Darboux Frame Research Articles

Electromagnetism and Maxwellian Evolution Equations in terms of Darboux Frame in Minkowski Space with Abnormalities

Abstract Electromagnetic wave propagation is often thought of as the transport of polarised light and this behaviour is well defined by Maxwell's equations when propagating in an optical fiber. In this paper, we examine the $\textbf{q}-direction$ and $\textbf{n}-direction$ Berry's phase equation along a Darboux-framed optical fibre in Minkowski space. Afterwards, $\textbf{q}-direction$ and $\textbf{n}-direction$ for the electromagnetic curves of Rytov parallel transport laws are determined to shown in the application section, the connections of the electromagnetic curve with the anholonomic coordinates for Maxwell's equation by Maxwellian evolution that cause are visualised and illustrated with the MAPLE program.

Characterizations of rectifying curves under conformal transformation using Darboux frame in Euclidean 3-Space

Characterizations of rectifying curves under conformal transformation using Darboux frame in Euclidean 3-Space

Darboux Frame of Pancake Curve

In this paper, we focus on such a rare curve: “Pancake curve” which is named by Robert Ferréol. We investigate the Darboux frame of the curve and we calculate the curvatures: geodesic curvature, normal curvature, and geodesic torsion. We discuss two cases, for instance pancake curve as an intersection of Plücker’s conoid and unit cylinder, and intersection of hyperbolic paraboloid and unit cylinder. Moreover, we give the relations between the Frenet and Darboux curvatures. Finally, we give some of examples related with Frenet and Darboux frame and curvatures.

Geometric Analysis of the NLS Surface with the Pseodu Null Darboux Frame of Spacelike Curve

Bu çalışmada, Minkowski uzayında birim hızlı spacelike eğri ile ilişkilendirilmiş dinamik bir sistem olarak ele alarak duman halkası denklemi olarak bilinen girdap filaman denkleminin araştırılması verilmiştir. Darboux çatısı kullanılarak, doğrusal olmayan Schrödinger (NLS) denklemine karşılık gelen soliton yüzeyinin diferensiyel geometrik özellikleri incelenmiştir. Bu yüzey genellikle NLS yüzeyi veya Hasimoto yüzeyi olarak adlandırılır. Ayrıca, NLS yüzeyinin Gauss ve ortalama eğrilikleri, Darboux eğrilikleri kullanılarak belirlenmiştir. Bu analiz, NLS yüzeyinin geometrik özelliklerine derinlemesine bir bakış sunarak, bu alandaki bilgiyi artırmaya ve potansiyel uygulamalara ışık tutmaya yönelik önemli bir katkı sağlayacaktır.

On null Cartan normal isophotic and normal silhouette curves on a timelike surface in Minkowski 3‐space

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature and constant geodesic torsion . We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non‐zero constant curvatures and have a remarkable property that they are general helices, relatively normal‐slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions.

On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$

In this paper, we introduce generalized Darboux frames of the first and the second kind along a null Cartan curve lying on a timelike surface in Minkowski space ${E}^{3}_{1}$ and define null Cartan rectifying isophotic and rectifying silhouette curves in terms of the vector field that belongs to generalized Darboux frame of the first kind. We investigate null Cartan rectifying isophotic and rectifying silhouette curves with constant geodesic curvature $k_g$ and geodesic torsion $\tau_g$ and obtain the parameter equations of their axes. We prove that such curves are the null Cartan helices and the null Cartan cubics. We show that the introduced curves with a non-zero constant curvatures $k_g$ and $\tau_g$ are general helices, relatively normal-slant helices and isophotic curves with respect to the same axis. In particular, we find that null Cartan cubic lying on a timelike surface is rectifying isophotic and rectifying silhouette curve having a spacelike and a lightlike axis. Finally, we give some examples.

New Parametrization of Bertrand Partner D-curves in $\mathbb{E}^{3}$

We define and study a new parametrization of a Bertrand partner D-curve \{$\alpha$,$\alpha^{*}$\} in Euclidean 3-space by not taking Darboux frame element $g^{*}$ of Bertrand partner D-curve $\alpha^{*}$ parallel to $\overrightarrow{\alpha \alpha^{*}}$. We obtain a necessary and sufficient condition for a curve to be such type of Bertrand D-curves. Also, we obtain a characterization of a new parametrization of asymptotic Bertrand D-curves and provide an example.

A NEW APPROACH TO HYPERBOLIC SPINOR B-DARBOUX EQUATIONS

In this research, spinor descriptions of the curves in surfaces have been given according to the B-Darboux frame in Lorentzian 3-space E_1^3. The relations between B-Darboux and Darboux frames have been specified via their spinor conceptions which expressed in both timelike and spacelike surfaces, separately. Additionally, all these spinor representations have been portrayed in the view of the B-Darboux frame (via the curvatures) in Lorentzian 3-space. The findings are supported by some theorems and corollaries.

Elliptical magnetic trajectories of a point on the elliptical 2-sphere

Elliptical trajectories play numerous significant roles in mathematics and physics. The focus of this work is to analyze the trajectories of a point on the ellipsoid S a 1 , a 2 , a 3 2 while it is under the influence of a killing vector field K. For this purpose, we introduce the generalized Darboux frame and the variational vector fields of S a 1 , a 2 , a 3 2 . We were able to get the trajectory equations of the charged particle on the ellipsoid by utilizing the elliptical structure and elliptic inner and cross product. We adopted a clearer and more comprehensible strategy in this way. With the aid of a mathematical program, we were able to display the paths we had determined based on the particle's velocity vector's position in the magnetic field. These trajectories are spiral, helical, and elliptical.

Curve-Surface Pairs on Embedded Surfaces and Involute D-Scroll of the Curve-Surface Pair in E3

Willmore defined embedded surfaces on f:S→E3, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of E3, inducing a Riemannian structure on f(S). The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature H2 on f(S). Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the D∼-scroll of the curve–surface pair and involute D∼-scroll of the curve–surface pair with some differential geometric elements and found D∼(α,M)(s) and D∗∼(α,M)(s)-scrolls of the curve–surface pair (α,M).

A SMARANDACHE CURVE ACCORDING TO THE DARBOUX FRAME AND THE KINEMATICS OF THIS CURVE IN MINKOWSKI SPACE

In this study, the curvatures and frames of the Smarandache curve formed by a timelike curve with unit velocity taken on a timelike surface according to the Darboux frame in Semi-Euclidean space were created. Then, the image curve on the screw surface was obtained by applying translational and rotational motion (screw motion) using the quaternion multiplication under the same frame of this curve..Finally, the curvature and torsion of the obtained image curve were calculated

Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space $ \mathrm{E}_1^3 $

<abstract><p>Using a common tangent vector field to a surface along a curve, in this study we discussed a new Darboux frame that we referred to as the rotation-minimizing Darboux frame (RMDF) in Minkowski 3-space. The parametric equation resulting from the RMDF frame for an imbricate-ruled surface was then provided. As a result, minimal (or maximal for timelike surfaces) ruled surfaces were derived, along with the necessary and sufficient criteria for imbricate-ruled surfaces to be developable. The surfaces also described the parameter curves of these surfaces' asymptotic, geodesic, and curvature lines. We also gave an example to emphasize the most significant results.</p></abstract>

Tubular surface generated by a curve lying on a regular surface and its characterizations

<abstract><p>In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.</p></abstract>

Investigation of Special Type-Π Smarandache Ruled Surfaces Due to Rotation Minimizing Darboux Frame in E3

This study begins with the construction of type-Π Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in E3. The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced.

On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in E13

The aim of this study is to examine Darboux frames and some other geometric properties (geodesic curvatures, geodesic torsions, normal curvatures, Darboux derivative formulas, Darboux vectors, angles, etc.) of the spherical indicatrices on Lorentzian unit sphere S_1^2 and hyperbolic unit sphere H_0^2 of the spacelike Salkowski curve with spacelike binormal in Lorentzian 3-space E_1^3. In this context, new and interesting results have been obtained for this curve. Thus, relationships between the newly obtained curvatures and torsions and the curvature and torsion of the original curve are given. Moreover, the matrix relationship between the Darboux and Frenet frames of these indicatrices is shown. Finally, the Darboux vectors belong to the Darboux frames and the Darboux vectors belong to the Frenet frames of these curves are compared.

On the curves lying on parallel-like surfaces of the ruled surface in $E^{3}$

In this paper, it has been researched curves lying on parallel-like surfaces $M^{f}$ of the ruled surface $M$ in $E^{3}$. Using the definition of parallel-like surfaces it has been found parametric expressions of parallel-like surface of the ruled surface and image curve of the directrix curve of the base surface. Moreover, obtaining Darboux frames of curves lying on surfaces $M$ and $M^{f}$, it has been compared the geodesic curvatures, the normal curvatures and the geodesic torsions of these curves.

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.

A geometric motion planning for a spin-rolling sphere on a plane

The paper deals with motion planning for a spin-rolling sphere when the sphere follows an optimal straight path on a plane. Since the straight line constrains the sphere’s motion, controlling the sphere’s spin motion is essential to converge to a desired full configuration of the sphere. In this paper, we show a new geometric-based planning approach that is based on a full-state description of this nonlinear system. First, the problem statement of the motion planning is posed. Next, we develop a geometric controller implemented as a virtual surface by using the Darboux frame kinematics. This virtual surface generates arc-length-based inputs for controlling the trajectories of the sphere. Then, an iterative algorithm is designed to tune these inputs for the desired configurations. Finally, the feasibility of the proposed approach is verified by simulations.

On Generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space

In this paper we define generalized Darboux frame of a a pseudo null curve $\alpha$ lying on a lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that $\alpha$ has two such frames and obtain generalized Darboux frame's equations. We obtain the relations between the curvature functions of $\alpha$ with respect to the Darboux frame and generalized Darboux frames. We also find parameter equations of the Darboux vectors of the Frenet, Darboux and generalized Darboux frames and give the necessary and the sufficient conditions for such vectors to have the same directions. Finally, we present related examples.

Image Curves on the Parallel-like Surfaces in E³

In this paper, it's introduced the curves lying on parallel-like surface M^{f} of a surface M in Euclidean space. Taking into account the definition of the parallel-like surface it's obtained parametric expression of these curves and examined the Darboux frame for these curves which we call image curves. And finally, the curves lying on the surfaces M and M^{f} are compared by considering their geodesic and normal curvatures, the geodesic torsion.