scholarly journals Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1111
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Seoung Dal Jung
Keyword(s):  
Frenet Frame ◽  
Geometric Properties ◽  
Null Curve ◽  
Gauss Maps

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme.

scholarly journals A Note on Parametric Surfaces in Minkowski 3-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Esra Betul Koc Ozturk
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Parametric Surface ◽  
Parametric Surfaces ◽  
Null Curve ◽  
Necessary And Sufficient

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.

scholarly journals A New Angular Measurement in Minkowski 3-Space

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 56 ◽  
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Jie Liu ◽  
Young Ho Kim
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Angular Measurement ◽  
Frenet Frame ◽  
Measuring Methods ◽  
Null Curve

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

scholarly journals A natural Frenet frame for null curves on the lightlike cone in Minkowski space $\mathbb{R} ^{4}_{2}$

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nemat Abazari ◽  
Martin Bohner ◽  
Ilgin Sağer ◽  
Alireza Sedaghatdoost ◽  
Yusuf Yayli
Keyword(s):  
Minkowski Space ◽  
Constant Curvature ◽  
Structure Functions ◽  
Curvature Function ◽  
Frenet Frame ◽  
Null Curve ◽  
Null Curves

Abstract In this paper, we investigate the representation of curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in Minkowski space $\mathbb {R}^{4}_{2}$ R 2 4 by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function $\kappa _{2}=0$ κ 2 = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

scholarly journals Quaternionic Serret-Frenet Frames for Fuzzy Split Quaternion Numbers

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Cansel Yormaz ◽  
Simge Simsek ◽  
Serife Naz Elmas
Keyword(s):  
Natural Extension ◽  
Frenet Frame ◽  
Real Numbers ◽  
Split Quaternion ◽  
Geometric Properties ◽  
Split Quaternions ◽  
Fuzzy Real Numbers

We build the concept of fuzzy split quaternion numbers of a natural extension of fuzzy real numbers in this study. Then, we give some differential geometric properties of this fuzzy quaternion. Moreover, we construct the Frenet frame for fuzzy split quaternions. We investigate Serret-Frenet derivation formulas by using fuzzy quaternion numbers.

Rail Geometry and Euler Angles

2006 ◽  
Vol 1 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Cheta Rathod ◽  
Ahmed A. Shabana
Keyword(s):  
Vehicle Dynamics ◽  
Euler Angles ◽  
Frenet Frame ◽  
Geometric Properties ◽  
Track Geometry ◽  
Railroad Vehicle Dynamics ◽  
Space Curves ◽  
Curvature And Torsion ◽  
The Relationship ◽  
Railroad Vehicle

In railroad vehicle dynamics, Euler angles are often used to describe the track geometry (track centerline and rail space curves). The tangent and curvature vectors as well as local geometric properties such as the curvature and torsion can be expressed in terms of Euler angles. Some of the local geometric properties and Euler angles can be related to measured parameters that are often used to define the track geometry. The Euler angles employed, however, define a coordinate system that may differ from the Frenet frame used in the classical differential geometry. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves is developed in this paper and is used to shed light on some of the formulas and identities used in the geometric description in railroad vehicle dynamics. The conditions under which the two frames (track and Frenet) become equivalent are presented and used to obtain expressions for the curvature and torsion in terms of Euler angles and their derivatives with respect to the arc length.

A Unified Conjugation Theory for Planar Transmission

2010 ◽  
Author(s):  
Huimin Dong ◽  
Kwun-Lon Ting ◽  
Jian Liu ◽  
Delun Wang
Keyword(s):  
General Theory ◽  
Direct Contact ◽  
Tooth Profile ◽  
Transmission Ratio ◽  
Frenet Frame ◽  
Geometric Properties ◽  
Profile Design ◽  
Center Distance

The paper presents a general and versatile model describing the kinematic and geometric properties relating the centrodes, the conjugate curves, and the path of contact in a transmission through direct contact between any bodies. From a prescribed motion transmission and a desired path of contact, the characteristics of the conjugate tooth profiles, such as sliding ratio, tooth curvature, induced tooth curvature, conjugate boundary, and interference can be explicitly expressed even before the tooth profiles are obtained. The model is made possible by expressing the path of contact with the Frenet frame on the centrodes. The paper establishes a general theory suitable for optimal tooth profile design. The model is applicable in situations with constant or various transmission ratio as well as constant or various center distance.

scholarly journals Local geometric properties of the lightlike Killing magnetic curves in de Sitter 3-space

2021 ◽  
Vol 6 (11) ◽  
pp. 12543-12559
Author(s):  
Xiaoyan Jiang ◽  
◽  
Jianguo Sun
Keyword(s):  
Magnetic Field ◽  
Local Structure ◽  
Frenet Frame ◽  
De Sitter ◽  
Geometric Properties ◽  
Geometrical Properties ◽  
Magnetic Curve ◽  
Gamma S

<abstract><p>In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.</p></abstract>

Rail Geometry and Euler Angles

2006 ◽  
Author(s):  
Cheta M. Rathod ◽  
Ahmed A. Shabana
Keyword(s):  
Vehicle Dynamics ◽  
Euler Angles ◽  
Frenet Frame ◽  
Geometric Properties ◽  
Track Geometry ◽  
Railroad Vehicle Dynamics ◽  
Space Curves ◽  
Curvature And Torsion ◽  
The Relationship ◽  
Railroad Vehicle

In railroad vehicle dynamics, Euler angles are often used to describe the track geometry (track centerline and rail space curves). The tangent and curvature vectors as well as local geometric properties such as the curvature and torsion can be expressed in terms of Euler angles. Some of the local geometric properties and Euler angles can be related to measured parameters that are often used to define the track geometry. The Euler angles employed, however, define a coordinate system that may differ from the Frenet frame used in the classical differential geometry. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves is developed in this paper and used to shed light on some of the formulas and identities used in the geometric description in railroad vehicle dynamics. The conditions under which the two frames (track and Frenet) become equivalent are presented and used to obtain expressions for the curvature and torsion in terms of Euler angles and their derivatives with respect to the arc length.

The pseudo-null geometric phase along optical fiber

2021 ◽  
Author(s):  
Nevin Ertuğ Gürbüz
Keyword(s):  
Optical Fiber ◽  
Light Wave ◽  
Null Space ◽  
Parallel Transport ◽  
Polarized Light ◽  
Tangent Vector ◽  
Polarization Vector ◽  
Frenet Frame ◽  
Null Curve ◽  
Vector Formula

In this study, a pseudo-null space curve in Minkowski 3-space is used to describe an optical fiber that is injected into monochromatic linear polarized light. The direction of the electric field vector with respect to the Frenet frame of a pseudo-null curve determines the state polarization of a monochromatic linearly polarized light wave traveling along an optical fiber. For the Frenet frame of a pseudo-null curve in Minkowski 3-space, the polarization vector [Formula: see text] is assumed to be perpendicular to the tangent vector [Formula: see text] with respect to anholonomic coordinates. Anholonomic coordinates for the Frenet frame of a pseudo-null curve are used to describe pseudo-null electromagnetic curves in the normal and binormal directions along an optical fiber. For the Frenet frame of the pseudo-null curve, Lorentz force equations in the normal and binormal directions along the optical fiber are presented. Pseudo-normal and binormal Rytov parallel transport laws for electric fields in the normal and binormal directions along with the optical fiber for the Frenet frame of the pseudo-null curve via anholonomic coordinates are presented. For anholonomic coordinates in Minkowski 3-space, rotations of the polarization planes of a light wave traveling in the normal and binormal directions along with the optical fiber with respect to the Frenet frame of the pseudo-null curve are obtained. Finally, a pseudo-null curve’s Maxwellian evolution is determined.

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