scholarly journals On in-plane drill rotations for Cosserat surfaces

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210158
Author(s):  
Maryam Mohammadi Saem ◽  
Peter Lewintan ◽  
Patrizio Neff
Keyword(s):  
Differential Geometry ◽  
Rotation Axis ◽  
Initial Surface ◽  
Identity Mapping ◽  
Shell Surface ◽  
The Given ◽  
Tangent Vectors

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C 2 -smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping.

scholarly journals On the Affine Image of a Rational Surface of Revolution

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2061
Author(s):  
Juan G. Alcázar
Keyword(s):  
Differential Geometry ◽  
Affine Transformation ◽  
Rational Surface ◽  
Affine Differential Geometry ◽  
Affine Mapping ◽  
Surfaces Of Revolution ◽  
Surface Of Revolution ◽  
Affine Image ◽  
Fixed Line ◽  
The Given

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space

2016 ◽  
Vol 13 (10) ◽  
pp. 1650116 ◽  
Author(s):  
Derya Kahvecí ◽  
Yusuf Yayli ◽  
Ísmaíl Gök
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Symmetric Matrix ◽  
Unit Length ◽  
Exponential Map ◽  
Spatial Displacement ◽  
Split Quaternions ◽  
Rodrigues Formula ◽  
Lightlike Axis ◽  
The Given

The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in [Formula: see text] and angle, the spatial displacement is represented by a [Formula: see text] semi-orthogonal rotation matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-skew-symmetric matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

scholarly journals Geometric and Meshing Properties of Conjugate Curves for Gear Transmission

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Dong Liang ◽  
Bingkui Chen ◽  
Yane Gao ◽  
Shuai Peng ◽  
Siling Qin
Keyword(s):  
Differential Geometry ◽  
Contact Angle ◽  
Power Transmission ◽  
Characteristic Point ◽  
Geometric Analysis ◽  
Gear Transmission ◽  
Axial Position ◽  
Gear Design ◽  
Conjugate Surfaces ◽  
The Given

Conjugate curves have been put forward previously by authors for gear transmission. Compared with traditional conjugate surfaces, the conjugate curves have more flexibility and diversity in aspects of gear design and generation. To further extend its application in power transmission, the geometric and meshing properties of conjugate curves are discussed in this paper. Firstly, general principle descriptions of conjugate curves for arbitrary axial position are introduced. Secondly, geometric analysis of conjugate curves is carried out based on differential geometry including tangent and normal in arbitrary contact direction, characteristic point, and curvature relationships. Then, meshing properties of conjugate curves are further revealed. According to a given plane or spatial curve, the uniqueness of conjugated curve under different contact angle conditions is discussed. Meshing commonality of conjugate curves is also demonstrated in terms of a class of spiral curves contacting in the given direction for various gear axes. Finally, a conclusive summary of this study is given.

scholarly journals NEW APPROACHES ON DUAL SPACE

2020 ◽  
pp. 437
Author(s):  
Olgun Durmaz ◽  
Busra Aktas ◽  
Halit Gündoğan
Keyword(s):  
Differential Geometry ◽  
Analytic Functions ◽  
Dual Space ◽  
Directional Derivatives ◽  
New Approaches ◽  
Basic Concepts ◽  
Tangent Vectors

In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.

scholarly journals A Distortion Theorem for Analytic Maps of Annuli

1965 ◽  
Vol 17 ◽  
pp. 185-198
Author(s):  
C. E. Castonguay ◽  
H. G. Helfenstein
Keyword(s):  
Differential Geometry ◽  
Extremal Property ◽  
Riemannian Metrics ◽  
Distortion Theorem ◽  
Point Of View ◽  
Covering Surface ◽  
Open Riemann Surface ◽  
Euclidean Metric ◽  
Analytic Maps ◽  
The Given

Every abstract open Riemann surface can be made "concrete" (in the terminology of (1)) by considering it as a covering surface (in general branched) of the complex plane by means of a suitable projection map p. Since this covering map is not unique, it seems natural to single out some such maps by an extremal property. The use of Riemannian metrics compatible with the conformai structure on the given surface for the study of $1 is well known ; from the point of view of differential geometry it suggests an investigation of the distortion caused by p between such a metric ds^ and the Euclidean metric of .

scholarly journals Crystal structure of an unknown solvate of bis(tetra-n-butylammonium) [N,N′-(4-trifluoromethyl-1,2-phenylene)bis(oxamato)-κ4O,N,N′,O′]nickelate(II)

2015 ◽  
Vol 71 (6) ◽  
pp. 578-581
Author(s):  
François Eya'ane Meva ◽  
Dieter Schaarschmidt ◽  
Tobias Rüffer
Keyword(s):  
Rotation Axis ◽  
Point Group ◽  
Solvent Molecule ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Acta Cryst ◽  
Coordination Environment ◽  
Dimensional Network ◽  
Square Planar ◽  
The Given

In the title compound, [N(C4H9)4]2[Ni(C11H3F3N2O6)] or [N(n-Bu)4]2[Ni(topbo)] [n-Bu =n-butyl and topbo = 4-trifluoromethyl-1,2-phenylenebis(oxamate)], the Ni2+cation is coordinated by two deprotonated amido N atoms and two carboxylate O atoms, setting up a slightly distorted square-planar coordination environment. The [Ni(topbo]2−anion lies on a twofold rotation axis. Due to an incompatibility with the point-group symmetry of the complete molecule, orientational disorder of the CF3group is observed. The tetrahedral ammonium cations and the anion are linked by weak intermolecular C—H...O and C—H...F hydrogen-bonding interactions into a three-dimensional network. A region of electron density was treated with the SQUEEZE procedure inPLATON[Spek (2015).Acta Cryst. C71, 9–18] following unsuccessful attempts to model it as plausible solvent molecule(s). The given chemical formula and other crystal data do not take into account the unknown solvent molecule.

scholarly journals Symmetry and rotation skills of prospective elementary mathematics teachers

2014 ◽  
Vol 28 (48) ◽  
pp. 383-402 ◽  
Author(s):  
Melih Turgut ◽  
Kürşat Yenilmez ◽  
Pınar Anapa
Keyword(s):  
Mathematics Teachers ◽  
Prospective Teachers ◽  
Elementary Mathematics ◽  
Rotation Axis ◽  
Structured Interview ◽  
Department Of Education ◽  
Measurement Tool ◽  
Western Turkey ◽  
The Given

The aim of this study was to investigate the skills and deficiencies of senior university students enrolled in a mathematics education program with the concepts of symmetry and rotation of geometric figures. The study was conducted with 32 prospective teachers in the Department of Education at a public university located in Western Turkey. This descriptive study was designed with a case study. A structured interview technique was used for data collection. A measurement tool consisting of 12 drawing problems testing symmetry (5 problems) and rotation (7 problems) ability was used in the study. Descriptive statistical methods were used for data analysis. The drawings were analyzed individually, and the students' mistakes and deficiencies were categorized. According to the results of the study, prospective elementary mathematics teachers did not have difficulty in drawing the symmetry of an object or in determining the symmetry axis. However, while they could rotate the figure when a rotation axis was provided, they failed to rotate it in the absence of an axis. In addition, prospective elementary mathematics teachers generally failed to find the center of the given rotated figures. The results of the present study were consistent with results in the literature.

Non-Geodesic Trajectories for Filament Wound Composite Truncated Conical Domes

2013 ◽  
Vol 281 ◽  
pp. 304-307 ◽  
Author(s):  
Lei Zu ◽  
Qin Xiang He ◽  
Jun Ping Shi ◽  
Hui Li
Keyword(s):  
Differential Geometry ◽  
Present Method ◽  
Pressure Vessels ◽  
Angle Distribution ◽  
Conical Shells ◽  
Filament Wound ◽  
Filament Wound Composite ◽  
The Given ◽  
Winding Angle ◽  
Wound Composite

The goal of this paper is to present non-geodesic trajectories for filament wound truncated conical domes for pressure vessels. The fiber trajectories for non-geodesically overwound truncated conical shells are obtained based on differential geometry and the non-geodesic winding law. The influence of the slippage coefficient on non-geodesic trajectories is evaluated in terms of the winding angle distributions. The non-geodesic trajectories corresponding to various initial winding angles are also illustrated for the given slippage coefficient. The results show that the winding angle distribution of non-geodesics on a truncated conical dome has an overall increase with the increase of the slippage coefficient or the initial winding angle. The present method can provide a significant reference for developing non-geodesically overwound conical structures.

scholarly journals Tangent Vectors on Tangent Euclidean Spaces

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Differential Geometry ◽  
Euclidean Spaces ◽  
Tangent Vectors

This is an article on differential geometry that connects tangent vectors and tangent Euclidean spaces.

scholarly journals Crystal structure of a sodium, zinc and iron(III)-based non-stoichiometric phosphate with an alluaudite-like structure: Na1.67Zn1.67Fe1.33(PO4)3

2015 ◽  
Vol 71 (6) ◽  
pp. 690-692 ◽  
Author(s):  
Jamal Khmiyas ◽  
Abderrazzak Assani ◽  
Mohamed Saadi ◽  
Lahcen El Ammari
Keyword(s):  
Title Compound ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Ionic Mobility ◽  
Solid State Reactions ◽  
Electrical Neutrality ◽  
Sodium Cations ◽  
Edge Sharing Octahedra ◽  
Structure Lead ◽  
The Given

The new title compound, disodium dizinc iron(III) tris(phosphate), Na1.67Zn1.67Fe1.33(PO4)3, which belongs to the alluaudite family, has been synthesized by solid-state reactions. In this structure, all atoms are in general positions except for four, which are located on special positions of theC2/cspace group. This structure is characterized by cation substitutional disorder at two sites, one situated on the special position 4e(2) and the other on the general position 8f. The 4esite is partially occupied by Na+[0.332 (3)], whereas the 8fsite is entirely filled by a mixture of Fe and Zn. The full-occupancy sodium and zinc atoms are located at the Wyckoff positions on the inversion center 4a(-1) and on the twofold rotation axis 4e, respectively. Refinement of the occupancy ratios, bond-valence analysis and the electrical neutrality requirement of the structure lead to the given composition for the title compound. The three-dimensional framework of this structure consists of kinked chains of edge-sharing octahedra stacked parallel to [10-1]. The chains are formed by a succession of trimers based on [ZnO6] octahedra and the mixed-cation FeIII/ZnII[(Fe/Zn)O6] octahedra [FeIII:ZnIIIratio 0.668 (3)/0.332 (3)]. Continuous chains are held together by PO4phosphate groups, forming polyhedral sheets perpendicular to [010]. The stacked sheets delimit two types of tunnels parallel to thecaxis in which the sodium cations are located. Each Na+cation is coordinated by eight O atoms. The disorder of Na in the tunnel might presage ionic mobility for this material.

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