Journal articles: 'Coordinate geometry'

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Dil, Emre, and Talha Zafer. "Transformation Groups for a Schwarzschild-Type Geometry in f(R) Gravity." Journal of Gravity 2016 (November 2, 2016): 1–8. http://dx.doi.org/10.1155/2016/7636493.

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We know that the Lorentz transformations are special relativistic coordinate transformations between inertial frames. What happens if we would like to find the coordinate transformations between noninertial reference frames? Noninertial frames are known to be accelerated frames with respect to an inertial frame. Therefore these should be considered in the framework of general relativity or its modified versions. We assume that the inertial frames are flat space-times and noninertial frames are curved space-times; then we investigate the deformation and coordinate transformation groups between a flat space-time and a curved space-time which is curved by a Schwarzschild-type black hole, in the framework of f(R) gravity. We firstly study the deformation transformation groups by relating the metrics of the flat and curved space-times in spherical coordinates; after the deformation transformations we concentrate on the coordinate transformations. Later on, we investigate the same deformation and coordinate transformations in Cartesian coordinates. Finally we obtain two different sets of transformation groups for the spherical and Cartesian coordinates.

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CAMACHO, A. "AHARONOV–BOHM EFFECT AND COORDINATE TRANSFORMATIONS." Modern Physics Letters A 14, no. 21 (July 10, 1999): 1445–53. http://dx.doi.org/10.1142/s0217732399001541.

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Resorting to a Gedankenexperiment which is very similar to the famous Aharonov–Bohm proposal it will be shown that, in the case of a Minkowskian space–time, we may use a nonrelativistic quantum particle and a noninertial coordinate system and obtain geometric information of regions that are, to this particle, forbidden. This shows that the outcome of a nonrelativistic quantum process is determined not only by the features of geometry at those points at which the process takes place, but also by geometric parameters of regions in which the quantum system cannot enter. From this fact we could claim that geometry at the quantum level plays a nonlocal role. Indeed, the measurement outputs of some nonrelativistic quantum experiments are determined not only by the geometry of the region in which the experiment takes place, but also by the geometric properties of space–time volumes which are, in some way, forbidden in the experiment.

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Terc, Michael. "Coordinate Geometry—Art and Mathematics." Arithmetic Teacher 33, no. 2 (October 1985): 22–24. http://dx.doi.org/10.5951/at.33.2.0022.

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Our sutudents cry for self-expression, for a chance to see mathematics in action. Frequently, however, the structure of mathematics does not lend itself to individual style or variation. Problem solving can tend to be dull and monotonous rather than exciting and stimulating.

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BHATTACHARYA, AMRITA, and ALEXANDER A. POTAPOV. "BENDING OF LIGHT IN ELLIS WORMHOLE GEOMETRY." Modern Physics Letters A 25, no. 28 (September 14, 2010): 2399–409. http://dx.doi.org/10.1142/s0217732310033748.

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A recent work by Dey and Sen derived the approximate light deflection angle α by an Ellis wormhole in terms of proper radial distance ℓ that covers the entire spacetime. On the other hand, Bodenner and Will calculated the expressions for light bending in Schwarzschild geometry using various coordinates and showed that they all reduce to a single formula when re-expressed in the coordinate independent language of "circumferential radius" rC identified with the standard radial coordinate rS. We shall argue that the coordinate invariant language for two-way wormholes should be ℓ rather than rS. Hence here we find the exact deflection α in Ellis wormhole geometry first in terms of ℓ and then in terms of rS. We confirm the latter expression using three different methods. We argue that the practical measurement scheme does not necessarily single out either ℓ or rS. Some errors in the literature are corrected.

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Shvets, S. V., and V. P. Astakhov. "Effect of Insert Angles on Cutting Tool Geometry." Journal of Engineering Sciences 7, no. 2 (2020): A1—A6. http://dx.doi.org/10.21272/jes.2020.7(2).a1.

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An analysis of publications has shown that mechanically clamped indexable inserts are predominantly used in modern tool manufacturing. Each insert has its shape and geometry in the tool coordinate system. The static system’s required geometry is achieved by the tilting of the insert pocket in the radial and axial directions. Therefore, it is of great importance in the tool design to know the relationships between the insert’s geometry parameters in the tool coordinate system where the geometry paraments of the insert are defined and working geometry parameters of the tool defined in the static coordinate system. The paper presents the developed methodology for determining the insert pocket base surface position to ensure the required values of the tool geometry parameters of the selected indexable insert in the static coordinate system. The graphs of the dependence of each of the angles of the insert geometry on the angles of rotation of this insert in the front and profile planes are presented as the level lines for practical use. Using these graphs, one can optimize all geometric insertion parameters in the static coordinate system. The model of the calculations of the mechanism of the insert clamping by a screw is developed. The basic size and tolerance of the output link determine the distance from the intersection line of the base surfaces to the thread axis on the pocket and the minimum amount of the screw stroke on the insert clamping in the pocket. Keywords: indexable insert, cutting tool, coordinate system, base surfaces, geometric parameters.

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Iofa, Mikhail Z. "Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells." Advances in High Energy Physics 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/5632734.

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Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates. We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications to the Israel junction conditions are discussed.

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Hawryluk, Marek, Marek Kuran, and Jacek Ziemba. "The use of replicas in the measurement of machine elements with use of contact coordinate measurements." Mechanik 91, no. 11 (November 12, 2018): 958–60. http://dx.doi.org/10.17814/mechanik.2018.11.169.

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Modern technology allows to design and manufacture machine elements with complex geometry that makes it difficult or even impossible to use coordinate measuring machines for verification of them. The article presents the possibility of using replicas of product geometry to control geometric features using contact measurements on a coordinate machine.

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Matsuura, Ryota, and Sarah Sword. "Illuminating Coordinate Geometry with Algebraic Symmetry." Mathematics Teacher 108, no. 6 (February 2015): 470–73. http://dx.doi.org/10.5951/mathteacher.108.6.0470.

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Westegaard, Susanne K. "Activities: Stitching Quilts into Coordinate Geometry." Mathematics Teacher 91, no. 7 (October 1998): 587–600. http://dx.doi.org/10.5951/mt.91.7.0587.

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Quilts—useful items, visually appealing, steeped in history, and an integral part of our culture—can also be the jumping-off point for many mathematical investigations” (Morrow and Bassarear 1996). The history behind many quilt patterns offers an opportunity for crossdisciplinary projects with art and social studies classes. Such children's books as Eight Hands Round (Paul 1991) and Selina and the Bear Paw Quilt (Smucker 1995) connect mathematics with literature.

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Wenxiang, Zhang. "Plücker Coordinate Geometry and its Applications to Studies of Instantaneous Axial Planes of Generating Gears." International Journal of Mechanical Engineering Education 22, no. 4 (October 1994): 235–44. http://dx.doi.org/10.1177/030641909402200401.

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After an introduction of the Plücker coordinate geometry, a discussion is made of the expression of screws in Plücker coordinates and the addition of screws. As a result, the geometry of generating gears is re-studied and a formula is derived for calculating the axode length of hypoid gears.

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Tiu, Janine, J. Neil Waddell, Basil Al-Amleh, Wendy-Ann Jansen van Vuuren, and Michael V. Swain. "Coordinate geometry method for capturing and evaluating crown preparation geometry." Journal of Prosthetic Dentistry 112, no. 3 (September 2014): 481–87. http://dx.doi.org/10.1016/j.prosdent.2013.11.012.

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Hutchison, A. R., A. Mitra, and D. A. Atwood. "The four coordinate geometric parameter: A new quantification of geometry for four coordinate aluminum and gallium." Main Group Chemistry 4, no. 3 (September 2005): 187–200. http://dx.doi.org/10.1080/10241220500324217.

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Nagy, Benedek, and Khaled Abuhmaidan. "A Continuous Coordinate System for the Plane by Triangular Symmetry." Symmetry 11, no. 2 (February 9, 2019): 191. http://dx.doi.org/10.3390/sym11020191.

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The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems. The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system.

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J. Dimitrijevic, Nebojsa. "CREATING GEOMETRIC SHAPES IN AUTODESK INVENTOR USING OF BORN (BASE ORPHAN REFERENCE NODE) TECHNIQUE." Knowledge International Journal 28, no. 4 (December 10, 2018): 1341–48. http://dx.doi.org/10.35120/kij28041341n.

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The basic concept of BORN (Base Orphan Reference Node) technique is the use of a Cartesian coordinate system as a first shape, which preceded the creation of any geometrical shape. After the establishment a Cartesian coordinate system, are obtained by three mutually normal working plane (XY, YZ and ZX), which can be used as a sketching plane, three working axes (X, Y, and Z) and a working point (the origin). Three working planes are used as a reference for dimensions and geometric constructions. Autodesk Inventor automatically adjust set of reference geometry, which consists of three working-plane, three working axes and the origin when we begin to create a new part. All of the following geometric shapes can be used by the coordinate system and / or reference geometry as sketching plane.

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Martin, R. R., H. Suzuki, and P. A. C. Varley. "Labeling Engineering Line Drawings Using Depth Reasoning." Journal of Computing and Information Science in Engineering 5, no. 2 (February 21, 2005): 158–67. http://dx.doi.org/10.1115/1.1891045.

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Automatic creation of B-rep models of engineering objects from freehand sketches would benefit designers. One step aims to take a line drawing (with hidden lines removed), and from it deduce an initial three-dimensional (3D) geometric realization of the visible part of the object, including junction and line labels, and depth coordinates. Most methods for producing this frontal geometry use line labeling, which takes little or no account of geometry. Thus, the line labels produced can be unreliable. Our alternative approach inflates a drawing to produce provisional depth coordinates, and from these makes deductions about line labels. Assuming many edges in the drawing are parallel to one of three main orthogonal directions, we first attempt to identify groups of parallel lines aligned with the three major axes of the object. From these, we create and solve a linear system of equations relating vertex coordinates, in the coordinate system of the major axes. We then inflate the drawing in a coordinate system based on the plane of the drawing and depth perpendicular to it. Finally, we use this geometry to identify which lines in the drawing correspond to convex, concave, or occluding edges. We discuss alternative realizations of some of the concepts, how to cope with nonisometric-projection drawings, and how to combine this approach with other labeling techniques to gain the benefits of each. We test our approach using sample drawings chosen to be representative of engineering objects. These highlight difficulties often overlooked in previous papers on line labeling. Our new approach has significant benefits.

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Wang, Shu, and Yongxin Wang. "The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates." Mathematics 8, no. 7 (July 21, 2020): 1195. http://dx.doi.org/10.3390/math8071195.

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This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

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Shragge, Jeffrey. "Solving the 3D acoustic wave equation on generalized structured meshes: A finite-difference time-domain approach." GEOPHYSICS 79, no. 6 (November 1, 2014): T363—T378. http://dx.doi.org/10.1190/geo2014-0172.1.

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The key computational kernel of most advanced 3D seismic imaging and inversion algorithms used in exploration seismology involves calculating solutions of the 3D acoustic wave equation, most commonly with a finite-difference time-domain (FDTD) methodology. Although well suited for regularly sampled rectilinear computational domains, FDTD methods seemingly have limited applicability in scenarios involving irregular 3D domain boundary surfaces and mesh interiors best described by non-Cartesian geometry (e.g., surface topography). Using coordinate mapping relationships and differential geometry, an FDTD approach can be developed for generating solutions to the 3D acoustic wave equation that is applicable to generalized 3D coordinate systems and (quadrilateral-faced hexahedral) structured meshes. The developed numerical implementation is similar to the established Cartesian approaches, save for a necessary introduction of weighted first- and mixed second-order partial-derivative operators that account for spatially varying geometry. The approach was validated on three different types of computational meshes: (1) an “internal boundary” mesh conforming to a dipping water bottom layer, (2) analytic “semiorthogonal cylindrical” coordinates, and (3) analytic semiorthogonal and numerically specified “topographic” coordinate meshes. Impulse response tests and numerical analysis demonstrated the viability of the approach for kernel computations for 3D seismic imaging and inversion experiments for non-Cartesian geometry scenarios.

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Cao, Anzhou, Yanqiu Gao, Jicai Zhang, and Xianqing Lv. "Trajectory Estimation of Aircraft in a Double-Satellite Passive Positioning System with the Adjoint Method." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/502610.

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A double-satellite passive positioning system is constructed based on the theory of space geometry, where two observation coordinate systems and a fundamental coordinate system exist. In each observation coordinate system, there exists a ray from the observation satellite to the aircraft. One difficulty lies in that these two rays may not intersect due to the existence of various errors. Under this situation, this work assumes that the middle point of common perpendicular between two rays is the actual position of aircraft. Based on the theory of space geometry, the coordinates of aircraft in the fundamental coordinate system can be determined. A dynamic model with the adjoint method is developed to estimate the trajectory of aircraft during the process of rocket propulsion. By assimilating observations, the trajectory of aircraft can be calculated. Numerical experiments are designed to validate the reasonability and feasibility of this model. Simulated results indicate that even by assimilating a small number of observations, the trajectory of aircraft can be estimated. In addition, the trajectory estimation can become more accurate when more observations are assimilated to the model.

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JANSE VAN RENSBURG, T., M. A. VAN WYK, and W. H. STEEB. "THREE-DIMENSIONAL GEOMETRY WITHIN A DRIVING SIMULATOR." International Journal of Modern Physics C 16, no. 06 (June 2005): 909–20. http://dx.doi.org/10.1142/s0129183105007650.

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Three-dimensional coordinate transformations are an essential part of the realistic visual display within a driving simulator. They are also used in other simulators such as flight simulators and for robotics. In this paper, the mathematical framework for implementing three-dimensional coordinate transformations is presented, provided with more detail for implementing it in a programming language such as C++. The realistic positioning of an observer for the "behind and above" view in a driving simulator will be discussed as an application of coordinate system transformations.

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Nielsen, P. M., I. J. Le Grice, B. H. Smaill, and P. J. Hunter. "Mathematical model of geometry and fibrous structure of the heart." American Journal of Physiology-Heart and Circulatory Physiology 260, no. 4 (April 1, 1991): H1365—H1378. http://dx.doi.org/10.1152/ajpheart.1991.260.4.h1365.

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We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

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Liu, Chun Feng, Shan Shan Kong, and Hai Ming Wu. "Research on a Single Camera Location Model and its Application." Applied Mechanics and Materials 50-51 (February 2011): 468–72. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.468.

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Digital cameras have been widely used in the areas of road transportation, railway transportation as well as security system. To address the position of digital camera in these fields this paper proposed a geometry calibration method based on feature point extraction of arbitrary target. Under the meaning of the questions, this paper first defines four kinds of coordinate system, that is the world coordinate system. The camera's optical center of the coordinate system is the camera coordinate system, using the same point in different coordinate system of the coordinate transformation to determine the relationship between world coordinate system and camera coordinate. And thus determine the camera's internal parameters and external parameters, available transformation matrix and translation vector indicated by the camera's internal parameters of the external parameters and the establishment of a single camera location model. According to the model, using the camera's external parameters to be on the target circle center point in the image plane coordinates.

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Eagle, Cassandra T., Nkongho Atem-Tambe, Kenneth K. Kpogo, Jennie Tan, and Fredricka Quarshie. "(3-Methylbenzonitrile-κN)tetrakis(μ-N-phenylacetamidato)-κ4N:O;κ4O:N-dirhodium(II)(Rh—Rh)." Acta Crystallographica Section E Structure Reports Online 69, no. 12 (November 6, 2013): m639. http://dx.doi.org/10.1107/s1600536813029838.

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In the title compound, [Rh2(C8H8NO)4(C8H7N)], the four acetamidate ligands bridging the dirhodium core are arranged in a 2,2-transmanner. One RhIIatom is five-coordinate, in a distorted pyramidal geometry, while the other is six-coordinate, with a disorted octahedral geometry. For the six-coordinate RhIIatom, the axial nitrile ligand shows a non-linear Rh–nitrile coordination with an Rh—N—C bond angle of 166.4 (4)° and a nitrile N—C bond length of 1.138 (6) Å. Each unique RhIIatom is coordinated by atranspair of N atoms and atranspair of O atoms from the four acetamide ligands. The Neq—Rh—Rh—Oeqtorsion angles on the acetamide bridge varies between 12.55 (11) and 14.04 (8)°. In the crystal, the 3-methylbenzonitrile ring shows a π–π interaction with an inversion-related equivalent [interplanar spacing = 3.360 (6) Å]. A phenyl ring on one of the acetamide ligands also has a face-to-face π–π interaction with an inversion-related equivalent [interplanar spacing = 3.416 (5) Å].

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Gielen, Steffen. "Group Field Theory and Its Cosmology in a Matter Reference Frame." Universe 4, no. 10 (October 2, 2018): 103. http://dx.doi.org/10.3390/universe4100103.

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While the equations of general relativity take the same form in any coordinate system, choosing a suitable set of coordinates is essential in any practical application. This poses a challenge in background-independent quantum gravity, where coordinates are not a priori available and need to be reconstructed from physical degrees of freedom. We review the general idea of coupling free scalar fields to gravity and using these scalars as a “matter reference frame”. The resulting coordinate system is harmonic, i.e., it satisfies the harmonic (de Donder) gauge. We then show how to introduce such matter reference frames in the group field theory approach to quantum gravity, where spacetime is emergent from a “condensate” of fundamental quantum degrees of freedom of geometry, and how to use matter coordinates to extract physics. We review recent results in homogeneous and inhomogeneous cosmology, and give a new application to the case of spherical symmetry. We find tentative evidence that spherically-symmetric group field theory condensates defined in this setting can reproduce the near-horizon geometry of a Schwarzschild black hole.

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Liu, Fang, Jing-Jing Zhang, Ming-Yuan Lei, and Qing-Fu Zhang. "A two-dimensional cadmium(II) coordination polymer based on 5-(pyridin-4-yl)isophthalic acid: poly[[tetraaquabis[μ3-5-(pyridin-4-yl)isophthalato]dicadmium(II)] pentahydrate]." Acta Crystallographica Section C Structural Chemistry 71, no. 9 (August 27, 2015): 834–38. http://dx.doi.org/10.1107/s2053229615015612.

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The title CdIIcompound, {[Cd2(C13H7NO4)2(H2O)4]·5H2O}n, was synthesized by the hydrothermal reaction of Cd(NO3)2·4H2O and 5-(pyridin-4-yl)isophthalic acid (H2L). The asymmetric unit contains two crystallographically independent CdIIcations, two deprotonatedL2−ligands, four coordinated water molecules and five isolated water molecules. One of the CdIIcations adopts a six-coordinate octahedral coordination geometry involving three O atoms from one bidentate chelating and one monodentate carboxylate group of two differentL2−ligands, one N atom of anotherL2−ligand and two coordinated water molecules. The second CdIIcation adopts a seven-coordinate pentagonal–bipyramidal coordination geometry involving four O atoms from two bidentate chelating carboxylate groups of two differentL2−ligands, one N atom of anotherL2−ligand and two coordinated water molecules. EachL2−ligand bridges three CdIIcations and, likewise, each CdIIcation connects to threeL2−ligands, giving rise to a two-dimensional graphite-like 63layer structure. These two-dimensional layers are further linked by O—H...O hydrogen-bonding interactions to form a three-dimensional supramolecular architecture. The photoluminescence properties of the title compound were also investigated.

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Miranda Jr., Gastão F., Gilson Giraldi, Carlos E. Thomaz, and Daniel Millàn. "Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning." International Journal of Natural Computing Research 5, no. 2 (April 2015): 37–68. http://dx.doi.org/10.4018/ijncr.2015040103.

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The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.

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TRÁNG, LÊ DŨNG, HÉLÈNE MAUGENDRE, and CLAUDE WEBER. "GEOMETRY OF CRITICAL LOCI." Journal of the London Mathematical Society 63, no. 3 (June 2001): 533–52. http://dx.doi.org/10.1017/s0024610701001995.

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Let(formula here)be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.For each α ∈ A, we define a rational number dα by(formula here)where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

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Zhou, Yan-Ling, Ming-Hua Zeng, and Seik Weng Ng. "Tris[2,5-bis(1H-benzimidazol-2-yl)pyridinato-κ2 N 1,N 2]cobalt(III) dihydrate." Acta Crystallographica Section E Structure Reports Online 62, no. 4 (March 3, 2006): m663—m665. http://dx.doi.org/10.1107/s1600536806007082.

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In the title compound, [Co(C19H12N5)3]·2H2O, three mono-deprotonated 2,5-bis(benzimidazolyl)pyridine heterocycles chelate to cobalt(III) through the N atom of one benzimidazolyl arm of the heterocycle as well as through the pyridyl N atom to form a fairly regular six-coordinate, octahedral geometry geometry for cobalt. A network of N—H...N, N—H...O and O—H...N hydrogen bonds involving the non-coordinated water molecules results in a layered structure.

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Smith, Robert F. "Let's Do It: Coordinate Geometry for Third Graders." Arithmetic Teacher 33, no. 8 (April 1986): 6–11. http://dx.doi.org/10.5951/at.33.8.0006.

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Post, R. L. "Gravitation in a Deformed 3-Space Coordinate Geometry." Physics Essays 19, no. 4 (December 1, 2006): 458–98. http://dx.doi.org/10.4006/1.3028853.

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Taback, Stanley F. "Coordinate Geometry: A Powerful Tool for Solving Problems." Mathematics Teacher 83, no. 4 (April 1990): 264–68. http://dx.doi.org/10.5951/mt.83.4.0264.

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In calling for reform in the teaching and learning of mathematics, the Curriculum and Evaluation Standards for School Mathematics (Standards) developed by NCTM (1989) envisions mathematics study in which students reason and communicate about mathematical ideas that emerge from problem situations. A fundamental premise of the Standards, in fact, is the belief that “mathematical problem solving … is nearly synonymous with doing mathematics” (p. 137). And the ability to solve problems, we are told, is facilitated when students have opportunities to explore “connections” among different branches of mathematics.

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Blight, Barry, Mike Cornelius, and Nick Lord. "Finding an angle of rotation in coordinate geometry." Mathematical Gazette 94, no. 529 (March 2010): 149–50. http://dx.doi.org/10.1017/s0025557200007324.

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Roman, Rodica. "‘Similar’ coordinate systems and the Roche geometry: application." Astrophysics and Space Science 335, no. 2 (June 21, 2011): 475–83. http://dx.doi.org/10.1007/s10509-011-0747-1.

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Ivanov, Vyacheslav N., and Alisa A. Shmeleva. "Geometric characteristics of the deformation state of the shells with orthogonal coordinate system of the middle surfaces." Structural Mechanics of Engineering Constructions and Buildings 16, no. 1 (December 15, 2020): 38–44. http://dx.doi.org/10.22363/1815-5235-2020-16-1-38-44.

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The aim of this work is to receive the geometrical equations of strains of shells at the common orthogonal not conjugated coordinate system. At the most articles, textbooks and monographs on the theory and analysis of the thin shell there are considered the shells the coordinate system of which is given at the lines of main curvatures. Derivation of the geometric equations of the deformed state of the thin shells in the lines of main curvatures is given, specifically, at monographs of the theory of the thin shells of V.V. Novozhilov, K.F. Chernih, A.P. Filin and other Russian and foreign scientists. The standard methods of mathematic analyses, vector analysis and differential geometry are used to receive them. The method of tensor analysis is used for receiving the common equations of deformation of non orthogonal coordinate system of the middle shell surface of thin shell. The equations of deformation of the shells in common orthogonal coordinate system (not in the lines of main curvatures) are received on the base of this equation. Derivation of the geometric equations of deformations of thin shells in orthogonal not conjugated coordinate system on the base of differential geometry and vector analysis (without using of tensor analysis) is given at the article. This access may be used at textbooks as far as at most technical institutes the base of tensor analysis is not given.

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Ferraro, Giarita, Tiziano Marzo, Maria Cucciolito, Francesco Ruffo, Luigi Messori, and Antonello Merlino. "Reaction with Proteins of a Five-Coordinate Platinum(II) Compound." International Journal of Molecular Sciences 20, no. 3 (January 26, 2019): 520. http://dx.doi.org/10.3390/ijms20030520.

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Stable five-coordinate Pt(II) complexes have been highlighted as a promising and original platform for the development of new cytotoxic drugs. Their interaction with proteins has been scarcely studied. Here, the reactivity of the five-coordinate Pt(II) compound [Pt(I)(Me) (dmphen)(olefin)] (Me = methyl, dmphen = 2,9-dimethyl-1,10-phenanthroline, olefin = dimethylfumarate) with the model proteins hen egg white lysozyme (HEWL) and bovine pancreatic ribonuclease (RNase A) has been investigated by X-ray crystallography and electrospray ionization mass spectrometry. The X-ray structures of the adducts of RNase A and HEWL with [Pt(I)(Me)(dmphen)(olefin)] are not of very high quality, but overall data indicate that, upon reaction with RNase A, the compound coordinates the side chain of His105 upon releasing the iodide ligand, but retains the pentacoordination. On the contrary, upon reaction with HEWL, the trigonal bi-pyramidal Pt geometry is lost, the iodide and the olefin ligands are released, and the metal center coordinates the side chain of His15 probably adopting a nearly square-planar geometry. This work underlines the importance of the combined use of crystallographic and mass spectrometry techniques to characterize, in detail, the protein–metallodrug recognition process. Our findings also suggest that five-coordinate Pt(II) complexes can act either retaining their uncommon structure or functioning as prodrugs, i.e., releasing square-planar platinum complexes as bioactive species.

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Shragge, Jeff. "Angle-domain common-image gathers in generalized coordinates." GEOPHYSICS 74, no. 3 (May 2009): S47—S56. http://dx.doi.org/10.1190/1.3103248.

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The theory of angle-domain common-image gathers (ADCIGs) is extended to migrations performed in generalized 2D coordinate systems. I have developed an expression linking the definition of reflection opening angle to differential traveltime operators and spatially varying weights derived from the non-Cartesian geometry. Generalized-coordinate ADCIGs can be calculated directly using Radon-based offset-to-angle approaches for coordinate systems satisfying the Cauchy-Riemann differentiability criteria. The canonical examples of tilted-Cartesian, polar, and elliptical coordinates can be used to illustrate the ADCIG theory. I have compared analytically and numerically generated image volumes for a set of elliptically shaped reflectors. Experiments with a synthetic data set showed that elliptical-coordinate ADCIGs better resolve the reflection opening angles of steeply dipping structure, relative to conventional Cartesian image volumes, because of improved large-angle propagation and enhanced sensitivity to steep structural dips afforded by coordinate system transformations.

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BURNETT, DAVID S. "RADIATION BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION FOR ELLIPSOIDAL, PROLATE SPHEROIDAL, OBLATE SPHEROIDAL AND SPHERICAL DOMAIN BOUNDARIES." Journal of Computational Acoustics 20, no. 04 (November 29, 2012): 1230001. http://dx.doi.org/10.1142/s0218396x12300010.

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One of the most popular radiation boundary conditions for the Helmholtz equation in exterior 3-D regions has been the sequence of operators developed by Bayliss et al.1 for computational domains with spherical exterior boundaries. The present paper extends those spherical operators to triaxial ellipsoidal boundaries by utilizing two mathematical constructs originally developed for ellipsoidal acoustic infinite elements.2 The two constructs are: (i) a radial/angular coordinate system for ellipsoidal geometry, and (ii) a convergent ellipsoidal radial expansion for exterior fields, analogous to the classical spherical multipole expansion. The ellipsoidal radial and angular coordinates are smooth generalizations of the traditional radial and angular coordinates used in spherical, prolate spheroidal and oblate spheroidal systems. As a result, all four coordinate systems and their corresponding radiation boundary conditions are included within this single ellipsoidal system, varying smoothly from one to the other. The geometric flexibility of this system enables the exterior boundary of the computational domain to closely circumscribe objects with a wide range of aspect ratios, thereby reducing the size and cost of 3-D computational models.

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Li, Xiao Cong, Tao Zheng, Zhi Jian Liang, and Jun Hua Xu. "Multi-Index Nonlinear Coordinated Control of TCBR and Hydro-Generator Excitation." Applied Mechanics and Materials 704 (December 2014): 199–203. http://dx.doi.org/10.4028/www.scientific.net/amm.704.199.

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As a new FACTS device, thyristor controlled braking resistor (TCBR) is one of the effective devices to enhance transient stability of generator. In this paper, a coordinated controller for TCBR and hydro-generator excitation system with the multi-index nonlinear coordinate control method based differential geometry theory is proposed. By means of Hartman-Grobman theorem, differential geometry multi-index nonlinear control (DGMINC) design method can reassign the closed-loop system eigenvalues of linear approximate system to the nonlinear system via appropriately selecting output function parameter matrix. Therefore, the system can get good control performance. The simulation results show that TCBR and generator excitation system controlled by this coordinate control strategy can decrease the generator acceleration area rapidly when the severe fault occurred and increase deceleration area significantly after the fault is cleared. So that, power system transient stability limitation improved significantly.

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BELLUCE, LAWRENCE P., ANTONIO DI NOLA, and GIACOMO LENZI. "ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS." Journal of Symbolic Logic 79, no. 4 (December 2014): 1061–91. http://dx.doi.org/10.1017/jsl.2014.53.

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AbstractIn this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

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Hua, Rong, Xiao-Liu Wu, and Jin-Ying Li. "Bis[2,6-bis(1H-pyrazol-1-yl)pyridine]decakis(μ2-3-nitrobenzoato)bis(3-nitrobenzoato)tetradysprosium(III): a linear tetranuclear dysprosium compound based on mixed N- and O-donor ligands." Acta Crystallographica Section E Structure Reports Online 70, no. 5 (April 2, 2014): m162—m163. http://dx.doi.org/10.1107/s1600536814006060.

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The title compound, [Dy4(C7H4NO4)12(C11H9N5)2] or Dy4(L1)12(L2)2, where HL1 = 3-nitrobenzoic acid and HL2 = 2,6-bis(1H-pyrazol-1-y1)pyridine, is a linear tetranuclear complex possessing inversion symmetry. The two central inversion-related DyIIIatoms are seven-coordinate, DyO7, with a monocapped triangular-prismatic geometry. The outer two DyIIIatoms are eight-coordinate, DyO5N3, with a bicapped triangular-prismatic geometry. The outer adjacent DyIIIatoms are bridged by threeL1−carboxylate groups, while the inner inversion-related DyIIIatoms are bridged by fourL1−carboxylate groups. TheL2 ligands are terminally coordinated to the outer DyIIIatoms in a tridentate manner. In the crystal, molecules are linkedviaC—H...O hydrogen bonds, forming a two-dimensional network parallel to (001). Two carboxylate O atoms, and N and O atoms of three nitro groups, are disordered over two positions, with a refined occupancy ratio of 0.552 (6):0.448 (6).

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Knutson, Allen, Thomas Lam, and David E. Speyer. "Positroid varieties: juggling and geometry." Compositio Mathematica 149, no. 10 (August 19, 2013): 1710–52. http://dx.doi.org/10.1112/s0010437x13007240.

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AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

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CORICHI, ALEJANDRO, and JOSÉ A. ZAPATA. "QUANTUM STRUCTURE OF GEOMETRY: LOOPY AND FUZZY?" International Journal of Modern Physics D 17, no. 03n04 (March 2008): 445–51. http://dx.doi.org/10.1142/s0218271808012115.

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In any attempt to build a quantum theory of gravity, a central issue is to unravel the structure of space–time at the smallest scale. Of particular relevance is the possible definition of physical coordinate functions within the theory and the study of their algebraic properties, such as noncommutativity. Here we approach this issue from the perspective of loop quantum gravity and the picture of quantum geometry that the formalism offers. In particular, as we argue here, this emerging picture has two main elements: (i) the nature of the quantum geometry at the Planck scale is one-dimensional, and polymeric with quantized geometrical quantities; and (ii) appropriately defined operators corresponding to coordinates by means of intrinsic, relational constructions become noncommuting. This particular feature of the operators, which operationally localize points on space, gives rise to an emerging geometry that is also, in a precise sense, fuzzy.

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Liu, Jian-Jun, Zuo-Yin Li, Xiong Yuan, Yao Wang, and Chang-Cang Huang. "A copper(I) coordination polymer incorporation the corrosion inhibitor 1H-benzotriazole: poly[μ3-benzotriazolato-κ3N1:N2:N3-copper(I)]." Acta Crystallographica Section C Structural Chemistry 70, no. 6 (May 23, 2014): 599–602. http://dx.doi.org/10.1107/s2053229614010390.

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The title complex, [Cu(C6H4N3)]n, was synthesized by the reaction of cupric nitrate, 1H-benzotriazole (BTAH) and aqueous ammonia under hydrothermal conditions. The asymmetric unit contains three crystallographically independent CuIcations and two 1H-benzotriazolate ligands. Two of the CuIcations, one with a linear two-coordinated geometry and one with a four-coordinated tetrahedral geometry, are located on sites with crystallographically imposed twofold symmetry. The third CuIcation, with a planar three-coordinated geometry, is on a general position. Two CuIcations are doubly bridged by two BTA−ligands to afford a noncentrosymmetric planar [Cu2(BTA)2] subunit, and two [Cu2(BTA)2] subunits are arranged in an antiparallel manner to form a centrosymmetric [Cu2(BTA)2]2secondary building unit (SBU). The SBUs are connected in a crosswise mannerviathe sharing of four-coordinated CuIcations, Cu—N bonding and bridging by two-coordinate CuIcations, resulting in a one-dimensional chain along thecaxis. These one-dimensional chains are further linked by C—H...π and weak van der Waals interactions to form a three-dimensional supramolecular architecture.

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Yu, Hefu, and Bo-Qiang Ma. "Unification of gravity and quantum field theory from extended noncommutative geometry." Modern Physics Letters A 32, no. 05 (February 7, 2017): 1750030. http://dx.doi.org/10.1142/s0217732317500304.

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We make biframe and quaternion extensions on the noncommutative geometry, and construct the biframe spacetime for the unification of gravity and quantum field theory (QFT). The extended geometry distinguishes between the ordinary spacetime based on the frame bundle and an extra non-coordinate spacetime based on the biframe bundle constructed by our extensions. The ordinary spacetime frame is globally flat and plays the role as the spacetime frame in which the fields of the Standard Model are defined. The non-coordinate frame is locally flat and is the gravity spacetime frame. The field defined in both frames of such “flat” biframe spacetime can be quantized and plays the role as the gravity field which couples with all the fields to connect the gravity effect with the Standard Model. Thus, we provide a geometric paradigm in which gravity and QFT can be unified.

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Ding, Ing-Jr, and Chong-Min Ruan. "A Study on Utilization of Three-Dimensional Sensor Lip Image for Developing a Pronunciation Recognition System." Journal of Imaging Science and Technology 63, no. 5 (September 1, 2019): 50402–1. http://dx.doi.org/10.2352/j.imagingsci.technol.2019.63.5.050402.

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Abstract The acoustic-based automatic speech recognition (ASR) technique has been a matured technique and widely seen to be used in numerous applications. However, acoustic-based ASR will not maintain a standard performance for the disabled group with an abnormal face, that is atypical eye or mouth geometrical characteristics. For governing this problem, this article develops a three-dimensional (3D) sensor lip image based pronunciation recognition system where the 3D sensor is efficiently used to acquire the action variations of the lip shapes of the pronunciation action from a speaker. In this work, two different types of 3D lip features for pronunciation recognition are presented, 3D-(x, y, z) coordinate lip feature and 3D geometry lip feature parameters. For the 3D-(x, y, z) coordinate lip feature design, 18 location points, each of which has 3D-sized coordinates, around the outer and inner lips are properly defined. In the design of 3D geometry lip features, eight types of features considering the geometrical space characteristics of the inner lip are developed. In addition, feature fusion to combine both 3D-(x, y, z) coordinate and 3D geometry lip features is further considered. The presented 3D sensor lip image based feature evaluated the performance and effectiveness using the principal component analysis based classification calculation approach. Experimental results on pronunciation recognition of two different datasets, Mandarin syllables and Mandarin phrases, demonstrate the competitive performance of the presented 3D sensor lip image based pronunciation recognition system.

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Yu, Hefu, and Bo-Qiang Ma. "Origin of fermion generations from extended noncommutative geometry." International Journal of Modern Physics A 33, no. 29 (October 20, 2018): 1850168. http://dx.doi.org/10.1142/s0217751x18501683.

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We propose a way to understand the three fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e. the emergence of three fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space.

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Pechenin, V. A., M. A. Bolotov, and N. V. Ruzanov. "А Model of Coordinate Measurements of Freeform Surfaces Geometry." Vestnik Tambovskogo gosudarstvennogo tehnicheskogo universiteta 21, no. 4 (2015): 675–85. http://dx.doi.org/10.17277/vestnik.2015.04.pp.675-685.

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Németh, Károly, and Matt Challacombe. "The quasi-independent curvilinear coordinate approximation for geometry optimization." Journal of Chemical Physics 121, no. 7 (August 15, 2004): 2877–85. http://dx.doi.org/10.1063/1.1771636.

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Mathis, P., and P. Schreck. "Coordinate-free geometry and decomposition in geometrical constraint solving." Computer-Aided Design 50 (May 2014): 51–60. http://dx.doi.org/10.1016/j.cad.2014.01.012.

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Shragge, Jeffrey Chilver. "Riemannian wavefield extrapolation: Nonorthogonal coordinate systems." GEOPHYSICS 73, no. 2 (March 2008): T11—T21. http://dx.doi.org/10.1190/1.2834879.

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Riemannian wavefield extrapolation (RWE) is used to model one-way wave propagation on generalized coordinate meshes. Previous RWE implementations assume that coordinate systems are defined by either orthogonal or semiorthogonal geometry. This restriction leads to situations where coordinate meshes suffer from problematic bunching and singularities. Nonorthogonal RWE is a procedure that avoids many of these problems by posing wavefield extrapolation on smooth, but generally nonorthogonal and singularity-free, coordinate meshes. The resulting extrapolation operators include additional terms that describe nonorthogonal propagation. These extra degrees of complexity, however, are offset by smoother coefficients that are more accurately implemented in one-way extrapolation operators. Remaining coordinate mesh singularities are then eliminated using a differential mesh smoothing procedure. Analytic extrapolation examples and the numerical calculation of 2D and 3D Green’s functions for cylindrical and near-spherical geometry validate the nonorthogonal RWE propagation theory. Results from 2D benchmark testing suggest that the computational overhead associated with the RWE approach is roughly 35% greater than Cartesian-based extrapolation.

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Peng, San-Jun, Cong-Shan Zhou, and Tao Yang. "Dichloro{2-[2-(ethylamino)ethyliminomethyl]phenolato}zinc(II)." Acta Crystallographica Section E Structure Reports Online 62, no. 5 (April 29, 2006): m1147—m1149. http://dx.doi.org/10.1107/s1600536806014759.

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In the title mononuclear zinc(II) complex, [ZnCl2(C11H16N2O)], the ZnII atom is coordinated by the imine N and phenolate O atoms of the Schiff base ligand and two chloride anions to give a four-coordinate tetrahedral geometry. In the crystal structure, the molecules are linked through intermolecular N—H...O, N—H...Cl and C—H...Cl hydrogen bonds, forming layers parallel to the ab plane.

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Journal articles on the topic 'Coordinate geometry'

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