Results in Mathematics

, Volume 59, Issue 1–2, pp 163–171 | Cite as

Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

  • Helmut Karzel
  • Mario Marchi
  • Sayed-Ghahreman Taherian


This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If \({(P,\mathfrak{G}, \equiv,\tau)}\) denotes an elliptic geometry ordered via a separation τ then there are polar points o and ∞ and on the line \({ \overline{K} := \overline{\infty,o}}\) there can be established an operation “+” such that \({(\overline{K},+)}\) becomes a commutative group and the map \({ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}\) is a motion on \({\overline{K}}\). The separation τ induces on \({\overline{K}}\) a cyclic order ω with [o, e, ∞] = 1 such that \({(\overline{K},+,\omega)}\) becomes a cyclic ordered group. For \({a,b \in K := \overline{K} {\setminus}\{\infty\}}\) we set \({a < b :\Longleftrightarrow [a,b,\infty] =1}\) and for all \({a\in K\,a < \infty}\). Then (K, <) is a totally ordered set. We show there is a surjective distance function:
$$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$
with “\({\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}\)”. We prove in the first part of our project like (cf. Gröger in Mitt Math Ges Hamburg 11:441–457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with λ(a, c), λ(b, c) < e then λ(a, b) ≤ λ(a, c) + λ(b, c) < ∞.

Mathematics Subject Classification (2000)

Primary 51M10 Secondary 51G05 


Elliptic geometry triangle-inequality cyclic ordered group 


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  1. 1.
    Gröger D.: Archimedisierung elliptischer Ebenen. Mitt. Math. Ges. Hamburg. 11, 441–457 (1987)MATHMathSciNetGoogle Scholar
  2. 2.
    Karzel H., Kroll H.-J.: Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt (1988)MATHGoogle Scholar
  3. 3.
    Karzel H., Sörensen K., Windelberg D.: Einführung in die Geometrie. UTB 184, Göttingen (1973)MATHGoogle Scholar
  4. 4.
    Karzel H., Maxson C.J.: Archimedisation of some ordered geometric structures which are related to kinematic spaces. Results Math. 19, 290–318 (1991)MATHMathSciNetGoogle Scholar
  5. 5.
    Sörensen K.: Elliptische Ebenen. Mitt. Math. Ges. Hamburg. 10, 277–296 (1976)Google Scholar
  6. 6.
    Sörensen K.: Elliptische Räume. Mitt. Math. Ges. Hamburg. 18, 159–167 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Helmut Karzel
    • 1
  • Mario Marchi
    • 2
  • Sayed-Ghahreman Taherian
    • 3
  1. 1.Zentrum MathematikTechnische Universität MünchenMunichGermany
  2. 2.Dipartimento di Matematica e FisicaUniversità CattolicaBresciaItaly
  3. 3.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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