SavedAdd your journal

Journals / Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi / 2017 / Cilt: 21 - Sayı: 2

On the Quaternionic Focal Curves

Nurten BAYRAK GÜRSES
ÖZCAN BEKTAŞ
SALİM YÜCE
Pages
357–366
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this study, a brief summary about quaternions and quaternionic curves are firstly presented.
📊
Key Findings & Data: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.
💡
Conclusion & Impact:Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Quaternions · Quaternionic curves · Osculating hypersphere

Özet

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.

🔖

Cite This Article

GÜRSES, N. B., BEKTAŞ, Ö., YÜCE, S. (2017). On the Quaternionic Focal Curves. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 21 - Sayı: 2, 357–366. https://ajindex.com/articles/310334

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  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 hypersurfa