Journal of Geometry

, Volume 108, Issue 1, pp 195–214 | Cite as

Evolutes of plane curves and null curves in Minkowski 3-space

  • Boaventura Nolasco
  • Rui Pacheco


We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski 3-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the corresponding plane curves. This leads to an alternative description of all plane curves which are Laguerre congruent to a given one.


Laguerre geometry Minkowski space null curves plane curves evolutes associated pairs of curves 

Mathematics Subject Classification

53A04 53A35 53B30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balgetir H., Bectass M., Inoguchi J.: Null Bertrand curves and their characterizations. Note Mat. 23(1), 7–13 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cartan, E.: Leçons sur la théorie des espaces à connexion projective. Paris, Gauthier-Villars (1937)Google Scholar
  3. 3.
    Cecil, T.E.: Lie Sphere Geometry. Universitext, Springer, New York (1992)Google Scholar
  4. 4.
    Choi J.H., Kim Y.H.: Note on Null Helices in \({\mathbf{E}_{1}^{3}}\), Bull. Korean Math. Soc. 50(3), 885–899 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Duggal K.L., Bejancu A.: Lightlike submanifolds of semi-Riemannian manifolds and applications. Kluwer Academic Publishers, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds, Birkhäuser (2010)Google Scholar
  7. 7.
    Ferrández A., Giménez A., Lucas P.: Null generalized helices in Lorentzian space forms. Internat. J. Modern Phys. A 16, 4845–4863 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fillmore J.P., Springer A.: New Euclidean theorems by the use of Laguerre transformations. J. Geom. 52, 74–90 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ghys, E., Tabachnikov, S., Timorin, V.: Osculating curves: around the Tait-Kneser theorem, Math. Intell. 35(1), (2013)Google Scholar
  10. 10.
    Graves L.K.: Codimension one isometric immersions between Lorentz spaces. Trans. Am. Math. Soc. 252, 367–392 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guggenheimer, H.W.: Differential Geometry. General Publishing Com., (1997)Google Scholar
  12. 12.
    Honda K., Inoguchi J.: Deformation of Cartan framed null curves preserving the torsion. Differ. Geom. Dyn. Syst. 5, 31–37 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Inoguchi J.-I., Lee S.: Null curves in Minkowski 3-space. Int. Electron. J. Geom. 1(2), 40–83 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    López R.: Differential geometry of curves and surfaces in Lorentz–Minkowski space. Int. Electron. J. Geom. 7(1), 44–107 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Olszak, Z.: A note about the torsion of null curves in the 3-dimensional Minkowski spacetime and the Schwarzian derivative. Filomat 29(3), 553–561 (2015) doi: 10.2298/FIL1503553O
  16. 16.
    Tait P.G.: Note on the circles of curvature of a plane curve. Proc. Edinburgh Math. Soc. 14, 403 (1896)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Instituto Superior de Ciências da EducaçãoLubangoAngola
  2. 2.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

Personalised recommendations