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Triangles in Euclidean Arrangements

  • Stefan Felsner
  • Klaus Kriegel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

The number of triangles in arrangements of lines and pseudolines has been Object of some research. Most results, however, concern arrangements in the projective plane. We obtain results for the number of triangles in Euclidean arrangements of pseudolines. Though the Change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs.

In 1926 Levi proved that a nontrivial arrangement – simple or not – of n pseudolines in the projective plane contains n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains n – 2 triangles, we had to find a completely different proof. On the other hand a non-simple arrangements of n pseudolines in the Euclidean plane tan have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and some extensions.

Mathematics Subject Classifications (1991) 52A10, 52ClO.

Keywords

Arrangement Euclidean plane pseudoline strechability triangle 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Stefan Felsner
    • 1
  • Klaus Kriegel
    • 1
  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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