Bishop Curvatures Research Articles

Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space

<abstract><p>The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.</p></abstract>

Electromagnetic waves along pseudo null curves in Minkowski space

Considering the importance of Minkowski space in physics, it is an incomplete approach to deal with EM waves only in Euclidean space. For this reason, this paper deals with EM waves along pseudo null curves in Minkowski space. The main purpose of this study is to examine electromagnetic waves by defining an adapted orthogonal frame along the EM wave which contains both electric and magnetic fields. For this purpose, the extended derivative formulas of pseudo null frame are obtained. Depending on the values of Bishop curvatures, the linear transformations between the pseudo null frame and EM wave vector fields are described in two cases. For all these cases, the relations between these frames are stated, respectively. Moreover, the derivative formulas EM wave vectors are stated by means of geometric phase. Furthermore, the necessary and sufficient conditions provided by the geometric phase are expressed for EM wave vectors to be parallel transportation of the pseudo null frame. Finally, an application is given to investigate the obtained results.

Null Cartan curve variations in 3D semi-Riemannian manifold

The aim of this study is to investigate the variations of the Bishop frame curvatures for null Cartan curves in semi-Riemannian manifolds. The Killing equations in terms of the variations of the Bishop curvatures along the null Cartan curve is especially derived. Killing equations are used to interpret the movement of the charged particles in a magnetic field. The charged particle motion along a null Cartan curve through the Killing equations is examined as an application in the study. It is found that the charged particle traces a trajectory in the form of the null cubic during its movement in the Killing magnetic vector field. According to the results obtained, an example of the null Cartan magnetic trajectories is presented.

New models of normal motions of the inextensible curves according to type-1 Bishop frame in ℝ3

In this work, we compute the time evolution equations (TEEs) of the type-1 Bishop frame of the curve. Also, we study the time evolution equations for type-1 Bishop curvatures (TEEBCs) as a system of partial differential equations (PDEs). Through this study, we give a necessary and sufficient condition for the normal and binormal Bishop velocities. Also, we construct new models of normal motions of inextensible curves in [Formula: see text]. These models are constructed for curves that move according to the type-1 Bishop frame. Also, we make a comparison study between surfaces obtained by the motions of inextensible curves according to the two frames, the type-1 Bishop frame and the Frenet frame.

Mannheim B-curve couples in Minkowski 3-space

In this paper, we define null Cartan and pseudo null Mannheim curves in Minkowski 3-space according to their Bishop frames. We obtain the necessary and the sufficient conditions for pseudo null curves to be Mannheim B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Mannheim B-curves, by considering the cases when their Mannheim B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Mannheim B-curve pairs.

Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space

UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space and investigate its relationship with the energy of the same curve in Bishop vector fields in . Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space .

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

In this paper, we introduce the Bishop frame of a pseudo null curve $\alpha$ in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using space-time geometric algebra, we derive the Darboux bivectors $D$ and $\tilde{D}$ of the Frenet and the Bishop frame of $\alpha$, respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of $\alpha$ in terms of areas of the projections of the corresponding Darboux bivectors onto the planes spanned by the frame vector's fields.

Lightlike hypersurfaces and lightlike focal sets with respect to Bishop frame in 4-dimensional Minkowski Space E1-4.

In this article, light-like hypersurfaces that is derived by null Cartan curves will be examined and discussed. The singularities of lightlike hypersurfaces and light-like focal sets are investigated by using the Bishop frame on the Null Cartan curves. We obtain that the types of these singularities and the order of contact between the null Cartan curves are closely related to the Bishop curvatures of the null Cartan curves. Moreover, two examples of light-like hypersurfaces and light-like focal sets are given to illustrate our theoretical results.

Null Cartan Helices in Lorentzian 3-Space: An Approximation

In this work, we investigate the null Cartan helices in Lorentzian 3-space. We derive the helices with the constant timelike, spacelike and lightlike Killing axis in Lorentzian 3-space. Then, we calculate the Bishop curvatures of the null Cartan helix and obtain the explicit parametric equations of these curves by using the Bishop curvatures. Finally, we present various related examples and draw their images using the Mathematica.

On k-type pseudo null slant helices due to the Bishop frame in Minkowski 3-space <i>E</i><sub>1</sub><sup>3</sup>

In this study, we examine k-type pseudo null slant helices due to the Bishop frame, where k∈{0, 1, 2}. There are two different cases of the Bishop frame of a pseudo null curve related to the Bishop curvatures. Based on these cases, we present that every pseudo null curve is a k-type pseudo null curve according to the Bishop frame in Minkowski 3-space E13. Then we obtain the axes of k-type pseudo null slant helices, and determine their causal characters.

Pseudo null curve variations for Bishop frame in 3D semi-Riemannian manifold

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

Asymptotic curves on B-surfaces according to type-2 bishop frame in the sol space

In this paper, we study B— surfaces of biharmonic constant Π2— slope curves according to type-2 Bishop in the . We characterize asymptotic curves on B— surfaces of biharmonic constant Π2— slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the .

<b>New type surfaces in terms of B-Smarandache Curves in Sol<sup>3

In this work, new type ruled surfaces in terms of B- Smarandache TM 1 curves of biharmonic B- slant helices in the SOL 3 are studied. We characterize the B- Smarandache TM 1 curves in terms of their Bishop curvatures. Additionally, we express some interesting relations.

Biharmonic constant Π₁-slope curves according to type-2 bishop frame in Heisenberg group Heis³

In this paper, we introduce constant Π₁- slope curves according to type-2 Bishop frame in the Heisenberg group Heis³. We characterize the biharmonic constant Π₁- slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Heisenberg group Heis³. Additionally, we illustrate our main theorem.

English

In this study, the relations between of the Bishop curvatures of involute-evolute curve couple were examined in . Also, the relation between of the Bishop frames of these curves was given. Moreover,  the relations between of curvature radii of these curves were investigated. Lastly, in the case of these curves were planar curves, some corollaries were obtained. Key words. Bishop frame, Bishop curvatures, involute-evolute curve couple.

Integral equations of biharmonic constant π₁-slope curves according to type-2 Bishop frame in the SOL space

In this paper, we study biharmonic constant Π₁- slope curves according to type-2 Bishop frame in the SOL³. We characterize the biharmonic constant Π₁- slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric integral equations in the SOL³.

B-Smarandache tm₂ curves of biharmonic new type b-slant helices according to Bishop frame in the Sol space Sol³

In this paper, we study b-Smarandache tm₂ curves of biharmonic new type b-slant helix in the Sol³. We characterize the b-Smarandache tm₂ curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Sol³.

One parameter family of b-m₁ developable surfaces of biharmonic new type b-slant helices according to Bishop frame in the sol space Sol³

In this paper, we study inextensible flows of b-m₁ developable surfaces of biharmonic new type b-slant helix in the Sol³. We characterize one parameter family of the b-m₁ developable surfaces in terms of their Bishop curvatures.

Cusps of Bishop Spherical Indicatrixes and Their Visualizations

The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspond to the point where Bishop curvatures and type-2 Bishop curvatures vanished and their derivatives are not equal to zero. As applications and illustration of the main results, two examples are given.

Construction of B-focal curves of spacelike biharmonic B-slant helices according to Bishop frame in E(1,1)

In this paper, we find parametric equations of B-focal curves of spacelike biharmonic B-slant helices according to Bishop frame in terms of Bishop curvatures in the Lorentzian group of rigid motions E(1,1).