Frenet Vectors Research Articles

SOME CLASSIFICATIONS FOR GAUSS MAP OF TUBULAR HYPERSURFACES IN E^4_1 CONCERNING LINEARIZED OPERATORS L_k

In this study, we deal with the Gauss map of tubular hypersurfaces in 4-dimensional Lorentz-Minkowski space concerning the linearized operators L_1 (Cheng-Yau) and L_2. We obtain the L_1 (Cheng-Yau) operator of the Gauss map of tubular hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on timelike or spacelike curves with non-null Frenet vectors in E^4_1 and give some classifications for these hypersurfaces which have generalized L_k 1-type Gauss map, first kind L_k-pointwise 1-type Gauss map, second kind L_k-pointwise 1-type Gauss map and L_k-harmonic Gauss map, k ∈ {1,2}.

A Generalization of Curve Mates: Normal Mate of a Curve

This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained in which particular case.

The Pedal Curves Generated by Alternative Frame Vectors and Their Smarandache Curves

In this paper, pedal-like curves are defined resulting from the orthogonal projection of a fixed point on the alternative frame vectors of a given regular curve. For each pedal curve, the Frenet vectors, the curvature and the torsion functions are found to provide the common relations among the main curve and its pedal curves. Then, Smarandache curves are defined by using the alternative frame vectors of each pedal curve as position vectors. The relations of the Frenet apparatus are also established for the pedal curves and their corresponding Smarandache curves. Finally, the expressions of the alternative frame apparatus of each Smarandache curves are given in terms of the alternative frame elements of the pedal curves. Thus, a set of new symmetric curves are introduced that contribute to the vast curve family.

Dual magnetic curves and flux ruled surfaces

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.

Pedal curves obtained from Frenet vector of a space curve and Smarandache curves belonging to these curves

<abstract><p>In this study, first the pedal curves as the geometric locus of perpendicular projections to the Frenet vectors of a space curve were defined and the Frenet vectors, curvature, and torsion of these pedal curves were calculated. Second, for each pedal curve, Smarandache curves were defined by taking the Frenet vectors as position vectors. Finally, the expressions of Frenet vectors, curvature, and torsion related to the main curves were obtained for each Smarandache curve. Thus, new curves were added to the curve family.</p></abstract>

Non-null slant ruled surfaces and tangent bundle of pseudo-sphere

<p>A slant ruled surface is a unique type of ruled surface composed by Frenet vectors that form a constant angle with each other and with specific directions in space. In this paper, the non-null slant ruled surface, which is generated by the striction curve of the natural lift curve, was constructed with a novel approximation in $ E^{3}_{1} $. To establish the approximation, E. Study mapping was then applied to determine the relationship between pseudo-spheres and non-null slant ruled surfaces that are generated by the striction curves of the natural lift curves. Furthermore, $ \vec{\bar{q}}-, \vec{\bar{h}}-, \vec{\bar{a}}- $ spacelike (resp., timelike) slant ruled surfaces were classified by using the striction curves of the natural lift curves in $ E^{3}_{1} $. We also provided examples to illustrate the findings.</p>

B-Lift curves and involute curves in Lorentzian 3-space

The involute of a curve is often called the perpendicular trajectories of the tangent vectors of a unit speed curve. Furthermore, the B-Lift curve is the curve acquired by combining the endpoints of the binormal vectors of a unit speed curve. In this study, we investigate the correspondences between the Frenet vectors of a curve’s B-lift curve and its involute. We also give an illustration of a helix that resembles space in Lorentzian 3-space and show how to visualize these curves by deriving the B-Lift curve and its involute.

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani’s curves and plot the graphics of these surfaces.

Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed.

Smarandache Curves of Involute-Evolute Curve According to Frenet Frame

In this paper, the invariants of the Smarandache curves, which consist of Frenet vectors of the involute curve, are calculated in terms of the evolute curve.

On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space

<abstract><p>In this study, the partner-ruled surfaces in Minkowski 3-space, which are defined according to the Frenet vectors of non-null space curves, are introduced with extra conditions that guarantee the existence of definite surface normals. First, the requirements of each pair of partner-ruled surfaces to be simultaneously developable and minimal (or maximal for spacelike surfaces) are investigated. The surfaces also simultaneously characterize the asymptotic, geodesic and curvature lines of the parameter curves of these surfaces. Finally, the study provides examples of timelike and spacelike partner-ruled surfaces and includes their graphs.</p></abstract>

On the Bertrand Mate of Cubic Bezier Curve by Using Matrix Representation in $\mathbf{E}^{3}$

In this study, we have examined Bertrand mate of a cubic Bezier curve based on the control points with matrix form in $E^3$. Frenet vector fields and also curvatures of Bertrand mate of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in $E^3$.

A New Generalization of Some Curve Pairs

In this study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $\gamma$ with the tangents of the curves. We examine the relations between torsions and curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=\cos (\theta/2)-\mathbf{i}\sin (\theta/2)\cos \gamma -\mathbf{j}\sin (\theta/2)\sin \gamma$, where $\theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.

Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

In this study, we define (1,3)-Bertrand-direction curve and (1,3)-Bertrand-donor curve in the 4-dimensional Euclidean space $E^{4}$. We introduce necessary and sufficient conditions for a special Frenet curve to have a (1,3)-Bertrand-direction curve. We introduce the relations between Frenet vectors and curvatures of these direction curves. Furthermore, we investigate whether (1,3)-evolute-donor curves in $E^{4}$ exist and show that there is no (1,3)-evolute-donor curve in $E^{4}$ .

Bertrand Curves and B-Lift Curves in Lorentzian 3-Space

In this article, based on Thorpe's definition, we define a new curve called the B-lift curve in Lorentzian 3-space and examine the Frenet vectors of the B-lift curve. Furthermore, we examine the relationship between the Frenet vectors of the Bertrand curve and the Frenet vectors of the natural lift curve. Finally, we give an example on these results.

Hasimoto Maps for Nonlinear Schrödinger Equations in Minkowski Space

In this paper, we study the vortex filament flow for timelike and spacelike curves in Minkowski 3-space. The vortex filament flow equations of the timelike and the spacelike curves are equivalent to the nonlinear Schrödinger equation and the heat equation, respectively. As a consequentce, we prove that a soliton of the nonlinear Schrödinger equations of the timelike curve gives a solution of a traveling wave on a line at infinity. Also, we study a solution of a traveling wave of the nonlinear Schrödinger equations of the spacelike curve in terms of a new complex frame. Finally, we discuss the method to find the exact shape of the timelike and the spacelike curves from the vortex filament by solving the Frenet vectors of these curves and provide applications to illustrate the method.

4-Boyutlu Lorentz Minkowski Uzayında Null Olmayan Ortak İzoasimptotik Eğrili Hiperyüzey Aileleri

In this study, the parametric equation of hypersurfaces passing through the spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with the help of this curve is Frenet frame. Furthermore, hypersurface families were created by giving necessary and sufficient conditions so that the spacelike and timelike curves on E_1^4 is common isoasymptotic on the hypersurface. Then, these conditions obtained with the help of deviation marching-scale functions were simplified. Finally, the study was supported with examples and the graphics were drawn using certain projection methods.

B-LIFT CURVES IN LORENTZIAN 3-SPACE

In this article, we define a new curve which is called B-Lift curve in 3-dimensional Lorentzian space and we obtain the Frenet vectors of the B-Lift curve. Also, we compare the Frenet vectors of the B-Lift curve and the natural lift curve.

Characterizations of Adjoint Curves According to Alternative Moving Frame

In this paper, the adjoint curve is defined by using the alternative moving frame of a unit speed space curve in 3-dimensional Euclidean space. The relationships between Frenet vectors and alternative moving frame vectors of the curve are used to offer various characterizations. Besides, ruled surfaces are constructed with the curve and its adjoint curve, and their properties are examined. In the last section, there are examples of the curves and surfaces defined in the previous sections.

Hypersurface families with common non-null geodesic in Minkowski 4-space

In this study, we create hypersurface families from each non-null isogeodesic, with non-null Frenet vectors, given in the Minkowski 4-space. We obtain parametric representations for hypersurface families whose members have the same curves as the given isogeodesic curves. By using the Frenet frame of each given geodesic curve, we create the hypersurfaces as a linear combination of this Frenet frame and obtain the necessary and sufficient conditions for these curves to be isogeodesic. Also, we give some examples so that the method presented is clear and understandable. In addition, the graphs of the surfaces formed by projecting the surfaces given with parametric equations to 3-dimensional space are drawn.