A new approach to Bertrand curves in Euclidean 3-space

Generalized Bertrand and Mannheim Curves in 3D Lie Groups

“…Corollary 3.8. If G is Abelian Lie group, the results obtained correspond to the generalized Bertrand curves given in study [10].…”

“…In this study, we examined generalized Bertrand and Mannheim curves in 3D Lie groups inspired by [10] and [11] studies. We have shown that we obtain the results in studies [10], [11], [14] and [15], especially considering the Abelian Lie groups.…”

“…In this study, we examined generalized Bertrand and Mannheim curves in 3D Lie groups inspired by [10] and [11] studies. We have shown that we obtain the results in studies [10], [11], [14] and [15], especially considering the Abelian Lie groups. In connection with this study, special curve types can be studied in Lie groups with different metric structures in the future.…”

“…Recently, C ¸. Camcı et.all and A. Uc ¸um et.all gave a generalization for Bertrand and Mannheim curves in 3D Euclidean space, respectively [10,11]. Lie groups are an important mathematical form because they have three different structures in mathematics such that S 3 , SO(3) and Abelian Lie groups.…”

Special Fractional Curve Pairs with Fractional Calculus

“…Let x : I → E 3 be a regular curve in E 3 according to Frenet frame {t, n, b} with non-zero curvatures κ, τ , and y : I 1 → E 3 be a Bertrand mate curve of x according to Frenet frame {t 1 , n 1 , b 1 } with non-zero curvatures κ 1 , τ 1 . Then the curve y can be written as[5], Let x(s) and y(s 1 ) be regular curves in E 3 with the Frenet frames {t, n, b} and {t 1 , n 1 , b 1 } according to arc-length parameter s and s 1 , respectively. If the pair (x, y) are the Bertrand curve pair, we can write the following results…”