Slant Helices Research Articles

On the Assocıated Curves of a Frenet Curve in R_1^4

In the present work, we have dealt with the properties of associated curves of a Frenet curve in R14. In addition to this, we define principal direction curve, B_1 -direction curve, B_2- direction curve of a given Frenet curve by using integral curves of 4-dimensional Minkowski space. Then we introduce some characterizations for general helix and slant helix. Finally, some new associated curves and theorems obtained for space-like curves and time- like curves in R14. Also, an example is given.

Associated curves of a Frenet curve in the dual Lorentzian space

In this work, we firstly introduce notions of principal directed curves and principal donor curves which are associated curves of a Frenet curve in the dual Lorentzian space D 3 1 D13 . We give some relations between the curvature and the torsion of a dual principal directed curve and the curvature and the torsion of a dual principal donor curve. We show that the dual principal directed curve of a dual general helix is a plane curve and obtain the equation of dual general helix by using position vector of plane curve. Then we show that the principal donor curve of a circle in $\mathbb{D}^{2}$ or a hyperbola in $\mathbb{D}_{1}^{2}$ and the principal directed curve of a slant helix in $\mathbb{D}_{1}^{3}$ are a helix and general helix, respectively. We explain with an example for the second case. Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .

Direction Curves of Spherical Indicatrices of A New Framed Curve

In this study, direction curves of spherical indicatrix curves (primarily tangent indicatrix) for a curve in three dimensional Euclidean space were investigated. Mentioned curve was given with a new alternative frame which is {N,C,W} frame. It is a new aspect that researching relations between direction curves and the main curve via that new alternative frame. The harmonic curvatures and geodesic curvatures of spherical image of principal normal indicatrix of direction curves were obtained. The general helix, slant helix and C-slant helix characterizations of direction curves were given. Finally these curves with their figures were exemplified.

Biharmonic Curves in a Strict Walker 3-Manifold

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.

On the quaternionic osculating direction curves

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields [Formula: see text] where [Formula: see text] for the spatial quaternionic curve [Formula: see text] and [Formula: see text] where [Formula: see text] for the quaternionic curve [Formula: see text]. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples.

Slant helices in Minkowski 3-space E31 with Sasai’s modified frame fields

In this paper, we study slant helix using modified orthogonal frame in Minkowski space E31 with timelike, lightlike and spacelike axes. We also study a general slant helix with the Killing vector field axis. Furthermore, we give a non-trivial example and find the relations for curvature and torsion of f-biharmonic slant helix.

On non-null relatively normal-slant helices in Minkowski 3-space

By using the Darboux frame |?, ?, ?| of a non-null curve lying on a timelike surface in Minkowski 3-space, where ? is the unit tangent vector of the curve, ? is the unit spacelike normal vector field restricted to the curve and ? = ?? ? ?, we define relatively normal-slant helices as the curves satisfying the condition that the scalar product of the fixed vector spanning their axis and the non-constant vector field ? is constant. We give the necessary and sufficient conditions for non-null curves lying on a timelike surface to be relatively normal-slant helices. We consider the special cases when non-null relatively-normal slant helices are geodesic curves, asymptotic curves, or lines of the principal curvature. We show that an asymptotic spacelike hyperbolic helix lying on the principal normal surface over the helix and a geodesic spacelike general helix lying on the timelike cylindrical ruled surface, are some examples of non-null relatively normal-slant helices in E31.

(k,m)-type Slant Helices According to Parallel Transport Frame in Euclidean 4-Space

In this work, we describe a Frenet frame in 4-dimensional Euclidean space and call this frame as parallel transport frame (PTF). PTF is an alternative approach to defining a moving frame. This frame is obtained by rotating the tangent vector and the first binormal vector of a unit speed curve by an euler angle and then we give curvature functions according to PTF of the curve. Also, we introduce $(k,m)$-type slant helices according to PTF in Euclidean 4-Space. Additionally, we obtain the characterization of slant helices according to this frame in $\mathbb{E}^{4}$ and give an example of our main result.

On the Geometrical Properties of the Lightlike Rectifying Curves and the Centrodes

This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves are the lightlike rectifying curves. Meanwhile, a detailed analysis between the N-type lightlike slant helices and the centrodes of lightlike curves is provided in this paper. We give the projections of the lightlike rectifying curves to the timelike planes.

Framed slant helices in Euclidean 3-space

In this paper, we define framed slant helices and give a necessary and sufficient condition for them in three-dimensional Euclidean space. Then, we introduce the spherical images of a framed curve. Also, we examine the relations between a framed slant helix and its spherical images. Moreover, we give an example of a framed slant helix and its spherical images with figures.

K-type slant helices on spacelike and timelike surfaces

We have derived a necessary and sufficient condition for a non-null normal spacelike curve lying in a spacelike or a timelike surface M ⊂ E13, so that the curve becomes a K-type spacelike slant helix with K ∈ {1,2,3}. We have used Darboux frame to define necessary and sufficient conditions. An example is given for a 1-type spacelike slant helix having a spacelike normal and a timelike binormal.

Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

Natural and conjugate mates of Frenet curves in three-dimensional Lie group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results.

Position Vectors of Curves Generalizing General Helices and Slant Helices in Euclidean 3-Space

Inthispaper,wegiveanewcharacterizationofak-slanthelixwhichisageneral- ization of general helix and slant helix. Thereafter, we construct a vector differential equation of the third order to determine the parametric representation of a k-slant helix according to standard frame in Euclidean 3-space. Finally, we apply this method to find the position vector of some examples of 2-slant helix by means of intrinsic equations.

Mannheim Curves in 3-Dimensional Euclidean Space

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature 𝜿(𝒔), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve 𝝉(𝒔)≠±∞ for all s.

Generalized Bertrand Curves in Minkowski 3-Space

We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves.

K-TYPE SLANT HELICES FOR SYMPLECTIC CURVE IN 4-DIMENSIONAL SYMPLECTIC SPACE

In this study, we have expressed the notion of $k$-type slant helix in $4$-symplectic space. Also, we have generated some differential equations for $k$-type slant helix of symplectic regular curves.

On Slant Helices And Manheim Curves in E^3

In this paper, the equation of the ones that provide the Mannheim curve feature in slant helices have been
 obtained and the intrinsic equation of Mannheim curves have been given for the first time.
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On geometry of isophote curves in Galilean space

In this paper, we introduce isophote curves on surfaces in Galilean 3-space. Apart from the general concept of isophotes, we split our studies into two cases to get the axis d of isophote curves lying on a surface such that d is an isotropic or a non isotropic vector. We also give the method to compute isophote curves of surfaces of revolution. Subsequently, we show the relationship between isophote curves and slant(general) helices on surfaces of revolution obtained by revolving a curve by Euclidean rotations. Finally, we give an example to compute isophote curves on isotropic surfaces of revolution.