Cartesian Reference Frame Research Articles

The use of special coordinate axes in direct and semi-direct implementations of second-order perturbation theory, including the derivation of a horizontal recurrence relation

A horizontal recurrence relation (HRR) is derived which may be used in the generation of two-electron repulsion integrals (ERIs). Also, special Cartesian coordinate axes may be used to reduce the number of intermediate terms. Heretofore, the use of local axis systems entailed rotations every time a shell quadruple of ERIs was computed. Recently, we pointed out the possibility of postponing the transformation using choices of Cartesian reference frames based on nuclear positions. In our pilot implementation of direct and semi-direct Møller–Plesset second-order perturbation theory (MP2) we construct the integrals in local coordinate systems and perform the back-transformation in the middle of the AO to MO integral transformation, rather than rotating the ERIs after the computation of each shell quadruple. The efficacy of this approach depends on whether the gains in ERI computation are greater than the losses incurred in transformation. Preliminary CPU profiles indicate that the increased time spent in rotating axes is quite small. Keywords: horizontal recurrence relation, second-order perturbation theory.

Global Warming as a Manifestation of a Random Walk

Global and hemispheric series of surface temperature anomalies are examined in an attempt to isolate any specific features of the structure of the series that might contribute to the global warming of about 0.5°C which has been observed over the past 100 years. It is found that there are no significant differences between the means of the positive and negative values of the changes in temperature from one year to the next; neither do the relative frequencies of the positive and negative values differ from the frequencies that would be expected by chance with a probability near 0.5. If the interannual changes are regarded as changes of unit magnitude and plotted in a Cartesian frame of reference with time measured along the x axis and yearly temperature differences along the y axis, the resulting path closely resembles the kind of random walk that occurs during a coin-tossing game. We hypothesize that the global and hemispheric temperature series are the result of a Markov process. The climate system is subjected to various forms of random impulses. It is argued that the system fails to return to its former state after reacting to an impulse but tends to adjust to a new state of equilibrium as prescribed by the shock. This happens because a net positive feedback accompanies each shock and slightly alters the environmental state.

On the orthogonality of surface wave eigenfunctions in cylindrical coordinates

SUMMARY The orthogonality of the Rayleigh wave eigenfunctions in laterally homogeneous, plane-stratified media is guaranteed by the structure of the ordinary differential equations describing elastic wave motion in cylindrical coordinates. This coupled first-order system is identical to that which characterizes 2-D plane wave propagation in a Cartesian coordinate reference frame. The orthogonality relation in 2-D can also be derived from energy considerations; however, an analogous argument in cylindrical coordinates has not hitherto been made. We derive the orthogonality relations for Rayleigh waves in 3-D from energy considerations and demonstrate that the standard (2-D) expression is, in fact, the generalization of a slightly more specific form. In addition, the cylindrical coordinate formulation permits the derivation of a functional orthogonality relation between Love and Rayleigh waves. The normalization of Love and Rayleigh wave eigenfunctions in cylindrical coordinates is shown to be related to the energy transport of a given outgoing Fourier-Bessel component across any surface which wholly encompasses the z-axis.

Review of relationships between different strain tensors

The relationships between different Lagrangian strain tensors are summarized on the basis of an illustrative tensor diagram emphasizing the equivalence of the definitions of lattice deformation in the crystallographic and the Cartesian reference frames.

A system for robot manipulation of electrical wires using vision

SUMMARYA prototype robot system for automated handling of flexible electrical wires of variable length is described. The handling process involves the selection of a single wire from a tray of many, grasping the wire close to its end with a robot manipulator, and either placing the end in a crimping press or, for tinning applications, dipping the end in a bath of molten solder. This system relies exclusively on the use of vision to identify the position and orientation of the wires prior to their being grasped by the robot end-effector. Two distinct vision algorithms are presented. The first approach utilises binary imaging techniques and involves object segmentation by thresholding followed by thinning and image analysis. An alternative general-purpose approach, based on more robust grey-scale processing techniques, is also described. This approach relies in the analysis of object boundaries generated using a dynamic contour-following algorithm. A simple Robot Control Language (RCL) is described which facilitates robot control in a Cartesian frame of reference and object description using frames (homogeneous transformations). The integration of this language with the robot vision system is detailed, and, in particular, a camera model which compensates for both photometric distortion and manipulator inaccuracies is presented. The system has been implemented using conventional computer architectures; average sensing cycle times of two and six seconds have been achieved for the grey-scale and binary vision algorithms, respectively.

An anatomy-based coordinate system for the description of the kinematic displacements in the human knee

The effects and interaction of the anatomical displacements in the human knee are a prerequisite to an accurate assessment and communication of the kinematic data. For the kinematic information to be used to improve diagnosis and treatment, and for better prosthetic design and installation, there must be clear, concise, and universal definitions of the displacements. In general, the displacements are defined as three translations and three rotations. In this paper, anatomic landmarks on the femur and on the tibia are used to define the locations and orientations of the six displacement axes; i.e. three translational and three rotational displacement axes. The most commonly accepted kinematic representation of the knee joint, in the literature, is a special geometry three-cylindric open chain in which the axes of the cylindric joints are defined according to the rotational displacement axes. The sequentially adjacent joint axes are assumed to not only intersect but to intersect at right-angles. The open chain permits a total of six degrees of freedom between a Cartesian reference frame attached to the femur and a Cartesian reference frame attached to the tibia. In this paper, the three rotational axes are shown to be skewed and off-set from each other, therefore, a three-cylindric open chain with skewed joint axes is proposed to measure the six displacements between the two reference frames. The authors believe that the proposed open chain is the most general to date and provides a more realistic representation of the displacements in the knee. To illustrate the significance of the reference frames on the interpretation of measured data, the anterior/posterior drawer is plotted against per cent gait cycle for three existing open chains and the proposed open chain.

Determining the representative composition of a set of sandstone samples

AbstractThe description of sandstone samples, defined by three components, is introduced through non-parametric sample density description. The composition of a particular sample defines a direction from the origin of the Cartesian reference frame. Radial difference between individual samples then provides the measure of their similarity. A sample set can be viewed as a bundle of compositional axes with a corresponding density distribution on both the ternary diagram (or unit constant sum plane) and the positive octant of the unit sphere. Because radial differences are disproportionately displayed on the ternary diagram, radial density calculations on the sphere are to be preferred. The mode of the sample set, the implied centre of greatest data concentration, is considered to be the most meaningful representative value, as the component joint mean can lie outside the data field with certain data configurations. In addition, the component joint mean cannot provide a useful representative composition for samples drawn from a bimodal distribution. Sample density distribution not only provides a data defined approach to describe sample dispersion, but also furnishes an indication of sample sufficiency in displaying the degree to which modes are supported by surrounding observations.

Angles measure compositional differences

The geometry of petrological sample space shows that a rock sample, defined by a particular composition or proportion of components, is uniquely described by a radial direction (= compositional axis) from the origin of a Cartesian reference frame where amounts of each component are measured along the axes. The original measured amounts determine the direction of the compositional axis defining a particular mixture of components. Normalization (i.e., expressing amounts as proportions of the whole) merely provides radial projection of the sample onto a proportions frame (three components = the ternary diagram) to permit a form of comparison of different compositions. Such comparisons become inconsistent when traditional parametric approaches are used because, as the constant sum problem has long revealed, numbers expressing proportions or percentages are not amenable to standard statistical methods. In addition, because of the disproportionate display of radial difference over the ternary diagram, it cannot be used as a basis for reliable calculations. Recognition of the fundamental radial character of compositional data reveals that they should, most logically, be analyzed on the sphere. Not only does such an approach bypass previously recognized problems in statistical treatment of compositional data, but also it opens the way for the application to petrological data of a whole class of established parametric and nonparametric statistical methods.

Perceptual organization as nested control.

It is argued here that perceptual organization is the structuring of environmental stimuli into nested structures of control. This view is derived from our proposal that the perceptual system forces any stimulus set to have the algebraic structure G of a machine, and that G is given a specific factorization sequence, G = G1 X G2 X ... X Gn, induced by dynamical systems criteria which the organism imposes on the environment. An investigation into these factorization sequences reveals that any such sequence is split into two subsequences, one structuring the way in which the stimulus set can change under external action, and the other describing the way in which the set is perceived as internally generated. An examination of the nested control structure of the latter yields a theory of grouping. The external and internal sequences are shown to be strongly related to each other in that the symmetry axes of the internal sequence are eigenspaces of the most stable subgroup factors of the external sequence. It is claimed that cartesian reference frames are subsequences of the full sequences. Using these principles, a unified theory is offered of several apparently quite separate perceptual areas; e.g. Marr-Nishihara shape perception, gestalt grouping, the orientation-and-form problem, and motion perception. The final claim is that planning hierarchies have the same dynamically-structured algebraic sequences and therefore that the study of perceptual organization should, in a very deep sense, be formally equivalent to the study of planning.

Dielectric currents in the low‐latitude boundary layer and geomagnetic tail

We consider a collisionless plasma with E (the electric field) parallel to the y‐axis of a cartesian frame of reference and B (the magnetic induction) parallel to the z‐axis. When there is a gradient, orthogonal to B, of the quantity ϵE², where ϵ = 1 + 4πρm c²/B², where ρm is the plasma mass density and c the speed of light, there is driven a (dielectric) current (j) given in the following two cases, by urn:x-wiley:00948276:media:grl2299:grl2299-math-0001 (i) When ▽(ϵE²) is parallel to E, j is in the direction of E × B, and (ii) when ▽(ϵE²) is in the direction E × B, j is in the direction of E. In the former case it follows that, in a steady state, urn:x-wiley:00948276:media:grl2299:grl2299-math-0002 along a line in the y‐direction, and in the latter case urn:x-wiley:00948276:media:grl2299:grl2299-math-0003 on a line in the x‐direction where p⊥ is the pressure perpendicular to B. The currents related to ϵE² are geophysically significant in the region of interaction of the solar wind and geomagnetic field in the low latitude boundary layer and geomagnetic tail. The last written equation enables a realistic estimate of the cross‐tail electric field to be made.

The Intrinsic Geometrical and Physical Properties of Plane Steady Gas Flow

The main purpose of the present paper is to establish a relation between the intrinsic geometrical and physical properties of plane steady gas flow, i.e., between the geometry of the flow pattern on one side and the flow variables on the other. This is accomplished by complete integration of the governing fluid dynamics equations in an a priori unknown system of curvilinear coordinates. The complete solution is presented by explicit expressions for the metric tensor components associated with the curvilinear system in terms of the flow Mach number and the Bernoulli constant along the streamlines. The curvilinear system relative to a Cartesian reference frame is a posteriors determined by quadratures. Several examples including subsonic, sonic, supersonic, rotational, and irrotational flow are discussed to illuminate the special technique and its potential for determining flow patterns that result from particular geometrical and/or physical specifications.

Observer frame rotation rates and magnetic fields in spatially homogeneous universes

We discuss how the rotation of an observer’s Cartesian reference frame is related to the precession of the shear tensor’s principal axes and to the rotation vector of the fluid. An approximate solution of Einstein’s equations illuminates this relationship further. In the case with a magnetic field and fluid flow a vorticity is allowed in Bianchi I cosmologies.

The tidal physics of headland flows and offshore tidal bank formation

The general tidal characteristics in the neighbourhood of a promotory are derived using a numerical model developed in plane polar coordinates. These same tidal features are found for a model of the tides off the Start Point promontory using a Cartesian reference frame. It is shown that promontories are characterised by strong tidal flows and it is suggested that vorticity generated near the headland plays a key role in the dynamics of offshore tidal bank formation.

The Cartesian frame of reference: a structure unifying the description of dyslexia.

It is shown that six phenomena often associated with dyslexia may be attributed to the lack of a visual, Cartesian frame of reference. These phenomena are (1) reading errors due to letter reversal, inversion, or rotation, (2) form recognition of all classes of mono-oriented objects which is independent of the figure's egocentric orientation, (3) defective visual sequential scanning, resulting in confused letter and word order (4) poor visual balance performance, (5) the failure to acquire nonspatial ordering relations, including temporal relations and tenses, and (6) the characteristics intellectual profile of severely retarded readers, namely, PIQ/VIQ greater than 1, where PIQ is the performance IQ score and VIQ the verbal IQ score on the Wechsler Intelligence Scale for Children (WISC). The latter phenomenon is dependent on (2) and (5) above. A strategy for further research is outlined.

Free Molecular Flow over a Rotating Sphere

6= freestream temperature =|5| = the x, y, and z components of the mass velocity of the freestream molecules with respect to the local coordinates, respectively = Cartesian stationary frame of reference = local Cartesian coordinates, fixed on the moving differential area = unit vectors in the x, y, and z directions, respectively = azimuth, measured from negative X axis = mn^, freestream density = momentum accommodation coefficients, Eq. (12) and Eq. (13) — tangential component of momentum flux, Eqs. (6b) and (6c), or shearing stress acting on the differential area, Eqs. (14b) and (14c) = latitude, measured from positive Y axis SubscriptsK = components in the X, Y, and Z directions, respectively i = quantity impinging to the sphere r = quantity reflected from the sphere w = value at sphere surface x, y, z = components in the x, y, and z directions, respectively.

Drawing Vibrational Modes by Computer

Spectroscopists by necessity utilize group theory and normal coordinate treatment (NCT) to describe the normal modes of vibration of molecules. As discussed previously, the normal coordinates can be transformed to a Cartesian reference frame yielding a table of Cartesian displacement coordinates. Using known bond distances and bond angles, the equilibrium Cartesian coordinates can be calculated. However, equilibrium and displacement coordinates are very difficult to comprehend unless the graphical equivalent (or plot) is drawn.

Invariant Piezoelectric Coefficients for Crystals

AbstractIn writing out tensors which represent physical properties of crystals, it is customary to choose a Cartesian frame of reference which has a specific orientation with respect to the crystallographic axes. A new formulation is proposed in which the physical properties (in particular the piezoelectric property) of crystals can be specified without reference to any coordinate system.

The incorporation of spirit-levelling into a three-dimensional cartesian reference frame

Abstract In modern first-order geodetic work it is customary to measure horizontal and vertical angles, spatial distances and astronomic azimuths, latitudes and longitudes. These are projected onto an ellipsoid of reference and are then adjusted to yield positional co-ordinates on that ellipsoid. The results of spirit-levelling, which yield heights above the geoid, are commonly adjusted as a separate entity. Thus the classical method produces two distinct sets of unrelated quantities. A method for rigorously incorporating the results of spirit-levelling into a three-dimensional Cartesian frame of reference is presented. The use of the geoid and of an ellipsoid is avoided.

Angular Dependence of Zeeman Spectra ofGd3+in CaF2at Cubic Sites

The optical Zeeman spectra of $^{6}P_{\frac{5}{2}}$ and $^{6}P_{\frac{7}{2}}$ multiplets of ${\mathrm{Gd}}^{3+}$ in Ca${\mathrm{F}}_{2}$ at a site of cubic symmetry show a pronounced angular variation with rotation of the sample in the magnetic field. This occurs even when the initial and final magnetic sublevels are rotationally invariant. This phenomenon may be explained by considering that the $^{8}S_{\frac{7}{2}}$ ground state, split only 0.149 ${\mathrm{cm}}^{\ensuremath{-}1}$ by the crystal field, is described at magnetic fields greater than about 4 kG by the projection quantum number ${J}_{z}$ referred to the magnetic field axis; and that the excited states are most conveniently described with the basis along the fourfold crystal ax is. The calculation of the dependence of the Zeeman patterns on crystal orientation, therefore, ${\mathrm{require}}^{\mathrm{S}}$ a transformation of the basis to a common Cartesian reference frame. This may be accomplished by the rotation matrix ${{R}_{{{J}_{z}}^{\ensuremath{'}}{J}_{z}}}^{J}$ acting on the ground-state basis, or by transformation of the crystal-field Hamiltonian for the excited states from the fourfold basis to the magnetic field. The crystal-field quantum numbers (irreducible representations) assigned to the $^{6}P_{\frac{7}{2}}$ and $^{6}P_{\frac{5}{2}}$ multiplets are in order of increasing energy: $^{2}\ensuremath{\Gamma}_{6}$, $^{4}\ensuremath{\Gamma}_{8}$, $^{2}\ensuremath{\Gamma}_{7}$ and $^{4}\ensuremath{\Gamma}_{8}$, $^{2}\ensuremath{\Gamma}_{7}$. The measured $g$ factors are 1.61\ifmmode\pm\else\textpm\fi{}0.06 and 1.76\ifmmode\pm\else\textpm\fi{}0.1 for the $^{6}P_{\frac{7}{2}}$ and $^{6}P_{\frac{5}{2}}$ levels, respectively.

An algebra for the point groups

A concept of composite numbers to denote coordinates of points in a cartesian frame of reference is developed in order to describe certain symmetry relationships between points in a crystal lattice. The laws of an algebra for use in the study of the crystallographic point groups are stated and are established on the basis of conventional matrix algebra.