Browse > Article
http://dx.doi.org/10.4134/BKMS.2015.52.6.2071

SIMILAR AND SELF-SIMILAR CURVES IN MINKOWSKI n-SPACE  

OZDEMIR, MUSTAFA (Department of Mathematics Akdeniz University)
SIMSEK, HAKAN (Department of Mathematics Akdeniz University)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.6, 2015 , pp. 2071-2093 More about this Journal
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;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. G. Alcazar, C. Hermosoa, and G. Muntinghb, Detecting similarity of rational plane curves, J. Comput. Appl. Math. 269 (2014), 1-13.   DOI
2 T. Aristide, Closed similarity Lorentzian Ane manifolds, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3697-3702.   DOI
3 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.   DOI
4 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.   DOI
5 A. Bejancu, Lightlike curves in Lorentz manifolds, Publ. Math. Debrecen 44 (1994), no. 1-2, 145-155.
6 M. Berger, Geometry I, Springer, New York 1987.
7 A. Brook, A. M. Bruckstein, and R. Kimmel, On similarity-invariant fairness measures, LNCS 3459, pp. 456-467, 2005.
8 K.-S. Chou and C. Qu, Integrable equations arising from motions of plane curves, Phys. D 162 (2002), no. 1-2, 9-33.   DOI
9 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.   DOI
10 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.   DOI
11 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.
12 R. Encheva and G. Georgiev, Shapes of space curves, J. Geom. Graph. 7 (2003), no. 2, 145-155.
13 R. Encheva and G. Georgiev, Similar Frenet curves, Results Math. 55 (2009), no. 3-4, 359-372.   DOI
14 K. Falconer, Fractal Geometry, Second Edition, John Wiley & Sons, Ltd., 2003.
15 A. Ferrandez, A. Gimenez, and P. Lucas, Null helices in Lorentzian space forms, Int. J. Mod. Phys. A 16 (2001), 4845-4863.   DOI
16 W. Greub, Linear Algebra, 3rd ed., Springer Verlag, Heidelberg, 1967.
17 R. Grigorchuk and Z. Sunic, Self Similarity an branching group theory, Volume 1, London Mathematical Society Lecture Note Series: 339, Groups St Andrews 2005.
18 M. Gurses, Motion of curves on two-dimensional surfaces and soliton equations, Phys. Lett. A 241 (1998), no. 6, 329-334.   DOI
19 J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747.   DOI
20 Y. Kamishima, Lorentzian similarity manifolds, Cent. Eur. J. Math. 10 (2012), no. 5, 1771-1788.   DOI
21 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.
22 S. Z. Li, Invariant representation, matching and pose estimation of 3D space curves under similarity transformation, Pattern Recognition 30 (1997), no. 3, 447-458.   DOI
23 B. B. Mandelbrot, The Fractal Geometry of Nature, New York: W. H. Freeman, 1982.
24 K. Nakayama, Motion of curves in hyperboloid in the Minkowski space, J. Phys. Soc. Japan 67 (1998), no. 9, 3031-3037.   DOI
25 V. Nekrashevych, Self-similar groups and their geometry, Sao Paulo J. Math. Sci. 1 (2007), no. 1, 41-95.   DOI
26 B. O'Neill, Semi-Riemannian Geometry, Academic Press Inc., London, 1983.
27 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.
28 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.
29 H. Sahbi, Kernel PCA for similarity invariant shape recognition, Neurocomputing 70 (2007), 3034-3045.   DOI
30 D. A. Singer and D. H. Steinberg, Normal forms in Lorentzian spaces, Nova J. Algebra Geom. 3 (1994), no. 1, 1-9.