On The slant helices according to Bishop frame of the timelike curve in Lorentzian space

Abstract

T.Ikawa obtained the following differential equation$$D_{T}D_{T}D_{T}T-KD_{T}T,K=\kappa ^{2}-\tau ^{2}$$for the c\i rcular helix which corresponds the case that the curvature $ \kappa $ and torsion $ \tau $ of timelike curve $ \alpha $ on the Lorentzian manifold $ M_{1} $ are constant [5]. In this paper, we have defined a slant helix according to Bishop frame of the timelike curve. Furthermore, we have given some necessary and sufficent conditions for the slant helix and T.Ikawa's result is generalized to the case of the general slant helix.

Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

“…Consider a timelike sweeping surface M represented by Eq. (16). Let x f (u) be the spacelike planar offset of the profile curve x(u) at distance f .…”

“…(30) are applied to Eq. (16), with attention of Eq. (6), we instantly find that the expression of the parabolic curve is…”

“…When we discuss a space curve, it is more convenient for us to use the Lorentzian Bishop frame along the curve as the basic tool than the Frenet-Serret type frame in Lorentzian space. There are several papers about Lorentzian Bishop frame [12][13][14][15][16][17].…”

Special Curves According to Bishop Frame in Minkowski 3-Space

“…It also provides a new way to control virtual cameras in computer animation [12]. Some applications of the Bishop frames in Minkowski spaces can be found in [3,4].…”

On Bishop frame of a pseudo null curve in Minkowski space-time

“…In Minkowski space-time E 4 1 the Bishop frame {T 1 , N 1 , N 2 , N 3 } of a null Cartan curve contains the tangent vector field T 1 of the curve and three vector fields whose derivatives N ′ 1 , N ′ 2 , and N ′ 3 with respect to pseudo-arc are collinear with N 2 [7]. Hence, they make a minimal rotations in the corresponding spaces [6], computer graphics [23], deformation of tubes [21], sweep surface modeling [16], and differential geometry in studying different types of curves (see for example [2,3,15,24]).…”