Terms Of Cartesian Coordinates Research Articles

Theory of spacetime impetus

This article introduces the theory of spacetime impetus (SI). The theory unites Newtonian theory (NT) and the theory of general relativity (GR). To develop SI, we reformulated NT in spacetime and replaced the particle primitive in NT with the fragment of energy primitive in field theory. SI replaces Newton’s second law F = ma governing the motion of particles, where F, m, and a are, respectively, interaction force, mass, and acceleration, with the change equation P = k governing the motion of fragments of energy, where P and k are, respectively, action force and the curvature of a path in spacetime. To verify SI, we conducted three tests: Test 1 predicted the precession angles of Mercury and Jupiter, test 2 predicted the bending angle of light as it grazes the surface of the sun, and test 3 predicted the radius of the photon sphere. All three tests were in agreement with GR, the third corresponding to strong Riemannian curvature in GR. The equations of motion in SI are in terms of Cartesian coordinates and time and are relatively simple to solve. Undergraduate students in science and engineering and others with similar mathematical skills can validate the results for themselves.

Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar

Multi-target tracking (MTT) generally requires either a network of Doppler radar receivers distributed at different locations or a phased array radar. The targets moving with small/no radial velocity or angular velocity only cannot be detected and localized completely by deploying Doppler radar without antenna arrays or multiple receivers. To resolve this issue, we present a new MTT algorithm based on 2-D velocity measurements, namely, radial and angular velocities, using dual-frequency interferometric radar. The contributions of the proposed research are twofold: First, we introduce the mathematical model and implementation of the proposed algorithm by explicitly establishing the relationship between 2-D velocity measurements and kinematic state of the target in terms of Cartesian coordinates. Based on 2-D velocity measurement function, the proposed MTT algorithm comprises the following steps: (i) data association using global nearest neighbor (GNN) method (ii) target state estimation using interacting multiple model (IMM) estimator combined with square-root cubature Kalman filter (SCKF) (iii) track management using rule-based M/N logic. Second, performance of the proposed algorithm is evaluated in terms of tracking accuracy, computational complexity and IMM mean model probabilities. Simulation results for different scenarios with multiple targets moving in different tracks have been presented to verify the effectiveness of the proposed algorithm.

On a new field theory formulation and a space-time adjustment that predict the same precession of Mercury and the same bending of light as general relativity

This article introduces a new field theory formulation. The new field theory formulation recognizes vector continuity as a general principle and begins with a field that satisfies vector continuity equations. Next, independent of the new formulation, this article introduces a new space-time adjustment. Then, we solve the one-body gravitational problem by applying the space-time adjustment to the new field theory formulation. With the space-time adjustment, the new formulation predicts precisely the same precession of Mercury and the same bending of light as general relativity. The reader will find the validating calculations to be simple. The equations of motion that govern the orbital equations are in terms of Cartesian coordinates and time. An undergraduate college student, with direction, can perform the validations.

Error propagation analysis for a static convergent beam triple LIDAR

We consider the matter of how to assess uncertainty propagation for the converging beam triple LIDAR technology, which is used for measuring wind velocity passing through a fixed point in space. Converging beam triple LIDAR employs the use of three non-parallel, non-coplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order to determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. One aim of this paper is to present the detailed formulae that would be required for the computation of the estimated variances of the reconstructed velocities, calculated on the basis of the error propagation formulae (and to highlight to the practitioner the conditions under which such estimated variances could be relied upon). The other main aim of this paper is to demonstrate that for (various wind profiles and, in particular) certain LIDAR configurations, which can be characterized by an associated parallelepiped with unit edge length, the estimated variances have the potential to be unsuitably large, thus indicating that the reconstructed velocity may be unreliable for gauging the value of the true wind velocity and that the practitioner should avoid such configurations in the measurement campaign.

Generalized modal element method: part-I—theory and its application to eight-node asymmetric and symmetric solid elements in linear analysis

In this paper, a new finite element method, termed the generalized modal element method (GMEM), is proposed. In GMEM, the element stiffness is derived by decomposing element deformation patterns into individual element generalized modes, where different methods are used to construct the generalized modes. Specifically, three different modal construction methods, including analytical method, assumed displacement method and traditional finite element technique, are proposed for developing the element generalized modes. The concept of modal local coordinate systems is also proposed to ensure the element frame invariance when using polynomial displacement functions, which successfully enables one to use the analytical solutions derived from governing differential equations to develop high accuracy element formulations. An asymmetric hexahedral solid element and a symmetric hexahedral solid element are subsequently derived by using GMEM. The displacement functions of the elemental 24 generalized modes are expressed in terms of Cartesian coordinates so that the element behavior is independent of mesh distortions. Furthermore, the first 21 generalized modes are derived from analytical method making the element capable of avoiding common locking phenomena. Several benchmark problems are performed to demonstrate the accuracy and performance of the new element formulations in linear static and frequency analysis.

Linear optimization of antenna array for radiation pattern defined on a planar surface

ABSTRACTIn this contribution, the antenna array synthesis for radiation pattern defined on a planar surface is described. This can be required when the electric field r-decay factor effect cannot be neglected, e.g. an antenna array mechanically tilted and a specific pattern defined in terms of cartesian coordinates. Differently from the classical sidelobe level reduction optimization problem, the case in which both a constant power bounded area and a sidelobe suppression region are defined and required to be synthesized is herein addressed. The general nonlinear optimization problem is first presented; then, the derivation of some sub-optimal linear problems based on practical implementation hypothesis is described, with the advantage of a reduced computational complexity. Numerical results validate the effectiveness of the proposed method.

Geometric generalised Lagrangian-mean theories

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward & Roberts (J. Fluid Mech., vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

JED: a Java Essential Dynamics Program for comparative analysis of protein trajectories

BackgroundEssential Dynamics (ED) is a common application of principal component analysis (PCA) to extract biologically relevant motions from atomic trajectories of proteins. Covariance and correlation based PCA are two common approaches to determine PCA modes (eigenvectors) and their eigenvalues. Protein dynamics can be characterized in terms of Cartesian coordinates or internal distance pairs. In understanding protein dynamics, a comparison of trajectories taken from a set of proteins for similarity assessment provides insight into conserved mechanisms. Comprehensive software is needed to facilitate comparative-analysis with user-friendly features that are rooted in best practices from multivariate statistics.ResultsWe developed a Java based Essential Dynamics toolkit called JED to compare the ED from multiple protein trajectories. Trajectories from different simulations and different proteins can be pooled for comparative studies. JED implements Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) as options to construct covariance (Q) or correlation (R) matrices. Statistical methods are implemented for treating outliers, benchmarking sampling adequacy, characterizing the precision of Q and R, and reporting partial correlations. JED output results as text files that include transformed coordinates for aligned structures, several metrics that quantify protein mobility, PCA modes with their eigenvalues, and displacement vector (DV) projections onto the top principal modes. Pymol scripts together with PDB files allow movies of individual Q- and R-cPCA modes to be visualized, and the essential dynamics occurring within user-selected time scales. Subspaces defined by the top eigenvectors are compared using several statistical metrics to quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes can be generated for both cPCA and dpPCA.ConclusionsJED offers a convenient toolkit that encourages best practices in applying multivariate statistics methods to perform comparative studies of essential dynamics over multiple proteins. For each protein, Cartesian coordinates or internal distance pairs can be employed over the entire structure or user-selected parts to quantify similarity/differences in mobility and correlations in dynamics to develop insight into protein structure/function relationships.

In brief

PAGE 1022 Researchers from the Institute of Image Processing and Pattern Recognition propose a simple algorithm with robust performance. It provides powerful image and video detail enhancement and is particularly useful for practical applications, being easily hardware implemented in field-programmable gate arrays. PAGE 1060 A team from the University of Nantes in France present an accurate model-based method for estimating the position of targets, using a bistatic MIMO radar system. They used factor analysis to estimate components and to precisely locate the targets in terms of Cartesian coordinates. The final enhanced images and videos tend to be more visible and natural looking PAGE 996 Collaboration between the Technical University of Denmark and Chungnam National University in Korea has produced a new circuit layout that enables precise matching of reactive values in small electric and magnetic spherical dipole antennas. The experimental results verified those of the circuit and electromagnetic simulations. GPR and indoor localisation are some typical applications of near field source localisation PAGE 1054 An integrated balancing transformer for a high frequency parallel AC-LED drive circuit is proposed. The team from Gyeongsang National University designed and implemented the transformer, with the experimental results showing the operation and current balancing performance of the proposed scheme. It is expected that the proposed technique will prove useful for high power AC-LED applications. The results of this work can be effectively used in the engineering fields that require small magnetic antennas with wide-band operation PAGE 1045 A team from the Vienna University of Technology suggest an analogue-digital circuit for thermal control of the resonance frequency of optical ring resonators. Compared to a purely analogue heater control, the approach dissipates 7% less power. The approach is appropriate for controlling the individual cells of large optical switch matrices. The engineering application of this work includes high powered LED lighting applications The approach has been successfully verified with a test chip and is very important for the advancement of electronic-photonic integration

Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.

Structure and Bonding Nature of the Strained Lewis Acid 3‐Methyl‐1‐boraadamantane: A Case Study Employing a New Data‐Analysis Procedure in Gas Electron Diffraction

Base-free 3-methyl-1-boraadamantane was synthesized by starting from its known THF adduct, transforming it to a butylate-complex with n-butyllithium, cleaving the cage with acetyl chloride to give 3-n-butyl-5-methyl-7-methylene-3-borabicyclo[3.3.1]nonane and closing the cage again by reacting the latter with dicyclohexylborane. The identity of 3-methyl-1-boraadamantane was proven by (1) H, (11) B and (13) C NMR spectroscopy and elemental analysis. The experimental equilibrium structure of the free 3-methyl-1-boraadamantane molecules has been determined at 100 °C by using gas-phase electron diffraction. For this structure determination, an improved method for data analysis has been introduced and tested: the structural refinement versus gas-phase electron diffraction data (in terms of Cartesian coordinates) with a set of quantum-chemically derived regularization constraints for the complete structure under optimization of a regularization constant, which maximizes the contribution of experimental data while retaining a stable refinement. The detailed analysis of parameter errors shows that the new approach allows obtaining more reliable results. The most important structural parameters are: r(e) (B-C)(av) =1.556(5) Å, angle(e) (C-B-C)(av) =116.5(2)°. The configuration of the boron atom is pyramidal with ∑ angle (C-B-C)=349.4(4)°. The nature of bonding was analyzed further by applying the natural bond orbital (NBO) and atoms in molecules (AIM) approaches. The experimentally observed shortening of the B-C bonds and elongation of the adjacent C-C bonds can be explained by the σ(C-C)→p(B) hyperconjugation model. Both NBO and AIM analyses predict that the B-C bonds are significantly bent in the direction out of the cage.

Nondegenerate superintegrable systems in n-dimensional Euclidean spaces

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained.

Study of floating zone profiles in materials grown by the laser‐heated pedestal growth technique under isostatic atmosphere

AbstractThe analysis of the melt profile in floating zone techniques has been made in the past years by raising differential equations in terms of Cartesian coordinates and numerically looking for its solution. Although this method permits to establish quite good approximations to the real melt profile, the solutions are deficient in the prediction of new information that can be related to the thermophysical properties of most materials. We investigated the applicability of a different approach that starting from a novel differential equation permits its fitting to the profile of a real situation and also the determination of some important growth parameters like the range of shape stability. Results concerning melt profiles of materials grown under controlled atmosphere are reported. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid

An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {λ,ϕ,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10−8 m2/s2 for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10−4m2/s2. Since 1.5× 10−4 m2/s2 is equivalent to 1.5×10−5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.

Development of a new quadrilateral thin plate element using area coordinates

A new quadrilateral thin plate bending element with 12 degrees of freedom, called element ACGCQ, has been developed by employing both the area coordinates and the generalized conforming approach. The deflection field of the element is expressed as a polynomial, in terms of area coordinates, which is corresponding to a fourth order polynomial in terms of Cartesian coordinates x and y. The 12 generalized conforming conditions implemented are four-point and four-line compatibility conditions for the deflection w, and four-line compatibility conditions for the normal slope ψ n . Since the relation between the area and Cartesian coordinates is linear, the formulation of the element stiffness matrix is straight forward. Several numerical examples were employed to test the proposed element. The results obtained show that it possesses the advantage of high accuracy and reliability, rapid convergence and insensitivity to geometric distortion.

Deflection of the vertical and refraction in three-dimensional adjustment of terrestrial networks

In this paper statistical tests are exploited in order to verify the hypotheses about the refraction and the deflection of the vertical pertaining to a geometrical model for the three-dimensional adjustment of terrestrial networks. The deflections of the vertical and the refraction coefficients can be assumed either as unknowns or fixed input data, at some or all the points of the network. The geometrical model, reported in the appendix for convenience, assumes as observables the slant distances, zenith and horizontal angles, without any reduction neither to the marks on the ground nor to the surface of reference. Further, the observation equations are derived and linearized in terms of Cartesian coordinates in Geocentric or Topocentric system; direction cosines of the vertical and of the ellipsoidal normal are adopted as the relevant direction parameters. Finally, an application to a network from Hradilek (1984), performed under different assumptions about the unknowns and the corrections of the angular observations due to the deflections of the vertical, shows the effectiveness of the proposed approach.

Free Vibration of a Class of Homogeneous Isotropic Solids

A Ritz approach, with simple polynomials as trial functions, is used to obtain the natural frequencies of vibration of a class of solids. Each solid is modeled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx, and xy orthogonal coordinate planes as well as by a fourth curved surface, which is defined by a polynomial expression in the coordinates x, y, and z. By exploiting symmetry, a number of three-dimensional solids previously considered in the open literature are treated, including a sphere, a cylinder and a parallelepiped. The versatility of the approach is then demonstrated by considering several solids of greater geometric complexity, including an ellipsoid, an elliptical cylinder, and a cone.

A higher order theory for vibration of shear deformable cylindrical shallow shells

The vibratory characteristics of thick cylindrical shallow shells of rȩctangular planform is investigated in this paper. A higher order shear deformation theory is proposed to analyse the effects of various shell geometries and boundary conditions on the vibration responses. The present higher order theory results in cubic expressions for the transverse shear distribution through shell thickness. The strain and kinetic integral energy expressions involving higher order shear deformation are derived in terms of Cartesian coordinates. An energy functional based on the principle of extremum energy is employed which yields a governing eigen-matrix equation. A set of orthogonally generated two-dimensional polynomial functions is adopted to approximate the deflections and rotations of the shells. The results, where possible, are compared with the available experimental and existing numerical results. Sets of new results for the hard simply supported and fully clamped cylindrical shallow shells are presented. Selected modes are illustrated in three-dimensional displacement plots.

Free vibration of a class of solids with cavities

The Ritz method is used to obtain an eigenvalue equation for the free vibration of a class of solids. Each solid is modelled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx and xy orthogonal coordinate planes as well as by two curved surfaces which are defined by polynomial expressions in the coordinates x, y and z. Simple algebraic polynomials which satisfy the boundary conditions at the five surfaces of the segment are used as trial functions. By exploiting symmetry, the range of problems which can be treated is substantially broadened and includes a variety of problems of significant interest in structural analysis, such as thick or very thick shells with various shaped cavities. In order to demonstrate the accuracy of the approach, natural frequencies are given for a sphere with a spherical cavity (a thick spherical shell) as calculated by using the present analysis and by using an exact formulation. The versatility of the approach is then demonstrated by the treatment of several other hollow solids of differing geometry, including a thick cylinder with end plates, a cubic box, a cube with a spherical cavity and a cylinder with a conical inclusion.