Cartesian Reference Frame Research Articles

Terrestrial monitoring of a radio telescope reference point using comprehensive uncertainty budgeting

During the 15-day-long global very long baseline interferometry campaign CONT14, a terrestrial monitoring campaign was carried out at the Onsala Space Observatory. The goal of these efforts was to monitor the reference point of the Onsala 20 m radio telescope during normal telescope operations. Parts of the local site network as well as a number of reflectors that were mounted on the 20 m radio telescope were observed in an automated and continual way using the in-house-developed software package HEIMDALL. The analysis of the observed data was performed using a new concept for a coordinate-based network adjustment to allow the full adjustment process in a true Cartesian global reference frame. The Akaike Information Criterion was used to select the preferable functional model for the network adjustment. The comprehensive stochastic model of this network adjustment process considers over 25 parameters, and, to describe the persistence of the observations performed during the monitoring with a very high measurement frequency, includes also time-dependent covariances. In total 15 individual solutions for the radio telescope reference point were derived, based on monitoring observations during the normal operation of the radio telescope. Since the radio telescope was moving continually, the influence of timing errors was studied and considered in the adjustment process. Finally, recursive filter techniques were introduced to combine the 15 individual solutions. Accuracies at the sub-millimeter level could be achieved for the radio telescope reference point. Thus, the presented monitoring concept fulfills the requirement proposed by the global geodetic observing system.

Invariant quantities of a Mueller matrix under rotation and retarder transformations

Mueller matrices are defined with respect to appropriate Cartesian reference frames for the representation of the states of polarization of the input and output electromagnetic probe beams. The polarimetric quantities that are invariant under rotations of the said reference frames about the respective directions of propagation (rotation transformations) provide particularly interesting physical information. Moreover, certain properties are also invariant with respect to the action of birefringent devices located on both sides of the medium under consideration (retarder transformations). The polarimetric properties that remain invariant under rotation and retarder transformations are calculated from any given Mueller matrix and are then analyzed and interpreted, providing significant parameterizations of Mueller matrices in terms of meaningful physical quantities.

Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model

We use a new exploratory model that simulates the evolution of sandy coastlines over decadal to centennial timescales to examine the behavior of crenulate-shaped bays forced by differing directional wave climates. The model represents the coastline as a vector in a Cartesian reference frame, and the shoreface evolves relative to its local orientation, allowing simulation of coasts with high planform-curvature. Shoreline change is driven by gradients in alongshore transport following newly developed algorithms that facilitate dealing with high planform-curvature coastlines. We simulated the evolution of bays from a straight coast between two fixed headlands with no external sediment inputs to an equilibrium condition (zero net alongshore sediment flux) under an ensemble of directional wave climate conditions. We find that planform bay relief increases with obliquity of the mean wave direction, and decreases with the spread of wave directions. Varying bay size over 2 orders of magnitude (0.1–16 km), the model predicts bay shape to be independent of bay size. The time taken for modeled bays to attain equilibrium was found to scale with the square of the distance between headlands, so that, all else being equal, small bays are likely to respond to and recover from perturbations more rapidly (over just a few years) compared to large bays (hundreds of years). Empirical expressions predicting bay shape may be misleading if used to predict their behavior over planning timescales.

English

This paper examines a consequence of spinodal curve condition on the Joule – Thomson inversion curves, for van der Waals real gas. To this end, we consider the differential form of Joule – Thomson coefficient, and concentrate on the parametrical algebraic equation of the corresponding family of curves in a P – V Cartesian frame of reference. The aim of this work is not the prediction of Joule – Thomson inversion curves for the van der Waals Eos, but to derive inequality relations amongst the variables T, V and the parameters which appear in this constitutive law. Here, we should clarify beforehand that all proposed expressions concern only the intersection points between the set of inversion curves and the isothermal spinodal lines, which are drawn in the same P – V frame of reference. Key words: JT inversion curves, spinodal curve condition, van der Waals Eos, P – V frame of reference.

A Generic Method for Azimuthal Map Projection

Map projections are mathematical methods for projecting spherical coordinates in the form of (φ, λ) to the map coordinates in the form of (X,Y) in Cartesian reference frame. Numerous methods for map projection have been derived and are being used for preparation of cartographic products. These map projections take into account specific position of the viewer on the datum surface for derivation of the map projections. A generic method for azimuthal map projection where the position of the viewer can be taken at an arbitrary point on the datum surface is derived. Using this generic method all the specific azimuthal map projections can be derived.

Pseudo steady state filters for target tracking with polar measurements

Target tracking under the linear and Gaussian assumption is a well known problem. In this case, the optimal state estimate is delivered by the standard Kalman filter. However, for real world systems, this assumption is not satisfied. A commonly encountered problem for the tracking community is: target tracking in a Cartesian frame of reference, with measurements delivered by radar in polar coordinates. In this paper we propose two novel static versions of the Debiased Converted Measurements Kalman Filter (DCMKF) and the Unscented Kalman Filter (UKF), that use the well known αβ filter. The proposed filters are: the Debiased Converted Measurement αβ Filter (DCMαβ) and the Unscented αβ filter (Uαβ). The RMSE calculated by means of Monte Carlo simulations is used to assess the performances of these new filters. The simulations results show that, the proposed filters are much simpler and thus 60% less computational time demanding and have acceptable RMSE compared to that of the classical ones. These filters are also suitable for parallel implementation, which makes them potential candidates for real time applications on embedded systems.

Helical vortex filament motion under the non-local Biot–Savart model

AbstractThe thin helical vortex filament is one of the fundamental exact solutions possible under the local induction approximation (LIA). The LIA is itself an approximation to the non-local Biot–Savart dynamics governing the self-induced motion of a vortex filament, and helical filaments have also been considered for the Biot–Savart dynamics, under a variety of configurations and assumptions. We study the motion of such a helical filament in the Cartesian reference frame by determining the curve defining this filament mathematically from the Biot–Savart model. In order to do so, we consider a matched approximation to the Biot–Savart dynamics, with local effects approximated by the LIA in order to avoid the logarithmic singularity inherent in the Biot–Savart formulation. This, in turn, allows us to determine the rotational and translational velocity of the filament in terms of a local contribution (which is exactly that which is found under the LIA) and a non-local contribution, each of which depends on the wavenumber, $k$, and the helix diameter, $A$. Performing our calculations in such a way, we can easily compare our results to those of the LIA. For small $k$, the transverse velocity scales as $k^{2}$, while for large $k$, the transverse velocity scales as $k$. On the other hand, the rotational velocity attains a maximum value at some finite $k$, which corresponds to the wavenumber giving the maximal torsion.

Alignment-free quantum key distribution by postselection

We propose an alignment-free photonic qubit transmission scheme and apply it in quantum key distribution schemes, in which the authorized parties do not have to calibrate their Cartesian reference frames in advance. The polarization states are labeled by time, and the receiver can obtain the original state in his own coordinate by postselecting a determinate time of arrival. The scheme does not resort to decoherence-free subspace based on multiple photon states, and it can be implemented with only passive linear optics, which make it efficient and practical with current technology. We demonstrate its application in quantum key distributions, and the scheme should find applications in long-distance quantum communication without frame alignment.

Terminal bang-bang impulsive control of linear time invariant dynamic systems

For a linear time-invariant dynamic system with bang-bang impulsive actuators whose position (off, on negative, or on positive) is determined by the switching points in time, an algorithm is developed that makes it possible to use methods of modal control for solving the terminal control problem. This algorithm is used to control the rendezvous of space vehicles with constant thrust engines arranged along the axes of a Cartesian reference frame.

A new approach to soil classification mapping based on the spatial distribution of soil properties

This study presents a new approach to classifying types of soil based on the probability classes of the relevant set of attributes. Two key ideas are addressed in this study: (i) the use of stochastic simulations to generate a local cumulative distribution function or extreme classes of each attribute and (ii) the use of a multidimensional scaling (MDS) technique to visualize and quantify the relative importance of each attribute in the classification process. After the simulated realizations, the weighted “distances” attributes extreme values (probability classes) of each grid node are calculated and the MDS algorithm is applied for the spatial representation of the grid nodes in a new Cartesian reference frame based on the “distances” of the probability classes of attributes. This allows the classification of soil types based on the clusters in the MDS space, after expert validation. In the second step, a sensitivity analysis of the attributes is performed with MDS: each attribute is made “neutral” one at a time, by assuming the median rather than the extreme values in each grid node before the distance evaluation, and the consequent impact on the shape and centroid displacement of the clusters (soil types) in the MDS reference frame is calculated. Hence, the spatial uncertainty of the soil type/classes and the influence of various properties are evaluated in the MDS reference frame. This method is applied to soils in a region of Brazilian in which the previous classification of soil types has been a crucial tool for precision agriculture management. Using the MDS algorithm, the selected attributes (horizon, textural gradient, colors, saturation, sand content, and clay content) were represented in a two-dimensional plot and grouped into eight clusters distinguished from each other by their characteristics. A sensitivity analysis shows that the horizon and saturation attributes had the greatest influence on determination of the clusters, i.e., the soil types.

Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall-Vinen model including superfluid friction

We determine conditions under which the fully nonlinear form of the local induction approximation (LIA) governing the motion of a vortex filament in superfluid 4He (that is, the Hall-Vinen model) in the Cartesian frame of reference permits the existence of self-similar solutions, even in the presence of superfluid friction parameters. Writing the Cartesian Hall-Vinen LIA in potential form for the motion of a vortex filament, we find that a necessary condition for self-similarity is that the normal-fluid component vanishes (which makes sense in the low temperature limit), and we reduce the potential form of the Hall-Vinen LIA to a complex nonlinear ordinary differential equation governing the behavior of a similarity solution. In the limit where superfluid friction parameters are negligible, we provide some analytical and asymptotic results for various regimes. While such analytical results are useful for determining the qualitative behavior of the vortex filament in the limit where superfluid friction parameters vanish, numerical simulations are needed to determine the true behavior of the filaments in the case of non-zero superfluid friction parameters. While the superfluid friction parameters are small, the numerical results demonstrate that the influence of the superfluid friction parameters on the self-similar vortex structures can be strong. We classify two types of filaments from the numerical results: singular filaments (which demonstrate growing oscillations and hold kink-type solutions as a special case) and bounded filaments (whose behavior is a bounded function of x). We also comment on how to include the case where there is a non-zero normal fluid, and we find a transformation of the self-similar solutions into non-similar solutions that can account for this.

Oblique circular torus, Villarceau circles, and four types of Bennett linkages

The oblique circular torus (OCT) and its main geometric properties are introduced. Intrinsic vector calculation is utilized to mathematically describe the OCT. The coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame. The OCT equation is used to confirm a theorem of Euclidean geometry. In a broad category of OCT, through any point five circles can be drawn on the surface, namely the parallel of latitude and four circular generatrices whose planes pass through the OCT center of symmetry. In the special case of a right circular torus, the Villarceau theorem is verified. Next, consider the four RRS open chains whose S spherical-joint centers move on the same OCT and their possible in-parallel assemblies in single-loop RRRS chains. From a category of the foregoing RRRS chains, a new derivation of the amazing Bennett 4R linkage is proposed. Two kinds of Bennett linkages are further verified and each kind contains two enantiomorphic or symmetric linkages. Four types of Bennett linkages associated with one OCT are established by uniquely specifying the link twist as an acute value. Two cases of special type, rectangular and equilateral configurations, are also confirmed.

Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation

We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.

Three-dimensional flow visualization and vorticity dynamics in revolving wings

We investigated the three-dimensional vorticity dynamics of the flows generated by revolving wings using a volumetric 3-component velocimetry system. The three-dimensional velocity and vorticity fields were represented with respect to the base axes of rotating Cartesian reference frames, and the second invariant of the velocity gradient was evaluated and used as a criterion to identify two core vortex structures. The first structure was a composite of leading, trailing, and tip-edge vortices attached to the wing edges, whereas the second structure was a strong tip vortex tilted from leading-edge vortices and shed into the wake together with the vorticity generated at the tip edge. Using the fundamental vorticity equation, we evaluated the convection, stretching, and tilting of vorticity in the rotating wing frame to understand the generation and evolution of vorticity. Based on these data, we propose that the vorticity generated at the leading edge is carried away by strong tangential flow into the wake and travels downwards with the induced downwash. The convection by spanwise flow is comparatively negligible. The three-dimensional flow in the wake also exhibits considerable vortex tilting and stretching. Together these data underscore the complex and interconnected vortical structures and dynamics generated by revolving wings.

Mechanics of curved surfaces, with application to surface-parallel cracks

[1] The surfaces of many bodies are weakened by shallow enigmatic cracks that parallel the surface. A re-formulation of the static equilibrium equations in a curvilinear reference frame shows that a tension perpendicular to a traction-free surface can arise at shallow depths even under the influence of gravity. This condition occurs if σ11k1 + σ22k2 > ρg cosβ, where k1 and k2 are the principal curvatures (negative if convex) at the surface, σ11 and σ22 are tensile (positive) or compressive (negative) stresses parallel to the respective principal curvature arcs, ρ is material density, g is gravitational acceleration, and β is the surface slope. The curvature terms do not appear in equilibrium equations in a Cartesian reference frame. Compression parallel to a convex surface thus can cause subsurface cracks to open. A quantitative test of the relationship above accounts for where sheeting joints (prominent shallow surface-parallel fractures in rock) are abundant and for where they are scarce or absent in the varied topography of Yosemite National Park, resolving key aspects of a classic problem in geology: the formation of sheeting joints. Moreover, since the equilibrium equations are independent of rheology, the relationship above can be applied to delamination or spalling caused by surface-parallel cracks in many materials.

Does gravity operate between galaxies? Observational evidence re-examined

The redshifts and luminosities of type 1A supernovae are conventionally fitted with the current paradigm, which holds that the galaxies are locally stationary in an expanding metric. The fit fails unless the expansion is accelerating; driven perhaps by ‘dark energy’. Is the recession of the galaxies slowed down by gravity or speeded up by some repulsive force? To shed light on this question the redshifts and apparent magnitudes of type 1A supernovae are re-analysed in a cartesian frame of reference omitting gravitational effects. The redshift is ascribed to the relativistic Doppler effect which gives the recession velocity when the light was emitted; if this has not changed, the distance reached and the luminosity follow immediately. This simple concept fits the observations surprisingly well with the Hubble constant H 0 =62.9±0.3 km s −1 Mpc −1 . It appears that the galaxies recede at unchanging velocities, so on the largest scale there is no significant intergalactic force. Reasons for the apparent absence of an intergalactic force are discussed.

Ambiguous figures and representationalism

Macpherson (Nous 40(1):82–117, 2006) argues that the square/regular diamond figure threatens representationalism, construed as the theory which holds that the phenomenal character is explained by the nonconceptual content of experience. Her argument is the claim that representationalism is committed to the thesis that differences in the experience of ambiguous figures, the gestalt switch, should be explained by differences in the NCC of perception of these figures. However, with respect to the square/regular diamond and some other ambiguous figure representationalism fails to offer a unified account of how representational content makes them ambiguous. In this paper, I aim, first, to offer a representationalist account of ambiguous figures and, second, to examine and rebut Macpherson’s arguments. My main point is that in each ambiguous figure Macpherson discusses there are differences in representational content that can explain differences in phenomenal character or content. The representational differences are due to the ways the Cartesian frame of reference in which perceptual content is always cast cuts the figure, underlying different properties of the figure with respect to the axes of the Cartesian frame of reference.

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various problems of satellite geodesy and celestial mechanics.

Extending the time scale in molecular dynamics simulations: Propagation of ripples in graphene

We show that it gives an exact solution of the temporal part of the MD equations for up to quadratic terms in atomic displacements in the interatomic potential. Higher-order terms are then included by iteration. We will refer to our technique as Green’s function in molecular dynamics GFMD. This technique should be applicable to all physical, chemical, and biological systems where MD is used. In certain class of problems in which the atoms vibrate about an equilibrium site, GFMD gives exact results in the harmonic approximation. Examples of such class of problems are phonon transport, thermal conduction etc., in disordered or finite solids and low-dimensional material systems for which MD has to be used because analytical solutions are not available even in the harmonic approximation. For nonlinear vibration problems, depending upon the anharmonicity, GFMD can accelerate MD by about 8 orders of magnitude, and model processes at microseconds. In other classes of problems in which the atoms are itinerant, such as diffusion or crystal growth, GFMD can be used iteratively and should still accelerate MD by a significant amount. Consider a set of N interacting atoms. We label the atoms by indices L and L. We assume a Cartesian frame of reference and denote the position vector of atom L by rL at t = 0. We denote the displacement and velocity of atom L at time t by uL ,t and cL ,t, respectively. As in classical MD, we need to solve the following equation for u:

The neighbor list algorithm for a parallelepiped box in molecular dynamics simulations

In classic molecular dynamics (MD) simulations, the conventional Verlet table, cell linked list and many other techniques have been adopted to increase the computational efficiency. However, these methods are only applicable in cubic systems. In this work, the above techniques along with the metric-tensor method are extended to handle NP ensembles, so that MD simulations can be carried out under the most general loading conditions. In order to do so, a particular spatial Cartesian reference frame is proposed to determine the scaling matrix. Also, a combination method, taking the advantages of the improved Verlet table and cell linked list, is established to identify the neighbor atoms very quickly in a parallelepiped box. An example using Lennard-Jones potential is presented to verify the validity of the proposed algorithm.