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Coordinate Systems and Vector Algebra
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
The three-dimensional coordinate systems can further be classified into (i) parallel coordinate systems wherein a point is visualized in n-dimensional space as a poly-line connecting points on n vertical lines; (ii) curvilinear coordinate systems that are generalized coordinate systems based on intersection of curves; (iii) circular (polar) coordinate systems that represent a point in a plane by an angle and a distance from the origin; (iv) Plücker coordinate systems, in which lines are represented in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates; (v) generalized coordinate systems that are used in a Lagrangian treatment of mechanics; (vi) canonical coordinate systems that are used in Hamiltonian treatment of mechanics; and (vii) orthogonal coordinate systems wherein the coordinate surfaces meet at right angles. This last category is commonly used in the study of field theory.
Further differentiation
Published in C.W. Evans, Engineering Mathematics, 2019
In the cartesian system, points are described relative to two fixed mutually perpendicular straight lines known as the axes. In the polar coordinate system, points are described relative to a point called the origin and a straight line emanating from the origin called the initial line.
Fundamentals to Geometric Modeling and Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
For rotation, we need to specify a particular point as the origin, and the rotation operation will be performed about it. For example, a 2D point at (x,y) is rotated about the origin by an angle θ to the new position (x′,y′). We can describe this rotation in the polar form as () x=ρcosϕ,x′=ρcos(θ+ϕ), () y=ρsinϕ,y′=ρsin(θ+ϕ).
A butt shear joint (BSJ) specimen for high throughput testing of adhesive bonds
Published in The Journal of Adhesion, 2023
Chao Kang, J. J. M. Machado, Yu Sekiguchi, Ming Ji, Chiaki Sato, Masanobu Naito
FEA models for the two types of shear tests are shown in Figure 5, with enlarged mesh distributions at the adhesives. The mesh size of the adhesives along the thickness direction was fixed at 10 μm for both models, and it gradually increased from the adhesive/adherent interface to the outer surface of the adherents. The origin of the Cartesian coordinate system was located at the center of the adhesives. As shown in Figure 5, the loading and adhesive thickness directions were along the z- and y-axes, respectively, for both models. 3D models were used mainly to compare the stress distributions of the adhesives in the SLJ and cylindrical BSJ specimens. Forces of 6 and 2 kN, which were determined according to the experimental results in Section 3.2, were applied to the SLJ and BSJ specimens along the z-axis, respectively, to evaluate the stress distribution and confirm the stress concentration of the adhesives at failure.
Spatially Variable Seismic Motions by a U-Shaped Canyon in a Multi-Layered Half-Space
Published in Journal of Earthquake Engineering, 2021
Lee and Cao [1989] introduced an approximate method for the scattering of the incident waves by a shallow circular tunnel with variable depth-to-width ratios. The approximation is assumed by replacing the flat half surface by the convex or concave circular surface of very large radius. Then the transformation of the wave potentials from one cylindrical coordinate system to the other can be performed in terms of the addition theorem. In this study, the approximate method is extended not only for the horizontal ground surface, but also for the horizontal layer interfaces, as shown in Fig. 3. The origin o of the Cartesian coordinate system (x, y) and polar coordinate system (r, θ) are placed below o1 with a very large distance of H0. The angle θ is measured positively from the vertical y-axis counterclockwise toward the x-axis. It is noted that the ratio is large enough to ensure the approximation of the curved circular surface approaching to the plane boundaries at the layer interface.
Direct 3D coordinate transformation based on the affine invariance of barycentric coordinates
Published in Journal of Spatial Science, 2021
Whether in the two-dimensional space or the three-dimensional space, to describe the position of a material point, it is an essential prerequisite to establish a corresponding coordinate system, such as a Cartesian coordinate system, a polar coordinate system or a spherical coordinate system, and it is also indispensable to specify the origin of the coordinate system and mutually orthogonal unit vectors. Unlike traditional practices, the barycentric coordinates can locate the position of a point through the existing points (also referred to as reference points) rather than the origin, which are referred to as local coordinates as well (Vince 2006). The barycentric coordinates were proposed by the German mathematician (Möbius 1827) and have been successfully applied to many fields, such as 2D datum transformation in geodetic networks (Ansari et al. 2018), texture mapping in computer graphics (Hormann and Sukumar 2017), PnP (Perspective-n-Points) in computer vision (Lepetit et al. 2009), LiDAR point cloud filtering (Gézero and Antunes 2018), unmixing of hyperspectral remote sensing images (Honeine and Richard 2012), and real-time path planning for unmanned aerial vehicles (Zollars et al. 2018).