Open Access
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Quaternionic osculating curves in Euclidean and semi-Euclidean space
*Özcan BektaşCorresponding authorOZCAN.BEKTAS@ERDOGAN.EDU.TRDEPARTMENT OF MATHEMATICSRECEP TAYYIP ERDOGAN UNIVERSITYRIZE, 53100, TURKEYView full profile → , Nurten (Bayrak) GürsesDEPARTMENT OF MATHEMATICSYILDIZ TECHNICAL UNIVERSITYISTANBUL, 34220, TURKEYView full profile → , Salîm YüceDEPARTMENT OF MATHEMATICSYILDIZ TECHNICAL UNIVERSITYISTANBUL, 34220, TURKEYView full profile →
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- Received:
- 01 Sep 2015
- Accepted:
- 12 Dec 2015
- Published Online:
- 02 Aug 2016
- Article type:
- Article
- Language:
- EN
- Article no.:
- 1726037X.2016.1177935
- Pages:
- 65–84
Abstract
In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 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 E4. 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 E4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E4. In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.
Keywords
Subject Classifications
11R5253A0453C50
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