On the Quaternionic Focal Curves

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In this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere.  By using this definition and the quaternionic osculating hyperspheres of these curves, the quaternionic focal curves in the spaces $\Q$ and $\Q_\nu$ with index $\nu=\{1,2\}$ are discussed. Some relations about spatial semi-real quaternionic curves and semi-real quaternionic curves are examined by using focal curvatures and "scalar Frenet equations" between the focal curvatures. Then, the notions: such as vertex, flattenings, a symmetry point are defined for these curves. Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented.

Anahtar kelimeler

Quaternions; Quaternionic curves; Osculating hypersphere; Focal curves; Semi-Euclidean space

Tam metin:


DOI: http://dx.doi.org/10.19113/sdufbed.14005


[1] Ward, J. P. 1997. Quaternions and Cayley Numbers, Kluwer Academic Publishers, Boston/London.

[2] Bharathi, K. and Nagaraj, M. 1987. Quaternion Valued Function of a Real Variable Serret-Frenet Formulae, Indian Journal of Pure and Applied Mathematics. 18 (6), 507–511.

[3] Tuna, A. 2002. Serret Frenet formulae for Quaternionic Curves in Semi Euclidean Space. Master Thesis, Süleyman Demirel University, Graduate School of Natural and Applied Science, Isparta, Turkey.

[4] Çöken, A. C. and Tuna, A. 2004. On the quaternionic inclined curves in the semi-Euclidean space E42, Applied Mathematics and Computation. 155 (2), 373–389.

[5] Kahraman, F., Gök, İ. and Hacısalihoğlu, H. H. 2012. On the quaternionic B2 slant helices in the semi- Euclidean space E42, Applied Mathematics and Computation. 218(11) , 6391–6400.

[6] Hacısalihoğlu, H. H. 1993. Diferensiyel Geometri, Faculty of Sciences University of Ankara Press.

[7] Sabuncuoğlu, A. 2010. Diferensiyel Geometri. Nobel Press.

[8] Struik, D. J. 2012. Lectures on Classical Differential Geometry. Second edition, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.

[9] Sağlam, D. 2012. On the Osculating Spheres of a Real Quaternionic Curve in the Euclidean Space E4, International Journal of Mathematical Combinatorics. 3, 46–53.

[10] Soytürk, E., İlarslan, K. and Sağlam, D. 2005. Osculating spheres and osculating circles of a curve in semi-Reimannian space, Communications, Faculty of Science. University of Ankara Series A1. 54 (2), 39–48.

[11] Bekta¸s, Ö., (Bayrak) Gürses, N. and Yüce, S. 2014. Osculating Spheres of a Semi Real Quaternionic Curves in E42, European Journal of Pure and Applied Mathematics. 7 (1), 86–96.

[12] Uribe-Vargas, R. 2005. On vertices, focal curvatures and differential geometry of space curves, Bull. Brazilian Math. Soc. 36 (3), 285–307.

[13] Özdemir, M. 2004. On the Focal Curvatures of Nonlightlike Curves in Minkowski (m+1)-Space, Fırat Üniversitesi Fen ve Mühendislik Bilimleri Dergisi. 16 (3), 401–409.

[14] Öztürk, G. and Arslan, K. 2016. On focal curves in Euclidean n-space Rn, Novi Sad Journal of Mathematics, 46 (1), 35-44.

[15] Şimşek, H. 2016. Focal curves and focal surfaces in finite dimensional minkowski space, Phd Thesis, Akdeniz University, 119 pages.

[16] Wang, Z., Pei, D., Chen, L., Kong, L. and Han, Q. 2012. Singularities of focal surfaces of null Cartan curves in Minkowski 3-space, Abstract and Applied Analysis, 1-20.

[17] Liu, X. and Wang, Z. 2015. On lightlike hypersurfaces and lightlike focal sets of null Cartan curves in Lorentz-Minkowski spacetime, Journal of Nonlinear Science and Applications, 8(5): 628-639.

[18] Şimşek, H. 2017. On focal curves of null Cartan curves, Turkish Journal of Mathematics, DOI: 10.3906/mat-1604-79.

[19] Asil, V., Ba¸s, S. and Körpınar, T. 2013. On Construction of D-Focal Curves in Euclidean 3-Space M3, Bol. Soc. Paran. Mat., (3s.) v. 31, 273-277.

[20] Körpınar, T., Turhan, E. and Bonilla, JL. 2014. Focal Curves of Bıharmonıc Curves in the SL R 2, International Journal of Mathematical Engineering and Science ISSN : 2277-6982 1(2).

[21] Körpınar, T. and Turhan, E. 2011. New representations of focal curves in the special Ricci Symmetric Para-Sasakian Manifold P, Revista Notas de Matemática, Vol.7(2), No. 320, 195-201.

[22] Tuna Aksoy, A. and Çöken, A. C. 2015. Serret-Frenet Formulae for Null Quaternionic Curves in Semi Euclidean 4-Space R41, Acta Physica Polonica A. 128 (2-B).

[23] Lopez, R. 2014. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, International Electronic Journal of Geometry, 7(1), 44–107.

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