arXiv research

In this paper we study slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field. We prove that for non-geodesic such curves there exists a unique Frenet frame for which the original parameter is distinguished. Moreover, we obtain …

Notes on Frenet-Serret formulas for curves in flat pseudo-hermitian manifolds.

Study of generalized Bishop frames on curves in 4D space.

Generalized Frenet frames for singular space curves

The Frenet frame generalizes the Park transform for multi-phase circuits.

The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in …

This paper defines directional derivatives and solves Maxwell's equations in curved 3D space.

Defines new curves from tangent indicatrix of curves, linking them to helices and slant helices.

A set of equations is developed to describe a curve in space given the curvature $κ$ and the angle of rotation $θ$ of the osculating plane. The set of equations has a solution (in terms of $κ$ and $θ$) that indirectly solves the Frenet-Serret equations, with a unique value of $θ$ for each specified value of $τ$. Explic…

Agrachev's problem on circle turns is solved for various topologies.

Generalizes Hasimoto transformation to arbitrary flows on space curves.

We consider a unit speed timelike curve $α$ in Minkowski 4-space $E_1^4$ and denote the Frenet frame of $α$ by $\{T,N,B_1,B_2\}$. We say that $α$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $E_1^4$. In this work we study those hel…

In this paper, we investigate some characterizations of involute -- evolute curves in dual space. Then the relationships between dual frenet frame and darboux vectors of these curves are found.

In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet frames of these timelike curves are not the same. We have moved the director vector …

In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymp…

Study constructs Frenet curves using semi-symmetric metric connection.

In this study, the new characterizations of special curves are investigated without using the curvatures of these special curves: general helices, slant helices, Bertrand curves, Mannheim curves. The curvatures are given by the help of the norms of the derivatives of Frenet vectors.

We study ruled surfaces in R3 which are obtained from dual spher- ical indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean curvatures of the ruled surfaces and give some results of being Wein- garten surface.

Study weak Frenet frame for non-smooth curves with finite curvature and torsion.

Study natural and conjugate mates of Frenet curves in Lie groups.

Equi-affine curvature of curves in 2-manifolds linked to Frenet curvature.

Rotation minimizing vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed …

In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…

We consider the Frenet-Serret geometry of null curves in a three and a four-dimensional Minkowski background. We develop a theory of deformations adapted to the Frenet-Serret frame. We exploit it to provide a Lagrangian description of the dynamics of geometric models for null curves.

Study on triharmonic curves in f-Kenmotsu manifolds.

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve, in [11]. We obtain the Frenet apparatus of these spherical indicatrix curves and the characterizations of being general helix and s…

The aim of this paper is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. Involute trajectory ruled surfaces generated by the Frenet trihedron, moving along spacelike involutes of a given timelike space curve, is stated according to Lorentzian timelike angle …

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…

The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle w…

The paper characterizes Legendre curves on trans-S-manifolds.

We consider a unit speed curve $α$ in Euclidean $n$-dimensional space $E^n$ and denote the Frenet frame by $\{v_1,...,v_n\}$. We say that $α$ is a cylindrical helix if its tangent vector $v_1$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms of …

A twisted curve in Euclidean 3-space E^3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E^3 and characterize such curves in terms of their curvature functions. Further, we obtain some re…

We consider a unit speed curve $α$ in Euclidean four-dimensional space $E^4$ and denote the Frenet frame by $\{T,N,B_1,B_2\}$. We say that $α$ is a slant helix if its principal normal vector $N$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms o…

Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ satisfies a vector differential equation of fourth order. The general so…

The paper explores equi-affine curvatures in pseudo-Riemannian manifolds.

Study on null helices in semi-Riemannian manifolds with special submanifolds.

The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.

In this paper, we give the characterizations of Mannheim Partner Curves in Minkowski 3-space . Firstly, we classify these curves in . Next, we give some relationships characterizing these curves and we show that Mannheim theorem is not valid for Mannheim partner curves in . Moreover, by considering the spherical indica…

The paper uses Frenet frame to unify electrical and geometric quantities.

Improved modeling of chaotic systems using time-delay embeddings and Frenet-Serret frame.

The paper studies holomorphic curves in a pseudo-Riemannian space and their moduli space.

The presented paper is devoted to study the curvature and torsion of slant Frenet curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non- Frenet slant curves (with null tangents or null normals) are obtained. The achieved results are illustrated by ex…

Study on triharmonic curves in 3D spaces, proving their existence and classification.

Study on Mannheim curves in 3D space with modified frame.

It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local moving frame of a holomorphic immersion of a Riemann surface into a complex Grassmann…