Quaternionic osculating curves in Euclidean and semi-Euclidean space

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 N1/2 (or N1/3) of its first binormal vector field N-2 (or N-3), where {T,N-1,N-2,N-3} 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.