Quaternionic Salkowski Curves and Quaternionic Similar Curves
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RESEARCH ARTICLE
Quaternionic Salkowski Curves and Quaternionic Similar Curves ¨ nder1 Mehmet O
Received: 27 July 2017 / Revised: 8 March 2019 / Accepted: 15 March 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic antiSalkowski and quaternionic similar curves in the Euclidean spaces E3 and E4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B2-slant helices.
In the differential geometry, special curves which satisfy some relationships between their curvatures and torsions have an important role. The most popular one of these curves is general helix which is defined by the property that the tangent of the curve makes a constant angle with a fixed straight line called the axis of the general helix [1]. Moreover, recently some new special curves have been defined and studied. Izumiya and Takeuchi [2] have defined slant helix which is a special curve whose principal normal vector makes a constant angle with a fixed direc¨ nder et al. [3] have considered the notion of slant tion. O helix in E4 and defined B2 -slant helix. Furthermore, Salkowski [4] defined the curves with constant curvature but
non-constant torsion by an explicit parametrization. Later, Monterde [5] has given some characterizations of Salkowski and anti-Salkowski curves. El-Sabbagh and Ali [6] have defined a new curve couple called similar curves whose arc-length parameters have a relationship and their tangents are the same. These curves have also been studied in different spaces [7, 8]. In this paper, we define quaternionic and spatial quaternionic Salkowski curves, anti-Salkowski curves and similar curves. We obtain the characterizations for these special quaternionic and spatial quaternionic curves. First, we give the basic elements of the theory of quaternions and quaternionic curves. A more complete elementary treatment of quaternions and quaternionic curves can be found in references [9–11]. A real quaternion q is an expression of the form q ¼ a1 e1 þ a2 e2 þ a3 e3 þ a4 e4 , where ai ; ð1 i 4Þ are real numbers, and ei ; ð1 i 4Þ, e4 ¼ 1 are quaternionic units which satisfy the non-commutative multiplication rules ei ei ¼ e4 ; ð1 i 3Þ; ei ej ¼ ej ei ¼ ek , ð1 i; j; k 3Þ where ðijkÞ is an even permutation of (123) in the Euclidean space. The algebra of the quaternions is denoted by Q, and its natural basis is given by fe1 ; e2 ; e3 ; e4 g. A real quaternion is given by the form q ¼ sq þ vq where sq ¼ a4 is scalar part and vq ¼ a1 e1 þ a2 e2 þ a3 e3 is vector part of q. The conjugate of q ¼ sq þ vq is defined by q ¼ sq vq . This defines the symmetric real-valued, non-degenerate, bilinear form as follows:
¨ nder & Mehmet O [email protected]
1 h : Q Q ! IR; ðq; pÞ ! h ðq; pÞ ¼ ðq p þ p qÞ; 2 ð1Þ
Keywords Quaternionic curve quaternionic frame Salkowski curve Similar curve Mathematics Subject Classific
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