Özcan BEKTAŞ, Salim YÜCE, Nurten BAYRAK GÜRSES

Kuaterniyonik Fokal Egriler Üzerine

Bu çalışmada, ilk olarak kuaterniyonlar ve kuaterniyonik egriler hakkında kısa bir özet sunulmuş, ayrıca fokal egri tanımı verilmiştir. Düzgün bir eğrinin fokal eğrisi oskülatör hiperküresinin merkezlerinden oluşur. Bu tanım ve bu egrilerin kuaterniyonik öskülatör hiperküreleri kullanılarak, ν = {1,2} olmak üzere Q ve Qν uzaylarında kuaterniyonik fokal egri tanımı ele alınmıştır. Uzaysal yarı-reel kuaterniyonik eğriler ve yarı-reel kuaterniyonik egriler hakkında bazı ilişkiler, fokal eğrilikler ve fokal eğriler arasındaki "skalar Frenet denklemleri" kullanılarak verilmiştir. Ayrıca bu egriler için tepe noktası, basıklık ve simetri noktası gibi kavramlar tanımlanmıştır. Bunların yanında bir kuaterniyonik egrinin Frenet elemanları ve onun kuaterniyonik fokal eğrisinin Frenet elemanları arasındaki ilişki sunulmuştur.

On the Quaternionic Focal Curves

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 Qν with index ν = {1,2} are discussed. Some relations about spatial semireal 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.

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Lopez, R. 2014. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, International Electronic Journal of Geometry, 7(1), 44–107.
una Aksoy, A. and Çöken, A. C. 2015. Serret-Frenet Formulae for Null Quaternionic Curves in Semi Euclidean 4-Space R 4 1 , Acta Physica Polonica A. 128 (2-B).
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.
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).
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.
Şim¸sek, H. 2017. On focal curves of null Cartan curves, Turkish Journal of Ma thematics, DOI: 10.3906/mat-1604-79.
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.
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.
Şim¸sek, H. 2016. Focal curves and focal surfaces in finite dimensional minkowski space, Phd Thesis, Akdeniz University, 119 pages.
Öztürk, G. and Arslan, K. 2016. On focal curves in Euclidean n-space R n , Novi Sad Journal of Mathematics, 46 (1), 35-44.
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.
Uribe-Vargas, R. 2005. On vertices, focal curvatures and differential geometry of space curves, Bull. Brazilian Math. Soc. 36 (3), 285–307.
Bekta¸s, Ö., (Bayrak) Gürses, N. and Yüce, S. 2014. Osculating Spheres of a Semi Real Quaternionic Curves in E 4 2 , European Journal of Pure and Applied Mathematics. 7 (1), 86–96.
Soytürk, E., İlarslan, K. and Saglam, D. 2005. Os- culating spheres and osculating circles of a curve in semi-Reimannian space, Communications, Faculty of Science. University of Ankara Series A1. 54 (2), 39– 48.
Saglam, D. 2012. On the Osculating Spheres of a Real Quaternionic Curve in the Euclidean Space E 4 , International Journal of Mathematical Combinatorics. 3, 46–53.
Struik, D. J. 2012. Lectures on Classical Differential Geometry. Second edition, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.
Sabuncuoglu, A. 2010. Diferensiyel Geometri. Nobel Press.
Hacısalihoglu, H. H. 1993. Diferensiyel Geometri, Faculty of Sciences University of Ankara Press.
Kahraman, F., Gök, İ. and Hacısalihoglu, H. H. 2012. On the quaternionic B2 slant helices in the semi- Euclidean space E 4 2 , Applied Mathematics and Computation. 218(11) , 6391–6400.
Çöken, A. C. and Tuna, A. 2004. On the quaternionic inclined curves in the semi-Euclidean space E 4 2 , Applied Mathematics and Computation. 155 (2), 373– 389.
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.
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.
Ward, J. P. 1997. Quaternions and Cayley Numbers, Kluwer Academic Publishers, Boston/London.