Unit Speed Curve Research Articles

Curves of Constant Ratio with Quasi frame in E^3

In the present study we handle a regular unit speed curve by means of the position vector given by the vectorial equation γ (s)=m0 t(s)+m1 nq (s)+m2 bq (s) where bq (s), nq (s) and t(s) are quasi frame vectors. Firstly, we analysis these curves and investigate to being constant ratio curve. Then, we give the parameterizations of T-constant and N- constant curve in accordance with quasi frame. Further, we get the conditions for a regular curve to correspond to be a W- curve in E^3.

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.

OSCULATING CURVES ACCORDING TO THE EXTENDED DARBOUX FRAME IN E4

In this paper, we define osculating curves according to the extended Darboux frame (ED-frame), in Euclidean 4-space. Then, according to Case 1 and Case 2 ED-frame fields, the necessary and sufficient condition for a unit speed curve lying on the hypersurface to be congruent to an osculating curve is obtained. For these necessary and sufficient conditions, the solutions of the differential equations have been investigated. Finally, an application has been made on the subject.

B-Lift curves and its ruled surfaces

In this paper, we have described the B-Lift curve in Euclidean space as a curve obtained by combining the endpoints of the binormal vector of a unit speed curve. Subsequently, we have explored the Frenet frames of the B-Lift curves. Moreover, we have introduced the tangent, normal and binormal surfaces of the B-Lift curve and examined the geometric invariants of these surfaces. Finally, we have investigated the singularities of these surface and visualized the surfaces with MATLAB program.

Differential geometric approach of Betchov-Da Rios soliton equation

In the present paper, we investigate differential geometric properties the soliton surface $M$ associated with Betchov-Da Rios equation. Then, we give derivative formulas of Frenet frame of unit speed curve $\Phi=\Phi(s,t)$ for all $t$. Also, we discuss the linear map of Weingarten type in the tangent space of the surface that generates two invariants: $k$ and $h$. Moreover, we obtain the necessary and sufficient conditions for the soliton surface associated with Betchov-Da Rios equation to be a minimal surface. Finally, we examine a soliton surface associated with Betchov-Da Rios equation as an application.

Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

In this paper, we give a generalization of the osculating curves to the $n$-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the ( n − 3 ) (n−3) th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type ( n − 3 ) (n−3) ”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n En . In particular, we characterize the osculating curves in E n En in terms of their curvature functions. Finally, we show that the ratio of the ( n − 1 ) (n−1) th and ( n − 2 ) (n−2) th curvatures of the osculating curve is the solution of an ( n − 2 ) (n−2) th order linear nonhomogeneous differential equation.

(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.

CONSTRUCTION OF BINORMAL MOTION AND CHARACTERIZATION OF CURVES ON SURFACE BY SYSTEM OF DIFFERENTIAL EQUATIONS FOR POSITION VECTOR

The main purpose of this paper is to investigate unit speed curve with constant geodesic, normal curvature and geodesic torsion of curve on a surface in the Euclidean 3-space. In accordance with this scope, the position vector of a curve is stated by a linear combination of its Darboux Frame with differentiable functions. Some special results have been obtained within the scope of this position curve and differentiable functions. As a physical application of obtained results, differential geometric properties of a surface with binormal motion of a given curve are given with the obtained characterization of the curve.

On rectifying curves in Minkowski 3-space

In this study, we investigate the rectifying curve and its centrode by using a dilation of a unit speed curve on pseudo-sphere or pseudo-hyperbolic space. Firstly, considering a causal character of any curve, we study the connection between Serret-Frenet apparatus of the curve on pseudo-sphere or pseudo-hyperbolic space and its dilation. Then, we extend necessary conditions in order to make the centrode a rectifying curve. Also, we examine some properties of centrode which is a rectifying curve.

Classification of f-Biharmonic Curves in Lorentz–Minkowski Space

In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski space L3 to be f-biharmonic. Then, using these equations, we classify such unit speed curves in L3.

Elementary Approachs on De Sitter Space

In this paper, we characterize the de Sitter space by means of spacelike and timelike curves that fullylies on it. For this purpose, we consider the tangential part of the second derivative of the unit speedcurve on the hypersurface, and obtain the vector equations of the geodesics. We find the geodesics ashyperbolas, ellipses, and helices. Moreover, we give an example of null curve with constant curvature in4−dimensional Minkowski space and we illustrate the geodesics of S11(r) × R .

Harmonicity and differential equation of involute of a curve in E3

In this paper, we first give necessary conditions in which we can decide whether a given curve is biharmonic or 1-type harmonic and differential equations characterizing the regular curves. Then we research the Frenet formulas of involute of a unit speed curve by making use of the relations between the involute of a curve and the curve itself. In addition we apply these formulas to define the essential conditions by which one can determine whether the involute of a unit speed curve is biharmonic or 1-type harmonic and then we write differential equations characterizing the involute curve by means of Frenet apparatus of the unit speed curve. Finally we examined the helix as an example to illustrate how the given theorems work.

Normal curves in n-dimensional Euclidean space

In this paper, we give a generalization of normal curves to n-dimensional Euclidean space. Then we obtain a necessary and sufficient condition for a curve to be a normal curve in the n-dimensional Euclidean space. We characterize the relationship between the curvatures for any unit speed curve to be congruent to a normal curve in the n-dimensional Euclidean space. Moreover, the differentiable function f ( s ) is introduced by using the relationship between the curvatures of any unit speed curve in E^{n}. Finally, the differential equation characterizing a normal curve can be solved explicitly to determine when the curve is congruent to a normal curve.

Characterizations of slant and spherical helices due to pseudo-Sabban frame

In this paper, we investigate that under which conditions of the geodesic curvature of unit speed curve $\gamma$ that lies on $S_1^2$ or $H^2,$ the curve $\alpha$ which is obtained by using $\gamma$, is a spherical helix or slant helix in Minkowski 3-space.

On Timelike Rectifying Slant Helices in Minkowski 3-Space

In this work, we study timelike rectifying slant helices in E31. First, we find general equations ofthe curvature and the torsion of timelike rectifying slant helices. After that, by solving second order linear differential equations, we obtain families of timelike rectifying slant helices that lieon cones.Helices arise in nanosprings, carbon nanotubes, DNA double and collagen triple helices. The double helix shape is commonly associated with DNA [1]. In differential geometry, a general helix in Euclidean 3-space is characterized by the property that the tangent lines make a constant angle with a fixed direction [12, 13]. Similarly, the notion of slant helix was introduced by Izuyama and Takeuchi by the property that the principal normal lines make a constant angle with a fixed direction [8, 9]. They showed that a space curve is a slant helix if and only if the geodesic curvature of the principal normal of the curve is a constant function. In [10, 11], Kula et al. studied the spherical images of slant helices. Later, Ahmet T. Ali studied slant helices in Minkowski 3-space [1, 2]. The notion of rectifying curve has been introduced by Chen [5, 6]. Chen proposed the conditions under which the position vector of a unit speed curve lies in its rectifying plane. Besides, he stated the importance of rectifying curves in Physics.

Natural mates of Frenet curves in Euclidean 3-space

For each Frenet curve $\alpha $ in the Euclidean 3-space $\mathbb{E}^{3}$, there exists a unique unit speed curve $\beta$ tangent to the principal normal vector field of $\alpha $. We simply call this curve $\beta $ the natural mate of $\alpha$. The main purpose of this paper is to prove some relationships between a Frenet curve and its natural mate. In particular, we obtain some necessary and sufficient conditions for the natural mate of a Frenet curve to be a helix, a spherical curve, or a curve of constant curvature. Several applications of our main results are also presented.

Characterization of Rectifying and Sphere Curves in ℝ3

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector

Slant Helix Curves and Acceleration Centers in Minkowski 3-Space Ε31

In this study, some basic concepts (e.g., instant screw axis (ISA), instantaneous pole points, acceleration pole points) will be given and analyzed about an alternative one-parameter motion of a rigid-body in 3-dimensional Minkowski space Ε31 obtained by moving coordinate frame {N, C, W} along a non-null unit speed curve α = α(t), where N, C and W correspond to unit principal normal vector field, derivative vector field of unit principal normal vector field and Darboux vector field (or angular-velocity vector field) of the non-null unit speed curve α, respectively.

Slant helices in the Euclidean 3-space revisited

In this paper, we study the surfaces whose geodesics are slant curves. We show that a unit speed curve $\gamma$ in the 3-dimensional Euclidean space is a slant helix if and only if it is a geodesic of a helix surface. We prove that the striction line of a helix surface is a general helix; as a consequence, slant helices are characterized as geodesics of the tangent surface of a general helix. Finally, we provide two methods for constructing slant helices in helix surfaces.

SOME CHARACTERIZATIONS OF SLANT AND SPHERICAL HELICES DUE TO SABBAN FRAME

In this paper, we are investigating that under which conditions of the geodesic curvature of unit speed curve that lies on the unit sphere, the curve c which is obtained by using , is a spherical helix or slant helix.