Quaternionic osculating curves in Euclidean and semi-Euclidean space. Issue 1 (2nd January 2016) - Bektaş, Özcan; (Bayrak) Gürses, Nurten; Yüce, Salîm
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.