have homogeneous coordinates \((x,y,z)\) and lines (that have non-empty intersection with \(D\)) have homogeneous coordinates \([u,v,w]\), with \((x,y,z)\) being incident with \([u,v,w]\) if and only if \(xu+yv+zw=0\), and with two lines \([u,v,w]\) and \([u',v',w']\) being orthogonal precisely if \(uu'+vv'-ww'=0\). If one regards the whole projective plane over an arbitrary field \(F\) of characteristic \(\neq 2\), lets points and lines be homogeneous triples as above, but excluding self-orthogonal lines, with incidence and orthogonality defined as above, one obtains a particular instance of a projective-metric plane with hyperbolic metric that we will refer to as \(P_{-1}(F)\).
The author of these two papers introduces \(P_{-1}(F)\) in an unfamiliar guise, in which the incidence of \((x,y,z)\) and \([u,v,w]\) is defined as \(xu+yv-zw=0\) (that they are isomorphic can be seen via the isomorphism sending every point \([x,y,z]\) to itself, and every line \((l,m,n)\) to the line \((l,m,-n)\)), and in which isotropic lines (called “null lines”) are part of the geometry. The reader is being misled on page 4 of this part II into believing that this particular form of incidence is “the characterizing equation for hyperbolic geometry” and that the usual definition of incidence(\(xu+yv+zw=0\)) is “a different convention” used “for spherical/elliptic geometry”. He claims that this is “hyperbolic geometry \(\ldots\) in a new and completely algebraic way (p. 215)”, and presents a long list of advantages of this “new approach”. Notions such as null lines, null points, quadrance (instead of distance between two points) and its line-dual spread, quadrea (instead of area), are defined and used in “theorems” bearing suggestive names, such as “Pythagoras’ theorem”, “Thales”, “Rightparallax”, “Napier’s rules”, “Pons Asinorum”, “Menelaus”, “Ceva”. No less than 92 “theorems” regarding these and other notions defined in this paper are proved in one paper and “visualized” by means of figures in the other.
Without “null lines” (better known as isotropic lines) and “null points,” this is thus a special case of the geometry of projective-metric planes with a hyperbolic metric (so that line-orthogonality is algebraically identical to that in the standard hyperbolic plane). This geometry has been axiomatized in Chapter 6, Section 7 of R. Lingenberg’s [Metric planes and metric vector spaces. New York etc.: John Wiley & Sons (1979; Zbl 0419.51001)] (see also his [Abh. Math. Semin. Univ. Hamb. 48, 241–263 (1979; Zbl 0412.51004)], §11 of [F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. ergänzte Aufl. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0254.50001)], as well as H. Struve and R. Struve [Z. Math. Logik Grundlagen Math. 34, No. 1, 79–88 (1988; Zbl 0639.51015)] and the reviewer’s [Bull. Pol. Acad. Sci., Math. 52, No. 3, 297–302 (2004; Zbl 1101.51007)]). The main novelty in the algebraic set-up consists in a change of sign that renders both incidence and perpendicularity algebraically identical. The concept of “quadrance” is not new: one can find it under the name of \(Q\)-Distanz or Cayley-Distanz in the general case of a projective-metric plane endowed with a quadratic form \(Q\) on page 144 of E. M. Schröder [Vorlesungen über Geometrie. Band 3: Metrische Geometrie. Mannheim etc.: BI-Wiss.-Verlag (1992; Zbl 0754.51004)]. The concepts of “quadrea” and “quadreal” appear to be new.
The entire presentation is algebraic in one paper and pictorial in the other, but there is no axiom system. The entire exercise seems to be of a rather pedagogical nature, the author aiming at introducing a less convoluted version of hyperbolic geometry to a novice audience. The main problem with this approach is that this is quite far from actual hyperbolic geometry, and that the criterion for the choice of the theorems (rather than an adequate set of axioms) or their significance is not indicated.


51M10 Hyperbolic and elliptic geometries (general) and generalizations
51M15 Geometric constructions in real or complex geometry
14N99 Projective and enumerative algebraic geometry
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