Classical Differential Geometry of Curves According to Type-2 Bishop Trihedra

Abstract

Abstract-In this work, we study classical differential geometry of the curves according to type-2 Bishop trihedra. First, we present some characterizations of a general helix, a helix, special cases and spherical curves. Thereafter, we investigate position vector of a regular curve by a system of ordinary differential equations whose solution gives the components of the position vector with respect to type-2 Bishop frame. Next we prove that the first vector field of the type-2 Bishop frame of a regular curve satisfies a vector differential equation of third order. Solutions of the mentioned system and vector differential equation have not been found. Therefore we present some special characterizations introducing special planes of three dimensional Euclidean space.

A study of tube-like surfaces according to type 2 Bishop frame in Euclidean space

“…In the present time a good deal of research has been done using Bishop frames [5,6,7,10,25]. Because of the importance of this frame, the authors in [29] introduced a new version of the Bishop frame and called it a type 2 Bishop frame which was studied in [11,19].…”

Minkowski 3-uzayda 2. tip Bishop çatısına göre null olmayan eğrilerin karakterizasyonlarına dair bir inceleme

“…First, this new version of Bishop frame was studied in Euclidean space by Yilmaz in [15]. Then Özyilmaz gave some characterizations of curves according to this new frame in Euclidean space [10]. Also, Ünlütürk and Yılmaz obtained the new version of Bishop frame for spacelike curves in [14].…”

“…That is, intuitively, it can be considered as a path traced out by a particle moving in 3 E . Position vectors of curves have been studied in Euclidean and its ambient spaces such as Minkowski and Galilean spaces by [1,3,4,5,[10][11][12][13]. Vectorial differential equation of third order characterizes regular curves of .…”

Motion of Bishop Frenet Offsets of Ruled Surfaces inE3

“…Classical differential geometry of the curves may be surrounded by the topics of general helices, involute-evolute curve couples, spherical curves, and Bertrand curves. Such special curves are investigated and used in some real world problems like mechanical design or robotics by the wellknown Frenet-Serret equations because we think of curves as the path of a moving particle in the Euclidean space [23]. Thereafter researchers aimed to determine another moving frame for a regular curve.…”