scholarly journals A special interpretation of the concept constant breadth for a space curve

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 371-382
Author(s):  
Tuba Agirman-Aydin
Keyword(s):  
Variable Coefficients ◽  
Space Curve ◽  
Normal Vector ◽  
Exit Point ◽  
Constant Breadth ◽  
Normal Vectors ◽  
Linear Differential ◽  
Curve Of Constant Breadth ◽  
Definition Of ◽  
Curvature And Torsion

The definition of curve of constant breadth in the literature is made by using tangent vectors, which are parallel and opposite directions, at opposite points of the curve. In this study, normal vectors of the curve, which are parallel and opposite directions are placed at the exit point of the concept of curve of constant breadth. In this study, on the concept of curve of constant breadth according to normal vector is worked. At the conclusion of the study, is obtained a system of linear differential equations with variable coefficients characterizing space curves of constant breadth according to normal vector. The coefficients of this system of equations are functions depend on the curvature and torsion of the curve. Then is obtained an approximate solution of this system by using the Taylor matrix collocation method. In summary, in this study, a different interpretation is made for the concept of space curve of constant breadth, the first time. Then this interpretation is used to obtain a characterization. As a result, this characterization we?ve obtained is solved.

Noncommutativity of Finite Rotations and Definitions of Curvature and Torsion

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Ahmed A. Shabana ◽  
Hao Ling
Keyword(s):  
Euler Angle ◽  
Rigid Body Dynamics ◽  
Space Curve ◽  
Euler Angles ◽  
Arc Length ◽  
Important Conclusion ◽  
Finite Rotations ◽  
Definition Of ◽  
Curvature And Torsion ◽  
Curve Geometry

The geometry of a space curve, including its curvature and torsion, can be uniquely defined in terms of only one parameter which can be the arc length parameter. Using the differential geometry equations, the Frenet frame of the space curve is completely defined using the curve equation and the arc length parameter only. Therefore, when Euler angles are used to describe the curve geometry, these angles are no longer independent and can be expressed in terms of one parameter as field variables. The relationships between Euler angles used in the definition of the curve geometry are developed in a closed-differential form expressed in terms of the curve curvature and torsion. While the curvature and torsion of a space curve are unique, the Euler-angle representation of the space curve is not unique because of the noncommutative nature of the finite rotations. Depending on the sequence of Euler angles used, different expressions for the curvature and torsion can be obtained in terms of Euler angles, despite the fact that only one Euler angle can be treated as an independent variable, and such an independent angle can be used as the curve parameter instead of its arc length, as discussed in this paper. The curve differential equations developed in this paper demonstrate that the curvature and torsion expressed in terms of Euler angles do not depend on the sequence of rotations only in the case of infinitesimal rotations. This important conclusion is consistent with the definition of Euler angles as generalized coordinates in rigid body dynamics. This paper generalizes this definition by demonstrating that finite rotations cannot be directly associated with physical geometric properties or deformation modes except in the cases when infinitesimal-rotation assumptions are used.

Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

2018 ◽  
Vol 19 (6) ◽  
pp. 621-626
Author(s):  
Amit Ujlayan ◽  
Ajay Dixit
Keyword(s):  
Fractional Derivative ◽  
Hybrid Method ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Order Linear Differential Equation ◽  
Conformable Fractional Derivative ◽  
Variation Of Parameters ◽  
Fractional Differential ◽  
Linear Differential ◽  
Definition Of

AbstractIn this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

Extracting Geometric Edges from 3D Point Clouds Based on Normal Vector Change

2014 ◽  
Vol 571-572 ◽  
pp. 729-734
Author(s):  
Jia Li ◽  
Huan Lin ◽  
Duo Qiang Zhang ◽  
Xiao Lu Xue
Keyword(s):  
Point Clouds ◽  
Detection Algorithm ◽  
Growth Strategy ◽  
Normal Vector ◽  
Geometric Morphometric ◽  
3D Point Clouds ◽  
Real Scene ◽  
Edge Points ◽  
Normal Vectors ◽  
Change Intensity

Normal vector of 3D surface is important differential geometric property over localized neighborhood, and its abrupt change along the surface directly reflects the variation of geometric morphometric. Based on this observation, this paper presents a novel edge detection algorithm in 3D point clouds, which utilizes the change intensity and change direction of adjacent normal vectors and is composed of three steps. First, a two-dimensional grid is constructed according to the inherent data acquisition sequence so as to build up the topology of points. Second, by this topological structure preliminary edge points are retrieved, and the potential directions of edges passing through them are estimated according to the change of normal vectors between adjacent points. Finally, an edge growth strategy is designed to regain the missing edge points and connect them into complete edge lines. The results of experiment in a real scene demonstrate that the proposed algorithm can extract geometric edges from 3D point clouds robustly, and is able to reduce edge quality’s dependence on user defined parameters.

scholarly journals Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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