Characterizations of Adjoint Curves According to Alternative Moving Frame

Abstract

In this paper, the adjoint curve is defined by using the alternative moving frame of a unit speed space curve in 3-dimensional Euclidean space. The relationships between Frenet vectors and alternative moving frame vectors of the curve are used to offer various characterizations. Besides, ruled surfaces are constructed with the curve and its adjoint curve, and their properties are examined. In the last section, there are examples of the curves and surfaces defined in the previous sections.

On Bishop Frames of Any Regular Curve in Euclidean 3-Space

“…As can be understood from their definitions, tangential vector fields of type-1 Bishop and Frenet frames, binormal vector fields of type-2 Bishop and Frenet frames, and principal normal vector fields of N-Bishop and Frenet frames are common. There are many studies on this new types of Bishop and alternative frame (Alıç and Yılmaz 2021, Çakmak and Şahin 2022, Damar et al 2017, Kızıltuğ et al 2013, Masal and Azak 2015, Ourab et al 2018, Samancı and Sevinç 2022, Şenyurt 2018, Şenyurt et al 2023, Yılmaz and Has 2022, Şenyurt and Kaya 2018. In these studies, the relationships between Frenet and various Bishop frames of a curve are given.…”

Bishop Frames of Salkowski Curves in E3

“…The Type-2 Bishop frame is obtained by rotating the Frenet frame of the curve around the B vector by a certain angle, while the N-Bishop frame is obtained by rotating the alternative frame of the curve around N by a certain angle. Some other studies on these frames are [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. On the other hand, Salkowski curves in 3 E are slant helix type curves introduced by Salkowski, [22].…”

Spinor Equations of Successor Curves

“…Another is the involute evolute curves obtained by establishing a special relationship between the tangents of any two curves [14]. There are a lot of studies about curve theory [4,17]. The curves that form the basis of this study are the successor curves.…”