Smarandache-Based Ruled Surfaces with the Darboux Vector According to Frenet Frame in $E^3$

Abstract

The paper introduces a new kind of special ruled surface. The base of each ruled surface is taken to be one of the Smarandache curves of a given curve according to Frenet frame, and the generator (ruling) is chosen to be the corresponding unit Darboux vector. The characteristics of these newly defined ruled surfaces are investigated by means of first and second fundamental forms and their corresponding curvatures. An example is provided by considering both the helix curve and the Viviani’s curve.

Bishop Frames of Salkowski Curves in E3

“…Proof: The proof is obvious that from (3), ( 17) and (23). Also, it is also obtained by comparing expressions (21) and (24).…”

“…Here, it is obvious that from (21). Also, it is also obtained by dividing the vector Proof: The proof is obvious that from ( 18) and ( 27).…”

“…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].…”

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

“…Finally, we show all these results on an example and plot the graphs of the surfaces. This study is another application of our previous paper: Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in E 3 , [32].…”

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.…”