New representations of Bertrand pairs in Euclidean 3-space

Abstract

In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indicatrices made new curve pairs in the means of Mannheim, involte-evolute and Bertrand pairs. Further more, we investigated the relations between the spherical images and introduced new representations of spherical indicatrices.

Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

“…The point p is called the center of S 2 1 (p, r), H 2 0 (p, r) and Q 2 1 (p). When p is the origin and r = 1, we simply denote them by S 2 1 , H 2 and Q 2 1 .…”

“…for some non-zero differentiable functions u(s) and v(s). In particular, if the associate normal curve b(s) = u(s)N(s) + v(s)B(s) lies on S 2 1 or H 2 , then u(s) and v(s) satisfy u(s)v(s) = 1 2 or u(s)v(s) = − 1 2 . Without lose of generality, we have the following definition.…”

“…In differential geometry, the space associate curves for which there exist some relations between their Frenet frames or curvatures compose a large class of fascinating subjects in the curve theory, such as Bertrand curve, Mannheim curve, central trace of osculating sphere, involute-evolute curves etc. [1][2][3]. Most of the researchers aimed to explore the relationships between the partner curves.…”

A New Generalization of Some Curve Pairs

“…Well-known pairs of curves are Involute-Evolute, Parallel, Bertrand, Mannheim, and Natural mates. Characterizations of these curves mates with some properties can be found in various papers : [2,6,12,13,14,15,20,23,25,27,31,32,39,42]. Some of these curves mates have been generalized to larger dimensions and have been studied by many authors [11,17,19,28,35].…”

Spherical indicatrices of a Bertrand curve in three dimensional Lie groups

“…The some relations between slant helices and their involutes, spherical images in three dimensional Lie groups are given in [12]. The some features of the spherical indicatrices of a Bertrand curve and its mate curve are presented in Euclidean 3-space in [13].…”