Minkowski Planes with Miquelian Pairs

Abstract: The miquelian Minkowski planes, i.e. the Minkowski planes which can be represented as geometry of the plane sections of a non-degenerate ruled quadric in a 3-dimensional projective space (cf. W. Benz: {\it Vorlesungen über Geometrie der Algebren.} Berlin-Heidelberg-New York 1973; Zbl.: 258.50024 ) are characterized by K. Dienst: {\it Minkowski-Ebenen mit Spiegelungen.} Mh. Math. 84 (1977), 197 - 208 p. 205; Zbl.: 378.50017) by the following property: For every cycle $C$ there is an inversion fixing $C$ pointwise. An inversion of a Minkowski plane is an improper automorphism $\sigma$ with the property, that for any two points $x,y$ where $y$ is joinable with $x$ and $\sigma (x)$ the points $x,y,\sigma (x), \sigma (y)$ are concyclic (cf. N. Percsy: {\it Les plans de Minkowski possedant des inversions sont miquellien.} Simon Stevin 57 (1983), 15 - 35; Zbl.: 541.51009). \par A 2-set $\{a,b\}$ of joinable points $a,b$ of an arbitrary Minkowski plane $\cal M$ is called miquelian, if the set $I(a,b)$ of all inversions interchanging $a$ and $b$ operates transitively on $X\setminus \{ a,b\}$ for all cycles $X$ containing $a,b$ . In his paper, N. Percsy studied Minkowski planes with miquelian pairs. His main result characterizes the miquelian Minkowski planes by the property that every 2-set $\{ a,b\}$ of joinable points $a,b$ is miquelian (N. Percsy, Theorem 1.3). \par For a group $\Gamma $ of automorphisms of ${\cal M}$ let $M(\Gamma )$ denote the set of all miquelian pairs $\{ a,b\}$ such that $I(a,b) \subset \Gamma $. In this paper, after recalling some basic definitions we will first study the configurations $M(\Gamma )$. Applying results of M. Klein: {\it Classification of Minkowski planes by transitive groups of homotheties.} J. Geom. 43 (1992), 116 - 128; Zbl.: 746.51009 we get a typification of the groups $\Gamma $ of automorphisms of a Minkowski plane ${\cal M}$ (Theorem 3.6). With $\Gamma = {\rm Aut}\, {\cal M}$ this means a typification of the Minkowski planes. In the last section we will improve N. Percsy's Theorem 1.3, namely a Minkowski plane is already miquelian if there exists two different miquelian pairs $\{ p,q\},\; \{ p',q'\}$ where $p,p'$ are points of a generator $G$ and $q,q'$ are points of a generator $F$ with $|G \cap F| = 1$ (Theorem 4.3). As a corollary we obtain an improvement of the above mentioned characterization of K. Dienst (Corollary 4.5).

Keywords: Minkowski plane, inversion, miquelian pair

Classification (MSC91): 51B20

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