Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry

Research output: Contribution to journalArticle

Abstract

With the aim of revealing their purely geometric nature, we rephrase two theorems of S. Yang and A. Fang [S. Yang, A. Fang, A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664] characterizing Möbius transformations as definability results in elementary plane hyperbolic geometry. We show not only that elementary plane hyperbolic geometry can be axiomatized in terms of the quaternary predicates λ or σ, with λ (a b c d) to be read as 'a b c d is a Lambert quadrilateral' and σ (a b c d) to be read as 'a b c d is a Saccheri quadrilateral', but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (∀ and ∃) and the connectives ∨ and ∧ in the definiens) in terms of λ or σ.

Original languageEnglish (US)
Pages (from-to)531-532
Number of pages2
JournalJournal of Mathematical Analysis and Applications
Volume346
Issue number2
DOIs
StatePublished - Oct 15 2008
Externally publishedYes

    Fingerprint

Keywords

  • Hyperbolic geometry
  • Lambert quadrilaterals
  • Möbius transformations
  • Saccheri quadrilaterals

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this