Results in Mathematics

, Volume 59, Issue 1–2, pp 163–171

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

• Helmut Karzel
• Mario Marchi
• Sayed-Ghahreman Taherian
Article

## Abstract

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

## Keywords

Elliptic geometry triangle-inequality cyclic ordered group

## References

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## 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