Quaternionic osculating curves in Euclidean and semi-Euclidean space. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Bektaş, Özcan; (Bayrak) Gürses, Nurten; Yüce, Salîm
• Is Part Of: Journal of dynamical systems and geometric theories. Volume 14:Issue 1(2016)
• Page Start: 65
• Page End: 84
• Publication Date: 2016-01-02
• Subjects: 11R52 -- 53A04 -- 53C50; Osculating curve -- Frenet equations -- Curvature -- Real quaternion; Differentiable dynamical systems -- Periodicals; Geometry -- Periodicals; Differentiable dynamical systems; Geometry; Periodicals; 515.39
• DOI: 10.1080/1726037X.2016.1177935
• Languages: English
• ISSNs: 1726-037X

Abstract

Abstract: In this study, the osculating curves in Euclidean space E 3 and E 4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E 3, then we portray the quaternionic osculating curve in E 4 as a quaternionic curve whose position vector every time reclines in the orthogonal complement N½ (or N⅓) of its first binormal vector field N2 (or N3 ), where {T, N1, N2, N3 } be the Frenet instrumentations of the quaternionic curve in the Euclidean space E 4 . We feature quaternionic oscu­lating curves from the point of view their curvature functions K, k and (r — K ) and serve the necessary and the sufficient conditions for arbitrary quaternionic curve in E 4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E 4 . In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.