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On the existence of Bertrand pairs

  • A. M. d’Azevedo BredaEmail author
  • F. J. Craveiro de Carvalho
  • Bernd Wegner
Original Paper
  • 62 Downloads

Abstract

In this note we introduce the Bertrand Group for a curve. While there is no mention of this group in it, Saban (Simon Stevin 57:37–45, 1983) contains a statement which would mean that for a simple, closed, twisted curve that group would be trivial. However there appears to be a flaw in the proof given there. Here we are able to show that for a simple, twisted curve that group is either trivial or Z 2. If this latter group can occur we do not know. We also show that for some classes of curves the non-triviality of that group forces the curve to be plane.

Keywords

Curves in Euclidean space Frenet-Serret frame Bertrand mates 

Mathematics Subject Classification (2000)

53A04 51H30 53A55 

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References

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Copyright information

© The Managing Editors 2011

Authors and Affiliations

  • A. M. d’Azevedo Breda
    • 1
    Email author
  • F. J. Craveiro de Carvalho
    • 2
  • Bernd Wegner
    • 3
  1. 1.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  2. 2.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  3. 3.Fachbereich MathematikBerlinGermany

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