Quaternionic osculating curves in Euclidean and semi-Euclidean space

Bektas O., Gurses N. (., YÜCE S.

JOURNAL OF DYNAMICAL SYSTEMS AND GEOMETRIC THEORIES, vol.14, no.1, pp.65-84, 2016 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 14 Issue: 1
  • Publication Date: 2016
  • Doi Number: 10.1080/1726037x.2016.1177935
  • Journal Name: JOURNAL OF DYNAMICAL SYSTEMS AND GEOMETRIC THEORIES
  • Journal Indexes: Emerging Sources Citation Index (ESCI)
  • Page Numbers: pp.65-84
  • Keywords: 11R52, 53A04, 53C50, Osculating curve, Frenet equations, Curvature, Real quaternion, HELICES
  • Recep Tayyip Erdoğan University Affiliated: Yes

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.