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.

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.

The form of the tangent-developable at points of zero torsion on space curves

1980 ◽  
Vol 88 (3) ◽  
pp. 403-407 ◽  
Author(s):  
J. P. Cleave
Keyword(s):  
Power Series ◽  
Space Curve ◽  
Arc Length ◽  
Space Curves ◽  
Normal Plane ◽  
Tangent Developable ◽  
Cuspidal Edge ◽  
Curvature And Torsion ◽  
Image Position ◽  
Classical Derivation

A tangent-developable is a surface generated by the tangent lines of a space curve. The intersection of a tangent-developable with the normal plane at a point P of the curve generally has a cusp at that point. Thus the tangent-developable of a space curve has a cuspidal edge along the curve. The classical derivation of this phenomenon takes the trihedron (t, n, b) at P as coordinate axes to which the curve is referred. Then the intersection of the part of the tangent-developable generated by tangent lines at points close to P with the normal plane at P (i.e. the plane through P containing n and b) is given parametrically by power serieswhere K, T are the curvature and torsion, respectively, of the curve at P and s is arc-length measured from P ((2) p. 68). It is tacitly understood in this analysis that curvature and torsion are both defined and non-zero.

On the curvature and torsion of an isolated vortex filament

1965 ◽  
Vol 22 (3) ◽  
pp. 471-479 ◽  
Author(s):  
Robert Betchov
Keyword(s):  
Vortex Filament ◽  
Inviscid Fluid ◽  
Arc Length ◽  
Local Curvature ◽  
Vortex Filaments ◽  
Non Linear ◽  
Curvature And Torsion

We consider a very thin vortex filament in an unbounded, incompressible and inviscid fluid. The filament is not necessarily plane. Each portion of the filament moves with a velocity that can be approximated in terms of the local curvature of the filament. This approximation leads to a pair of intrinsic equations giving the curvature and the torsion of the filament, as functions of the time and the arc length along the filament. It is found that helicoidal vortex filaments are elementary solutions, and that they are unstable.The intrisic equations also suggest a linear mechanism that tends to produce concentrated torsion and a non-linear mechanism tending to disperse such singularities.

scholarly journals Cone Algorithm of Spinning Vehicles under Dynamic Coning Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Shuang-biao Zhang ◽  
Xing-cheng Li ◽  
Zhong Su
Keyword(s):  
Pitch Angle ◽  
Euler Angle ◽  
Roll Angle ◽  
Angle Error ◽  
Attitude Error ◽  
Spinning Process ◽  
Cone Algorithm ◽  
Definition Of ◽  
Yaw Angle ◽  
The Relationship

Due to the fact that attitude error of vehicles has an intense trend of divergence when vehicles undergo worsening coning environment, in this paper, the model of dynamic coning environment is derived firstly. Then, through investigation of the effect on Euler attitude algorithm for the equivalency of traditional attitude algorithm, it is found that attitude error is actually the roll angle error including drifting error and oscillating error, which is induced directly by dynamic coning environment and further affects the pitch angle and yaw angle through transferring. Based on definition of the cone frame and cone attitude, a cone algorithm is proposed by rotation relationship to calculate cone attitude, and the relationship between cone attitude and Euler attitude of spinning vehicle is established. Through numerical simulations with different conditions of dynamic coning environment, it is shown that the induced error of Euler attitude fluctuates by the variation of precession and nutation, especially by that of nutation, and the oscillating frequency of roll angle error is twice that of pitch angle error and yaw angle error. In addition, the rotation angle is more competent to describe the spinning process of vehicles under coning environment than Euler angle gamma, and the real pitch angle and yaw angle are calculated finally.

scholarly journals Charged Particle Trajectories in Static Electric and Magnetic Fields

1961 ◽  
Vol 14 (2) ◽  
pp. 310
Author(s):  
KJ Ausburn
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Magnetic Fields ◽  
Analytical Solution ◽  
Arc Length ◽  
Intrinsic Interest ◽  
Field Distributions ◽  
Electric And Magnetic Fields ◽  
Electric And Magnetic Field ◽  
Curvature And Torsion

A trajectory passing through a given point in a given direction is completely determined if its curvature and torsion are known functions of its arc length. Relativistic expressions for the curvature and torsion in terms of the electric and magnetic field distributions are derived below. Besides their intrinsic interest these expressions may be useful in the analytical solution of some simple trajectory problems.

Authoritarianism in West European Electorates

2021 ◽  
pp. 41-73
Author(s):  
Erik R. Tillman
Keyword(s):  
Western Europe ◽  
Descriptive Analysis ◽  
Political Trust ◽  
Important Conclusion ◽  
Cultural Attitudes ◽  
Traditional Gender Roles ◽  
And Gender ◽  
Definition Of ◽  
The Relationship ◽  
West European

This chapter engages in a descriptive analysis of authoritarianism in Western Europe and its relationship to economic, social, and political attitudes. It considers the definition of authoritarianism and how it is distinct from related concepts such as conservatism. The descriptive analysis addresses several important questions. First, it examines the distribution of authoritarianism in West European societies, along with its relationship to education, age, and gender. Then, it examines the relationship between authoritarianism and socio-cultural, political, and economic attitudes. The analysis finds that authoritarianism is closely related to socio-cultural attitudes on matters such as acceptance of same-sex marriage, endorsement of traditional gender roles, immigration, and ethnocentrism. Authoritarianism also correlates with attitudes towards democracy and political trust. However, high authoritarians are not meaningfully different from low authoritarians on economic questions. These findings point to an important conclusion for this book’s argument. Because high authoritarians vary from low authoritarians most on socio-cultural attitudes, it is likely that the factors driving the worldview issue are socio-cultural rather than economic in nature.

Understanding Hezbollah symbolism through symbolic convergence theory

2019 ◽  
Vol 7 (1) ◽  
pp. 43-60
Author(s):  
Jonathan Matusitz
Keyword(s):  
Convergence Theory ◽  
Important Conclusion ◽  
Symbolic Convergence ◽  
Symbolic Convergence Theory ◽  
The World ◽  
Terrorist Organization ◽  
Significant Change ◽  
Definition Of ◽  
Individual Self

This article applies symbolic convergence theory (SCT) to the type of symbolism used by Hezbollah, a Shia terrorist organization. By and large, SCT posits that sharing group fantasies contributes to symbolic convergence. The success of symbolic convergence hinges on three important concepts: symbolic cue, fantasy type and saga. An example of a symbolic cue is Hezbollah’s yellow-and-green logo that symbolizes global militancy against infidels. In regards to the organization’s fantasy types, eight of them were identified (e.g. Al-Quds Day, definition of enemies and individual self-sacrifice). Apropos of Hezbollah’s sagas, two figureheads, Ruhollah Khomeini (the Grand Ayatollah in Iran) and Hassan Nasrallah (the current Hezbollah leader), have become idols of worship by hundreds of thousands of Hezbollah members and millions of Shia Muslims. An important conclusion is that events can be more important for their symbolism than their very happening or existence. Hence, through symbolism, Hezbollah can strike a chord within its followers so as to produce significant change in the world.

scholarly journals Modeling Parallel Robot Kinematics for 3T2R and 3T3R Tasks Using Reciprocal Sets of Euler Angles

Robotics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 68 ◽  
Author(s):  
Moritz Schappler ◽  
Svenja Tappe ◽  
Tobias Ortmaier
Keyword(s):  
Degrees Of Freedom ◽  
Kinematic Model ◽  
Parallel Robot ◽  
Parallel Robots ◽  
Robot Kinematics ◽  
Euler Angles ◽  
Gradient Descent Algorithm ◽  
Kinematic Models ◽  
Rotational Degrees Of Freedom ◽  
Definition Of

Industrial manipulators and parallel robots are often used for tasks, such as drilling or milling, that require three translational, but only two rotational degrees of freedom (“3T2R”). While kinematic models for specific mechanisms for these tasks exist, a general kinematic model for parallel robots is still missing. This paper presents the definition of the rotational component of kinematic constraints equations for parallel robots based on two reciprocal sets of Euler angles for the end-effector orientation and the orientation residual. The method allows completely removing the redundant coordinate in 3T2R tasks and to solve the inverse kinematics for general serial and parallel robots with the gradient descent algorithm. The functional redundancy of robots with full mobility is exploited using nullspace projection.

scholarly journals Methodology of the cam mechanism design of rotary rakes

2012 ◽  
Vol 60 (1) ◽  
pp. 137-144
Author(s):  
Vladimír Šmíd ◽  
Stanislav Bartoň
Keyword(s):  
Mathematical Formulation ◽  
Space Curve ◽  
Transition Curve ◽  
Operational Parameters ◽  
Mathematical Functions ◽  
Cam Mechanism ◽  
Execution Phase ◽  
Optimum Formulation ◽  
Mathematical Equations ◽  
Definition Of

This paper describes the method of derivation of mathematical description of a cam orbit in rotary rakes (hay aggregators). At first, the authors describe basic construction elements of the rotary hay rake mechanism and their mutual links and relationships. Thereafter, they define the origin and the orientation of the system of coordinates, in which all calculations are carried out. In the next step they define basic requirements concerning the assurance of an optimum functioning of cam mechanisms as well as their transformation into mathematical equations. These requirements represent a base for the mathematical formulation of an optimum transition curve and it is emphasized that an optimum formulation of parameters of this curve is very important. In the course of calculation, they use also a normalized transition curve, which is used for the optimizations of the total number of parameters of the transition curve. Thereafter, they take into account mechanical and operational parameters of the hay aggregators and convert the optimum transition curve to that part of the space curve, which agrees at best with these parameters. Finally, the whole cam orbit is constructed using individual segments and presented as a sequentially defined space curve. Its individual parts concur sequentially to the level of the second derivation and are described as explicit mathematical functions of mechanical and operational parameters of the hay aggregator. The definition of the system of coordinates, the execution phase of calculations and the final shape of the cam orbit are illustrated in graphs.

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