DOI QR코드

DOI QR Code

SIMILAR AND SELF-SIMILAR CURVES IN MINKOWSKI n-SPACE

  • OZDEMIR, MUSTAFA (Department of Mathematics Akdeniz University) ;
  • SIMSEK, HAKAN (Department of Mathematics Akdeniz University)
  • Received : 2014.11.11
  • Published : 2015.11.30

Abstract

In this paper, we investigate the similarity transformations in the Minkowski n-space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion of ${\mathbb{E}}_1^n$. We determine the parametrizations of non-null self-similar curves in ${\mathbb{E}}_1^n$.

Keywords

Lorentzian similarity geometry;similarity transformation;similarity invariants;similar curves;self-similar curves

References

  1. S. Z. Li, Invariant representation, matching and pose estimation of 3D space curves under similarity transformation, Pattern Recognition 30 (1997), no. 3, 447-458. https://doi.org/10.1016/S0031-3203(96)00089-1
  2. B. B. Mandelbrot, The Fractal Geometry of Nature, New York: W. H. Freeman, 1982.
  3. K. Nakayama, Motion of curves in hyperboloid in the Minkowski space, J. Phys. Soc. Japan 67 (1998), no. 9, 3031-3037. https://doi.org/10.1143/JPSJ.67.3031
  4. V. Nekrashevych, Self-similar groups and their geometry, Sao Paulo J. Math. Sci. 1 (2007), no. 1, 41-95. https://doi.org/10.11606/issn.2316-9028.v1i1p41-95
  5. B. O'Neill, Semi-Riemannian Geometry, Academic Press Inc., London, 1983.
  6. M. Ozdemir, On the focal curvatures of non-lightlike curves in Minkowski (m+1)-space, F. U. Fen ve Muhendislik Bilimleri Dergisi 16 (2004), no. 3, 401-409.
  7. H. Sahbi, Kernel PCA for similarity invariant shape recognition, Neurocomputing 70 (2007), 3034-3045. https://doi.org/10.1016/j.neucom.2006.06.007
  8. D. A. Singer and D. H. Steinberg, Normal forms in Lorentzian spaces, Nova J. Algebra Geom. 3 (1994), no. 1, 1-9.
  9. D. Xu and H. Li, 3-D curve moment invariants for curve recognition, Lecture Notes in Control and Information Sciences, 345, pp. 572-577, 2006.
  10. M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 243-275. https://doi.org/10.1098/rspa.1985.0057
  11. M. F. Barnsley, J. E. Hutchinson, and O. Stenflo, V-variable fractals: Fractals with partial self similarity, Adv. Math. 218 (2008), no. 6, 2051-2088. https://doi.org/10.1016/j.aim.2008.04.011
  12. A. Bejancu, Lightlike curves in Lorentz manifolds, Publ. Math. Debrecen 44 (1994), no. 1-2, 145-155.
  13. M. Berger, Geometry I, Springer, New York 1987.
  14. A. Brook, A. M. Bruckstein, and R. Kimmel, On similarity-invariant fairness measures, LNCS 3459, pp. 456-467, 2005.
  15. K.-S. Chou and C. Qu, Integrable equations arising from motions of plane curves, Phys. D 162 (2002), no. 1-2, 9-33. https://doi.org/10.1016/S0167-2789(01)00364-5
  16. K.-S. Chou and C. Qu, Motions of curves in similarity geometries and Burgers-mKdV hierarchies, Chaos Solitons Fractals 19 (2004), no. 1, 47-53. https://doi.org/10.1016/S0960-0779(03)00060-2
  17. Q. Ding and J. Inoguchi, Schrodinger ows, binormal motion for curves and the second AKNS-hierarchies, Chaos Solitons Fractals 21 (2004), no. 3, 669-677. https://doi.org/10.1016/j.chaos.2003.12.092
  18. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Volume 364 of Mathematics and its Aplications. Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996.
  19. J. G. Alcazar, C. Hermosoa, and G. Muntinghb, Detecting similarity of rational plane curves, J. Comput. Appl. Math. 269 (2014), 1-13. https://doi.org/10.1016/j.cam.2014.03.013
  20. T. Aristide, Closed similarity Lorentzian Ane manifolds, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3697-3702. https://doi.org/10.1090/S0002-9939-04-07560-4
  21. R. Encheva and G. Georgiev, Shapes of space curves, J. Geom. Graph. 7 (2003), no. 2, 145-155.
  22. R. Encheva and G. Georgiev, Similar Frenet curves, Results Math. 55 (2009), no. 3-4, 359-372. https://doi.org/10.1007/s00025-009-0407-8
  23. K. Falconer, Fractal Geometry, Second Edition, John Wiley & Sons, Ltd., 2003.
  24. A. Ferrandez, A. Gimenez, and P. Lucas, Null helices in Lorentzian space forms, Int. J. Mod. Phys. A 16 (2001), 4845-4863. https://doi.org/10.1142/S0217751X01005821
  25. W. Greub, Linear Algebra, 3rd ed., Springer Verlag, Heidelberg, 1967.
  26. R. Grigorchuk and Z. Sunic, Self Similarity an branching group theory, Volume 1, London Mathematical Society Lecture Note Series: 339, Groups St Andrews 2005.
  27. M. Gurses, Motion of curves on two-dimensional surfaces and soliton equations, Phys. Lett. A 241 (1998), no. 6, 329-334. https://doi.org/10.1016/S0375-9601(98)00151-0
  28. J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. https://doi.org/10.1512/iumj.1981.30.30055
  29. Y. Kamishima, Lorentzian similarity manifolds, Cent. Eur. J. Math. 10 (2012), no. 5, 1771-1788. https://doi.org/10.2478/s11533-012-0076-9
  30. S. Z. Li, Similarity invariants for 3D space curve matching, In Proceedings of the First Asian Conference on Computer Vision, pp. 454-457, Japan 1993.

Cited by

  1. Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space vol.15, pp.06, 2018, https://doi.org/10.1142/S0219887818500925