On the Smarandache Curves of Spatial Quaternionic Involute Curve
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On the Smarandache Curves of Spatial Quaternionic Involute Curve Su¨leyman S¸ enyurt1
•
Ceyda Cevahir1 • Yasin Altun1
Revised: 12 September 2019 / Accepted: 16 October 2019 Ó The National Academy of Sciences, India 2019
Abstract In this study, the spatial quaternionic curve and the relationship between Frenet frames of involute curve of spatial quaternionic curve are expressed by using the angle between the Darboux vector and binormal vector of the basic curve. Secondly, the Frenet vectors of involute curve are taken as position vector and curvature and torsion of obtained Smarandache curves are calculated. The calculated curvatures and torsions are given depending on Frenet apparatus of basic curve. Finally, an example is given and the shapes of these curves are drawn by using Mapple program. Keywords Quaternionic curves Involute curve Quaternionic Smarandache curves Mathematics Subject Classification 53A04 53C26
1 Introduction The quaternion first time was introduced by Irish mathematic William Rowan Hamilton in 1843. His initial attempt to generalize the complex numbers by introducing a three-
& Su¨leyman S¸ enyurt [email protected] Ceyda Cevahir [email protected] Yasin Altun [email protected] 1
Department of Mathematics, Faculty of Arts and Sciences, Ordu Universty, Ordu, Turkey
dimensional object failed in the sense that the algebra he constructed for these three-dimensional objects did not have the desired properties. In 1987, Bharathi and Nagaraj defined the quaternionic curves in E3 , E4 and studied the differential geometry of space curves and introduced Frenet frames and formulae by using quaternions [1]. Following, quaternionic-inclined curves have been defined and harmonic curvatures studied by Karadag˘ and Sivridag˘ [2]. In [3], Tuna and C ¸ o¨ken have studied quaternion-valued functions and quaternionic-inclined curves in the semiEuclidean space E2 4 . In [4], Eris¸ ir and Gu¨ngo¨r have obtained some characterizations of semi-real spatial quaternionic rectifying curves in IR1 3 . Moreover, by the aid of these characterizations, they have investigated semi-real quaternionic rectifying curves in semi-quaternionic space. In [5], after general definition of quaternions, relations between real quaternions and Serret–Frenet formulas have been investigated. Although real quaternions are represented by four basis elements, vectors can also be expressed by using their three basis elements that have complex nature. On the other hand, the difference of quaternion product than the well-known vector product is not an obstacle to obtain Serret–Frenet formulas by real quaternions. In this study, an alternative formulation has been developed for the representation of Serret–Frenet formulas. In the literature, G. Darboux defined the DArboux vector and many studies have been done in the light of this definition. Fenchell gave more importance to Darboux vector interpretation initiated by G. Darboux and he enhanced [6]. The relationship between the Frenet frames of the
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