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Journal articles on the topic 'Polar coordinates system'

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Published: 4 June 2021
Last updated: 1 February 2022

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1

Banachowicz, Andrzej, and Adam Wolski. "Determination of Ship Approach Parameters in the Polar Coordinates System." Reports on Geodesy and Geoinformatics 96, no. 1 (June 1, 2014): 1–8. http://dx.doi.org/10.2478/rgg-2014-0001.

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Abstract An essential aspect of the safety of navigation is avoiding collisions with other vessels and natural or man made navigational obstructions. To solve this kind of problem the navigator relies on automatic anti-collision ARPA systems, or uses a geometric method and makes radar plots. In both cases radar measurements are made: bearing (or relative bearing) on the target position and distance, both naturally expressed in the polar coordinates system originating at the radar antenna. We first convert original measurements to an ortho-Cartesian coordinate system. Then we solve collision avoiding problems in rectangular planar coordinates, and the results are transformed to the polar coordinate system. This article presents a method for an analysis of a collision situation at sea performed directly in the polar coordinate system. This approach enables a simpler geometric interpretation of a collision situation
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2

Walton, Karen Doyle, and J. Doyle Walton. "Microcomputer-Assisted Mathematics: Computer-Assisted Polar Graphing." Mathematics Teacher 80, no. 3 (March 1987): 246–50. http://dx.doi.org/10.5951/mt.80.3.0246.

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Faced with the dilemma of how to teach polar coordinates? Whether the subject is being introduced at the secondary school or college level, the problem is the same. At either level, the objectives include the following; (1) introduce a new coordinate system, relating (x, y) rectangular coordinates to (r, Q) polar coordinates; (2) enable the students to plot polar equations; (3) acquaint students with standard types of polar graphs; and (4) find points of intersection of two polar graphs. This article presents a method of using the microcomputer to teach polar coordinates and graphing in an effective, interesting way, avoiding the drudgery of having students plot hundreds of points.
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3

Wade, W. R. "A Walsh system for polar coordinates." Computers & Mathematics with Applications 30, no. 3-6 (September 1995): 221–27. http://dx.doi.org/10.1016/0898-1221(95)00100-x.

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4

Lei, Kin, Dongxu Qi, and Xiaolin Tian. "A New Coordinate System for Constructing Spherical Grid Systems." Applied Sciences 10, no. 2 (January 16, 2020): 655. http://dx.doi.org/10.3390/app10020655.

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In astronomy, physics, climate modeling, geoscience, planetary science, and many other disciplines, the mass of data often comes from spherical sampling. Therefore, establishing an efficient and distortion-free representation of spherical data is essential. This paper introduces a novel spherical (global) coordinate system that is free of singularity. Contrary to classical coordinates, such as Cartesian or spherical polar systems, the proposed coordinate system is naturally defined on the spherical surface. The basic idea of this coordinate system originated from the classical planar barycentric coordinates that describe the positions of points on a plane concerning the vertices of a given planar triangle; analogously, spherical area coordinates (SACs) describe the positions of points on a sphere concerning the vertices of a given spherical triangle. In particular, the global coordinate system is obtained by decomposing the globe into several identical triangular regions, constructing local coordinates for each region, and then combining them. Once the SACs have been established, the coordinate isolines form a new class of global grid systems. This kind of grid system has some useful properties: the grid cells exhaustively cover the globe without overlapping and have the same shape, and the grid system has a congruent hierarchical structure and simple relationship with traditional coordinates. These beneficial characteristics are suitable for organizing, representing, and analyzing spatial data.
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Kaffka, Károly J., and László S. Gyarmati. "Investigating the Polar Qualification System." Journal of Near Infrared Spectroscopy 6, A (January 1998): A191—A200. http://dx.doi.org/10.1255/jnirs.193.

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A new, rapid qualification method was introduced at the 3rd International Conference on Near Infrared Spectroscopy according to which a “quality point” was defined on a two-dimensional “quality plane”. The quality point of the investigated material was given by the center of its spectrum represented in a polar coordinate system. The method was further developed, three interpretations were given for the “center” of the polar spectrum, resulting in three different formulas for determining the x and y coordinates of the quality point. The effect of the change in the amplitude of the absorption peaks, the effect of the noise of the spectrum, the effect of the shifting and tilting the base-line of the spectrum on the location of the quality point were investigated using the three formulas. The results of the investigation and the characteristic features of the three formulas are introduced.
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6

Kilner, Steven J., and David L. Farnsworth. "Parabolic coordinates." Mathematical Gazette 105, no. 563 (June 21, 2021): 226–36. http://dx.doi.org/10.1017/mag.2021.51.

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An important first step in understanding or solving a problem can be the selection of coordinates. Insight can be gained from finding invariants within a class of coordinate systems. For example, an important feature of rectangular coordinates is that the Euclidean distance between two points is an invariant of a change to another rectangular system by a rigid motion, which consists of translations, rotations and reflections. Indeed, the form of the distance function is an invariant. In calculus courses, students learn about polar coordinates, so that useful curves can be simply expressed and more easily studied.
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Yousif, Sami R., and Frank C. Keil. "The Shape of Space: Evidence for Spontaneous but Flexible Use of Polar Coordinates in Visuospatial Representations." Psychological Science 32, no. 4 (March 15, 2021): 573–86. http://dx.doi.org/10.1177/0956797620972373.

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What is the format of spatial representation? In mathematics, we often conceive of two primary ways of representing 2D space, Cartesian coordinates, which capture horizontal and vertical relations, and polar coordinates, which capture angle and distance relations. Do either of these two coordinate systems play a representational role in the human mind? Six experiments, using a simple visual-matching paradigm, show that (a) representational format is recoverable from the errors that observers make in simple spatial tasks, (b) human-made errors spontaneously favor a polar coordinate system of representation, and (c) observers are capable of using other coordinate systems when acting in highly structured spaces (e.g., grids). We discuss these findings in relation to classic work on dimension independence as well as work on spatial representation at other spatial scales.
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8

Odo, Satoru, and Kiyoshi Hoshino. "Hand Shape Recognition using Higher Order Local Autocorrelation Features in Log Polar Coordinate Space." Journal of Robotics and Mechatronics 15, no. 3 (June 20, 2003): 286–92. http://dx.doi.org/10.20965/jrm.2003.p0286.

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The friendly communication can be more promoted between the human and computer if the function of gesture recognition is implemented to the computer system as the input interface along with the keyboards and mice. We propose a mouse-like function for estimating hand shape from input images with a monocular camera, with which a computer user feels no restraint or awkwardness. Our system involves conversion of sequential images from Cartesian coordinates to log-polar coordinates. Temporal and spatial subtractions and color information are used to extract the hand region. The origin of log-polar coordinates is chosen as the center of the acquired image, but once the hand has been extracted, the estimated centroid position of the hand region in the next frame, obtained from the current hand position and speed, is used as the origin to convert. Recognition of the hand shape is carried out by multiple regression analysis using higher order local autocorrelation features of log-polar coordinate space. Mouse-like functions are realized according to the hand shape and motion trajectory. Compared to conventional Cartesian coordinates, conversion to log-polar coordinates enables us to reduce image date and computation time, remove the variability by the scaling, and improve antinoise characteristics.
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9

Mohanty, R. K., Rajive Kumar, and Vijay Dahiya. "Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates." ISRN Mathematical Physics 2012 (February 12, 2012): 1–11. http://dx.doi.org/10.5402/2012/234516.

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Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.
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Zakarevičius, Algimantas, Vladislovas Česlovas Aksamitauskas, Algimantas Jakučionis, and Arminas Stanionis. "DETERMINATION OF GEOGRAPHIC POSITION OF OBJECT BY APPLYING 3D POLAR OBSERVATIONS." Aviation 14, no. 2 (June 30, 2010): 43–48. http://dx.doi.org/10.3846/aviation.2010.07.

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The geographic position of an object is determined by geodetic latitude, geodetic longitude, geodetic and normal height, and height of the object above the surface of the earth. To determine the geographic position of an object, a technologic scheme is recommended: by applying 3D polar observations, the 3D Cartesian coordinates of the object in the local horizontal coordinate system (for example, with an airport runway tied system) are determined; local horizontal 3D Cartesian coordinates are recomputed into the system of geocentric equatorial 3D Cartesian coordinates; the geodetic coordinates (geodetic latitude, geodetic longitude, and geodetic height) are computed from the geocentric equatorial 3D Cartesian coordinates; based on information about geodetic height and the digital geoid model, the normal height of the object is computed; and object height above the earth is computed from normal height and the digital terrain model. Algorithms for the realisation of this technologic scheme are presented. Santrauka Objekto geografine padetis apibūdinama geodezine platuma, geodezine ilguma, geodeziniu bei normaliniu aukščiais ir objekto aukščiu virš Žemes paviršiaus. Objekto geografinei padečiai nustatyti rekomenduojama tokia technologine schema: taikant erdvinius polinius matavimus, nustatomos objekto erdvines stačiakampes koordinates vietineje (pvz., oro uosto) horizontineje koordinačiu sistemoje; vietines horizontines erdvines stačiakampes koordinates perskaičiuojamos i geocentriniu ekvatoriniu erdviniu stačiakampiu koordinačiu sistema; pagal geocentrines ekvatorines erdvines stačiakampes koordinates apskaičiuojamos geodezines koordinates (geodezine platuma, geodezine ilguma ir geodezinis aukštis); žinant geodezini aukšti ir turint skaitmenini geoido modeli, skaičiuojamas objekto normalinis aukštis; turint normalini aukšti ir skaitmenini reljefo modeli, apskaičiuojamas objekto aukštis virš Žemes paviršiaus. Pateikiami algoritmai šiai technologinei schemai realizuoti.
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11

HILL, J. M., and Y. M. STOKES. "A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES." ANZIAM Journal 59, no. 3 (January 2018): 335–48. http://dx.doi.org/10.1017/s144618111700058x.

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There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.
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12

Bramson, Brian. "Newtonian quantum gravity: axisymmetric, spinning systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2078 (October 17, 2006): 503–20. http://dx.doi.org/10.1098/rspa.2006.1775.

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Consider a rotating, gravitating system whose mass centre and intrinsic spin define a natural axis of symmetry. A pair of quantum–mechanical polar coordinates, a continuous radial coordinate and an angular coordinate with discrete eigenvalues, are tied to the system's geometry. The gravitational scalar potential generated by the system is a quantum operator that is a function of the two polar coordinates. Exact expressions for the potential's gradient and Laplacian are derived, which involve forward, backward and central differences. The system's binding energy, an integral over Euclidean 3-space, comprises a radial part and an integral over the unit 2-sphere. The latter is shown to amount to a summation over the angular eigenvalues. The result is applied to the Newtonian analogues of Kerr black holes, a class of gravitational potentials with classical, ring singularities. It is shown that any fermionic, Kerr – Newton particle has a finite binding energy. This is remarkable because the classical binding energy is divergent. Furthermore, the binding energy tends to a constant in the limit of large spin.
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13

Ivanova, E. V., C. M. Volk, O. Riediger, H. Klein, N. M. Sitnikov, A. E. Ulanovskii, V. A. Yushkov, F. Ravegnani, T. Möbius, and U. Schmidt. "A quasi-Lagrangian coordinate system based on high resolution tracer observations: implementation for the Antarctic polar vortex." Atmospheric Chemistry and Physics Discussions 8, no. 4 (August 25, 2008): 16123–73. http://dx.doi.org/10.5194/acpd-8-16123-2008.

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Abstract. In order to quantitatively analyse the chemical and dynamical evolution of the polar vortex it has proven extremely useful to work with coordinate systems that follow the vortex flow. We propose here a two-dimensional quasi-Lagrangian coordinate system {χi, Δχi}, based on the mixing ratio of a long-lived stratospheric trace gas i, and its systematic use with i = N2O, in order to describe the structure of a well-developed Antarctic polar vortex. In the coordinate system {χi, Δχi} the mixing ratio χi is the vertical coordinate and Δχi = χi(Θ)−χivort(Θ) is the meridional coordinate (χivort(Θ) being a vertical reference profile in the vortex core). The quasi-Lagrangian coordinates {χi, Δχi} persist for much longer time than standard isentropic coordinates, potential temperature Θ and equivalent latitude φe, do not require explicit reference to geographic space, and can be derived directly from high-resolution in situ measurements. They are therefore well-suited for studying the evolution of the Antarctic polar vortex throughout the polar winter with respect to the relevant chemical and microphysical processes. By using the introduced coordinate system {χN2O, ΔχN2O} we analyze the well-developed Antarctic vortex investigated during the APE-GAIA (Airborne Polar Experiment – Geophysica Aircraft in Antarctica – 1999) campaign (Carli et al., 2000). A criterion, which uses the local in-situ measurements of χi=χi(Θ) and attributes the inner vortex edge to a rapid change (δ-step) in the meridional profile of the mixing ratio χi, is developed to determine the (Antarctic) inner vortex edge. In turn, we suggest that the outer vortex edge of a well-developed Antarctic vortex can be attributed to the position of a local minimum of the χH2O gradient in the polar vortex area. For a well-developed Antarctic vortex, the ΔχN2O-parametrization of tracer-tracer relationships allows to distinguish the tracer inter-relationships in the vortex core, vortex boundary region and surf zone and to examine their meridional variation throughout these regions. This is illustrated by analyzing the tracer-tracer relationships χi : χN2O obtained from the in-situ data of the APE-GAIA campaign for i = CFC-11, CFC-12, H-1211 and SF6. A number of solitary anomalous points in the CFC-11 : N2O correlation, observed in the Antarctic vortex core, are interpreted in terms of small-scale cross-isentropic dispersion.
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14

Camp, Dane R. "Sharing Teaching Ideas: Starship." Mathematics Teacher 88, no. 2 (February 1995): 113–15. http://dx.doi.org/10.5951/mt.88.2.0113.

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Every year in my precalculus classes, I introduce polar coordinates. Though students seem to have little trouble with the concept of using a different coordinate system, they do have difficulty with the fact that a location has multiple representations. For instance, the point (2, 30†) can also be expressed as (2, −330†), (−2, 210†), (−2, −150†), (2, 390†), and so on. Knowing how to manipulate these alternative forms is crucial to understanding polar graphs.
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15

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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Kroon, Cindy. "Back Page: My Favorite Lesson: Arctic Search and Destroy." Mathematics Teacher 108, no. 1 (August 2014): 80. http://dx.doi.org/10.5951/mathteacher.108.1.0080.

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“You sunk my battleship!”Many of us have fond memories of the Milton Bradley game Battleship®. Using rectangular coordinates to identify and sink an opponent's ships remains a classic childhood pastime. A similar activity can be used to help students as they learn to plot positions in the polar coordinate system. The Common Core State Standards for Mathematics (CCSSM) suggest that students “[r]epresent complex numbers on the complex plane in rectangular and polar form, and explain why the rectangular and polar forms of a given complex number represent the same number” (CCSSM Standard N-CN.4).
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Wang, Shu Ying, Zhan Min Yang, and Yi Wang. "A Range of Involute (Evolute) Transformation Model of Gear Characteristic Curve under Scroll Coordinate System." Advanced Materials Research 505 (April 2012): 494–500. http://dx.doi.org/10.4028/www.scientific.net/amr.505.494.

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By tangent and normal use of the circle scroll coordinate system is defined. Through regular, long pieces, short pieces of involute (evolute) process analysis, a range of involute (evolute) transformation model is established to achieve the high degree unity of a rectangular coordinate curves and polar coordinates (rotation) curve. For the practical application of engineering technology, the paper presented a method of continuous circular gear tooth profile curve, non-circular section of the conjugate gear involute curve rate curve for the rectangular coordinate, providing an effective way for the research of the rotation curve and the design of gear transmission.
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18

Gratus, J., and T. Banaszek. "The correct and unusual coordinate transformation rules for electromagnetic quadrupoles." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2213 (May 2018): 20170652. http://dx.doi.org/10.1098/rspa.2017.0652.

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Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.
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19

Seidelman, Kennet P., M. F. A'HEARN, D. P. CRUIKSHANK, J. L. HILTON, H. U. KELLER, R. J. OBERST, J. L. SIMON, P. STOOKE, D. J. THOLEN, and P. C. THOMAS. "Working Group on Cartographic Coordinates and Rotational Elements." Proceedings of the International Astronomical Union 1, T26A (December 2005): 181. http://dx.doi.org/10.1017/s1743921306004492.

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The 2003 report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements has appeared in Celestial Mechanics and Dynamical Astronomy(2005)volume 91, part 3–4, pages 203–215. The Working Group continues to maintain and update the information for cartographic coordinates and rotational elements for the Sun, Moon, planets, satellites, asteroids, and comets. A report is published treannually. A web site is maintained at http://extranet.astrogeology.wr.usgs.gov/Projects/WGCCRE with the up-to-date information. It is anticipated that the next report will include updates for Saturn (rotation rate) and all of the Saturnian moons, additions for the newly visited comets and asteroids, and a clarification of the Moon's coordinate systems (the mean Earth/polar axis system needs to be defined more precisely).
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Luo, Ya Bo. "A virtual layout system integrated with polar coordinates-based genetic algorithm." International Journal of Computer Applications in Technology 35, no. 2/3/4 (2009): 122. http://dx.doi.org/10.1504/ijcat.2009.026589.

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21

Trochimczuk, Roman, and Marek Gawrysiak. "Analysis of Polar Positioning System in Laser Device for Glass Engraving." Solid State Phenomena 147-149 (January 2009): 107–12. http://dx.doi.org/10.4028/www.scientific.net/ssp.147-149.107.

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In the work the new concept of polar positioning system, alternative for Cartesian one, is presented. The mathematical model which allows calculation of the rotation angle of polar positioning system is considered. The Cartesian coordinates of formed points of object structure, received after discretization process of object are the inputs data to calculation. Output data however are rotation angles i.e. both working arm and rotation table angles. The influence of the stiffness of mechanical parts on the precision of instrument positioning and productivity of the presented polar device is defined.
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22

Tenzer, Robert, and Vladislav Gladkikh. "Application of Möbius coordinate transformation in evaluating Newton's integral." Contributions to Geophysics and Geodesy 41, no. 2 (January 1, 2011): 95–115. http://dx.doi.org/10.2478/v10126-011-0004-1.

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Application of Möbius coordinate transformation in evaluating Newton's integralWe propose a numerical scheme which efficiently combines various existing methods of solving the Newton's volume integral. It utilises the analytical solution of Newton's integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky's truncation coefficients to a spectral representation of Newton's integral. The weak singularity of Newton's integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the Möbius transformation with an enhanced integration grid resolution at vicinity of the observation point.
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Okon, Tomasz, and Kazimierz Wilkosz. "Impact of Quadrature Booster on Power-System State Estimation in Polar Coordinates." Energies 14, no. 18 (September 21, 2021): 5992. http://dx.doi.org/10.3390/en14185992.

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The paper concerns the estimation of the state of a power system in which there is a phase shifter called a quadrature booster. The aim of the paper is a comparative analysis of two different cases including the quadrature booster in the state estimation. In the first case, the quadrature booster is represented by a model consisting of two real voltage sources, one in series with a power line and the other in a shunt branch. In the second case, in the power system model, the real branch with the quadrature booster is represented as off at the end where the considered quadrature booster is actually installed. The state estimation is assumed to be carried out in the polar coordinate system. The properties of the state estimation are characterized by: the number of iterations in the calculation process, the index of conditioning of the matrix of coefficients in the equations to be solved (cond(G)), and ratio Je/Jm, which is a measure of the accuracy of the estimation. Using IEEE 14-bus test system, investigations are carried out in such a way as to cover the entire state space of the power system as possible. In the investigations, Monte Carlo experiments are carried out for each of the considered cases of the state estimation. Each of these cases is also analyzed from the point of view of the assumed definition of the state estimation. Investigations show that in the first of the previously described cases, the state estimation is more accurate, but there are more iterations in the calculations and worse conditioning of the estimation process. The comparative analysis also shows that, the accuracy of the results obtained in each of the considered cases is practically independent of the coordinate system in which the estimation calculations are performed. Taking into account the number of iterations in the estimation process and index cond(G), it can be concluded that the implementation of each of the above-mentioned estimation cases in the rectangular coordinate system is more reasonable.
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Huang, Yue Hua, Bing Sen Xu, Yu Huan Jiang, and Yan Zou. "Another Method for the Polar Coordinates Newton Raphson Power Flow Calculation." Applied Mechanics and Materials 701-702 (December 2014): 227–30. http://dx.doi.org/10.4028/www.scientific.net/amm.701-702.227.

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With the development of the society, the power system flow calculation becomes more important. Another new method for the polar coordinates Newton Raphson power flow calculation was proposed according to the power system operating characteristics. This research eliminated the vacant of the Jacobian matrix completely, reduced the amount of computer memory requirements greatly and also canceled the inversion completely.
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Zhang, W. X., and Y. Bai. "The Symplectic Method in Polar Coordinates for Linear Viscoelastic Materials." Open Materials Science Journal 9, no. 1 (September 23, 2015): 130–34. http://dx.doi.org/10.2174/1874088x01509010130.

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In this study, the symplectic method is applied to a two-dimensional annular-sector viscoelastic domain under the polar coordinate system. By applying variable separation approach, all fundamental solutions are derived in analytical form. Furthermore, using the method of variable substitution, lateral conditions are transformed into finding a particular solution for the governing equations, and this particular solution is derived with the use of eigensolution expansion. In the numerical example, the boundary condition problem is discussed in detail to analyze the stress response of viscoelastic solids.
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Paul, Gour Chandra, Md Masum Murshed, Md Mamunur Rasid, and Md Morshed Bin Shiraj. "Approximation of a Complex Geometric Domain in Polar Coordinates." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 105–18. http://dx.doi.org/10.3329/ganit.v38i0.39790.

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In this study, a complex geometric domain having a colour picture is approximated through a stair- step representation of the coastal and island boundaries to make it suitable for implementing finite difference method in solving shallow water equations (SWEs) in polar coordinates. As a complex domain, we choose the coastal region of Bangladesh situated at the northern tip of the Bay of Bengal (BOB). To cover the whole coastal region, the pole is selected at the point in the plane assuming it on the mean sea level (MSL). Along the tangential direction, 265 uniformly distributive straight lines are considered through the pole and 959 circular grid lines centered at are drawn towards the radial direction covering up to latitude in the BOB. Firstly, a matrix with 960´265 computational grids is constructed from the colour information of the picture. By representing the grids with suitable notations, a proper stair-step algorithm is employed to the matrix obtained with the 960´265 grids to approximate the coastal and island boundaries to the nearest finite difference grid lines using an Arakawa C-grid system. The whole procedure is done with our developed MATLAB program. The grids representing the coastal stations are also identified closely in the obtained approximated domain. Such a type of presentation of the coastal geometry of the region of interest is found to incorporate its complexities properly with minimum computational grid points. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 105-118
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Brodsky, Alexander, and Natan Kaplan. "Analytical and practical implementation of anamorphic beam shaping in polar coordinates, Spinner system." Optics & Laser Technology 107 (November 2018): 408–14. http://dx.doi.org/10.1016/j.optlastec.2018.06.015.

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Zhang, Songyuan, Hongji Zhang, and Yili Fu. "Leg Locomotion Adaption for Quadruped Robots with Ground Compliance Estimation." Applied Bionics and Biomechanics 2020 (September 22, 2020): 1–15. http://dx.doi.org/10.1155/2020/8854411.

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Locomotion control for quadruped robots is commonly applied on rigid terrains with modelled contact dynamics. However, the robot traversing different terrains is more important for real application. In this paper, a single-leg prototype and a test platform are built. The Cartesian coordinates of the foot-end are obtained through trajectory planning, and then, the virtual polar coordinates in the impedance control are obtained through geometric transformation. The deviation from the planned and actual virtual polar coordinates and the expected force recognized by the ground compliance identification system are sent to the impedance controller for different compliances. At last, several experiments are carried out for evaluating the performance including the ground compliance identification, the foot-end trajectory control, and the comparison between pure position control and impedance control.
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Chang, Jiang, and Qing Xin Meng. "Stabilization Control of Nonholonomic Wheeled Mobile Service Robots." Key Engineering Materials 419-420 (October 2009): 593–96. http://dx.doi.org/10.4028/www.scientific.net/kem.419-420.593.

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Nonholonomic wheeled mobile service robot’s posture error model denoted by polar coordinates in global coordinates is established .Based on the inherent discontinuousness of the closed-loop system model, a novel nonlinear state feedback stabilization control law is proposed，which causes closed-loop system state space equation of robot to have isolated equilibrium state at origin. By Lyapunov candidate function method，this paper concludes that the closed-loop system is global uniformly asymptotically stable at origin. Simulation results indicate that the proposed control law is effective.
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Shkolnyy, E. P. "Normal law of distribution of wind velocity vector in polar coordinates." Ukrainian hydrometeorological journal, no. 20 (October 29, 2017): 5–17. http://dx.doi.org/10.31481/uhmj.20.2017.01.

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It is impossible to organize wind energy systems without studying of wind speed regime at the surface layer of the atmosphere within a specific area and at climatic scales. Such studies are often accompanied by approximations of probabilities of wind speed performed in the form of normal law of a system of random values presented by a zonal u and a meridional u which are components of a wind speed vector. It is suggested that, for the purposes of wind energy, display of a wind speed vector in polar coordinates (r, α) where r is a module of wind speed and α is a polar angle appears to be more preferable. The article shows a transform from a normal law of distribution of probabilities with density f(u,u) to a normal law distribution with density f(r,α) completed by means of functional transformation with elliptic dispersion in place. Based on a normal law of distribution f(r,α) and through integration with respect to corresponding variables individual distributions of probabilities f(r) and f(α) as well as conditional distributions of probabilities f(r/α) and f(α/r) were obtained in the areas of their existence. The article shows individual distributions in case of circular and elliptic dispersion of a wind speed vector. It shows that an individual distribution of a wind speed probability in case of circular dispersion and in the absence of correlated dependence turns into the Rayleigh's distribution and a conditional distribution of a polar angle degenerates in an even distribution. The cases of distributions with dispersions of a wind speed module having elliptic properties subject to availability of correlated connection between wind speed components were also studied. Calculation of probabilities of a polar angle being within different sections of the area 0≤α≤2π with set values of a wind speed module also took place. Numerical experiments proved the advantage of such modeling of distributions of wind speed vector.
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Shahmin, Nikolay V. "The Eurasian project in a multi-polar system of coordinates: Formulation of the problem." LOCUS people society cultures meaning 11, no. 2 (2020): 135–44. http://dx.doi.org/10.31862/2500-2988-2020-11-2-135-144.

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In this article the analysis of process, tendencies of change of unipolar model of a world order on multipolar is carried out. The most essential characteristics of liberalism as the most characteristic expression of the unipolar model of the world order are given. In the course of the study, special attention is paid to the analysis of the most significant events that mark the process of changing the model of the world order. The “Order of Large Spaces” as a supposed concept of legitimacy of the new world order is presented as a proposed structure of the world order. The author pays special attention to the potential of Russia to consolidate the Large Eurasian space and to the existential necessity of this process. The result of this study can be considered the justification of the Eurasian project as an expression of the essence of a Large space, the process of integration of the post-Soviet space, the key civilizational and foreign policy of Russia.
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32

Benchikha, A., and M. Merad. "Energy-dependent harmonic oscillator in noncommutative space: A path integral approach." International Journal of Modern Physics A 32, no. 32 (November 20, 2017): 1750194. http://dx.doi.org/10.1142/s0217751x17501949.

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In the context of noncommutative quantum mechanics, the energy-dependent harmonic oscillator problem is solved via path integral approach. The propagator of the system is calculated using polar coordinates. The normalized wave functions and the energy eigenvalues are obtained in two different cases.
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Kornilov, Yu N., and O. S. Tsareva. "Perfecting the methods of monitoring the buildings and structures deformation." Geodesy and Cartography 958, no. 4 (May 20, 2020): 9–18. http://dx.doi.org/10.22389/0016-7126-2020-958-4-9-18.

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The authors present the peculiarities of evaluating the parameters (module, direction and coordinates of the application point) of the displacement vector of the object’s deformation mark, its stabilitydegree is to be evaluated, and on this basis technologies simplifying the processes of performing field work necessary to obtain relevant information are discussed. Various options to implementpolar spatial intersection during observations from fixed stations are considered. In particular, if there aren’t any connecting marks between the instrument’s standing points, it is proposed to use a remote target or measure directional angles (magnetic azimuths) of reference directions. Methods for obtaining the coordinates and displacement vectors of deformation marks in the same coordinate system are considered, depending on the implementation of the polar spatial intersection. An example of implementing the proposed observation methodology on the example of the Hydrocorpus-1 building located at 29 Polytechnicheskaya st., St. Petersburg, Russia is presented. The results obtained indicate the correctness of the proposed methodology.
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Wang, Tong, K. Jiang, Shu Qiang Xie, and Shuang Shuang Hao. "Motion Analysis and Simulation of Complex Curved Surface WEDM System." Key Engineering Materials 392-394 (October 2008): 151–55. http://dx.doi.org/10.4028/www.scientific.net/kem.392-394.151.

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In this paper, the characteristics and general laws of cutting complex curved surface by wire electrical discharge machining (WEDM) system are studied. Based on analysis of motion parameters the universal mathematical model of polar coordinates is derived. Moreover, the simulation of WEDM system is introduced, which is carried out by using language Visual C++ and the three dimensional graph software OpenGL.This simulation method is helpful in improving machining quality and productivity of complex curved surfaces, and is fundation for establishing CAD/CAPP/CAM technology in WEDM.
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Wang, Tong, and Zhang Kun Hu. "Simulation of Complex Curved Surface WEDM System Based on OpenGL." Applied Mechanics and Materials 10-12 (December 2007): 364–68. http://dx.doi.org/10.4028/www.scientific.net/amm.10-12.364.

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This paper introduces the characteristics and general laws of cutting complex curved surface by Wire-cut Electric Discharge Machining (WEDM) system. Taking the three-axle and double-rotating linkage system as an example, the universal mathematical model of polar coordinates is derived. Moreover, the simulation of WEDM system is introduced, which utilizing language Visual C++ and the three dimensional graph software OpenGL.This simulation method is helpful in improving machining quality and productivity of complex curved surfaces, and is fundation for establishing CAD/CAPP/CAM technology in WEDM.
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Sárközi, Réka, Péter Iványi, and Attila Béla Széll. "Formex algebra adaptation into parametric design tools and rotational grids." Pollack Periodica 15, no. 2 (August 2020): 152–65. http://dx.doi.org/10.1556/606.2020.15.2.14.

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Abstract:This paper describes the adaptation of the formex configuration processing to the computer program Grasshopper 3D and focuses on the applied mathematical solutions. Formex algebra is a mathematical system, primarily used for planning structural systems like truss-grid domes and vaults, together with the programming language Formian. The goal of the research is to allow architects to plan truss-grid structures easily with parametric design tools based on the versatile formex algebra mathematical system. To produce regular structures, coordinate system transformations are used. Owing to the abilities of the parametric design software, it is possible to apply further modifications on the structures and gain special forms. The paper covers the basic dome types, and it introduces additional dome-based structures using special coordinate-system solutions based on a spherical coordinate system, vault structures and their modifications based on a cylindrical coordinate system and circular structures and their modifications based on polar coordinates. Moreover two rotational grid tools are introduced, which uses coordinate system transformations on a unique way to create surfaces of revolutions based on the given generating curve and create grid structures on these surfaces. It also describes the solution technique to implement the triangular grid version of every one of these tools based on diamatic domes. The adaptation of formex algebra and the parametric workflow of Grasshopper together give the possibility of quick and easy design and optimization of special truss-grid domes.
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Zarrinkamar, S., K. Jahankohan, and H. Hassanabadi. "The spin–orbit interaction in minimal length quantum mechanics; the case of a (2+1)-dimensional Dirac oscillator." Canadian Journal of Physics 93, no. 12 (December 2015): 1638–41. http://dx.doi.org/10.1139/cjp-2015-0403.

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We solve the (2+1)-dimensional Dirac oscillator in the presence of spin–orbit interaction within the framework of minimal length quantum mechanics. To report an exact analytical solution, we transform the problem into the momentum space and polar coordinates and thereby obtain the eigenfunctions and eigenenergies of the system.
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38

Lee, Byoung Koo, Sang Jin Oh, Guangfan Li, and Kou Moon Choi. "Free Vibration Analysis of Parabolic Arches in Cartesian Coordinates." International Journal of Structural Stability and Dynamics 03, no. 03 (September 2003): 377–90. http://dx.doi.org/10.1142/s021945540300094x.

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The differential equations governing free vibrations of the elastic, parabolic arches with unsymmetric axes are derived in Cartesian coordinates rather than in polar coordinates. The formulation includes the effects of axial extension, shear deformation and rotatory inertia. Frequencies and mode shapes are computed numerically for arches with clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged ends. The convergent efficiency is highly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters are reported as functions of four non-dimensional system parameters: the rise to chord length ratio, the span length to chord length ratio, the slenderness ratio and the shear parameter. Typical mode shapes of vibrating arches are also presented.
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39

Calvet, Ramon González. "On a New Analytic Theory of the Moon's Motion II: Orbit and Length of Months." Journal of Geometry and Symmetry in Physics 58 (2020): 13–54. http://dx.doi.org/10.7546/jgsp-58-2020-13-54.

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The differential equation in polar coordinates of the Moon's orbit is outlined from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed with relative coordinates and accelerations. The orbit of the Moon calculated this way is similar to Clairaut's modified orbit and has better parameters than those previously published. An improvement to this orbit is proposed based on theoretical arguments. With help of this new orbit, the variations in the draconic, synodic and anomalistic months are also computed showing very good agreement with observations.
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40

Legkostup, V. V., and V. E. Markevich. "Methodology of determining of the transfer function of engagement kinematics of accelerations of an aircraft and its elliptic coordinates used for thr guidance based on time difference of arrival." «System analysis and applied information science», no. 1 (April 26, 2021): 12–20. http://dx.doi.org/10.21122/2309-4923-2021-1-12-20.

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This paper discusses the problem of determining a kinematics (in terms of transfer function, as far as possible) of parameters of the motion of an aircraft expressed in the curvilinear coordinate system and control accelerations expressed in a rectangular coordinate system. Examples of curvilinear coordinate systems using in practice can be polar, biangular, two-center bipolar, elliptic, parabolic cylindrical, spherical, ellipsoidal, coordinate systems. A technique for obtaining a kinematic link for the control problem of an unmanned aerial vehicle in the elliptic coordinate system was described. It allowed to obtain simpler view of the kinematic link which could provide navigation an aircraft along the hyperbola deriving from the time difference of arrival navigation system. It can. As a result, it is possible to reduce the number of the navigation radio beacons.
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41

Bosakov, S. V., and O. V. Kozunova. "METHOD B.N. ZHEMOCHKINA IN CALCULATION OF PLATES IN THE FORM OF A PART OF A CIRCLE OR A RING ON AN ARBITRARY ELASTIC BASE." STRUCTURAL MECHANICS AND ANALYSIS OF CONSTRUCTIONS, no. 1 (February 25, 2021): 40–44. http://dx.doi.org/10.37538/0039-2383.2021.1.40.44.

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Abstract. Problems of calculating slabs on an elastic foundation in polar coordinates in the traditional formulation without taking into account the shear stresses in the contact zone are considered. The shape of the plates is taken in the form of a sector of a circle with an arbitrary angle or part of a ring. Analytical solutions for such problems are known for slabs of a circular or annular plan. The calculation is carried out by the Zhemochkin method, therefore, the deflections of the slab are first determined in the form of a sector of a circle with an arbitrary angle or a part of a ring with a clamped normal. This stage of the calculation is performed by the Ritz method, where the terms of the series by the product of the powers of the radius by the trigonometric functions of the angular coordinate are taken as the coordinate functions. The expressions obtained for the deflections of a slab with a clamped normal make it possible to form a system of resolving equations of the Zhemochkin method, the solution of which is the linear and angular displacements of the introduced clamping and the distribution of reactive stresses under the slab. Further, by known methods, the movements of the slab on the elastic foundation and the forces in it are determined. Two examples are given for slabs in the form of a semicircle and an annular sector with a right angle on an elastic half-space. The results obtained can find application in the calculation of circular and ring foundations for non-axisymmetric loads and slab foundations of complex shapes in polar coordinates.
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42

Smith, G. Louis, and David R. Doelling. "Computation of Radiation Budget on an Oblate Earth." Journal of Climate 27, no. 19 (September 24, 2014): 7203–6. http://dx.doi.org/10.1175/jcli-d-14-00058.1.

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Abstract The effects of the earth’s oblateness on computation of its radiation budget from satellite measurements are evaluated. For the Clouds and the Earth’s Radiant Energy System (CERES) data processing, geolocations of the measurements are computed in terms of the geodetic coordinate system. Using this system accounts for oblateness in the computed solar zenith angle and length of day. The geodetic and geocentric latitudes are equal at the equator and poles but differ by a maximum of 0.2° at 45° latitude. The area of each region and zone is affected by oblateness as compared to geocentric coordinates, decreasing from zero at the equator to 1.5% at the poles. The global area receiving solar radiation is calculated using the equatorial and polar axes. This area varies with solar declination by 0.0005. For radiation budget computations, the earth oblateness effects are shown to be small compared to error sources of measuring or modeling.
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43

Lim, Kyung-Hee, and Duane H. Smith. "Experimental test of catastrophe theory in polar coordinates: Emulsion inversion for the ethanol/benzene/water system." Journal of Colloid and Interface Science 142, no. 1 (March 1991): 278–90. http://dx.doi.org/10.1016/0021-9797(91)90057-f.

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44

Huang, Hsu-Chih, and Jing-Jun Xu. "Evolutionary Machine Learning for Optimal Polar-Space Fuzzy Control of Cyber-Physical Mecanum Vehicles." Electronics 9, no. 11 (November 18, 2020): 1945. http://dx.doi.org/10.3390/electronics9111945.

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This paper contributes to the development of evolutionary machine learning (EML) for optimal polar-space fuzzy control of cyber-physical Mecanum vehicles using the flower pollination algorithm (FPA). The metaheuristic FPA is utilized to design optimal fuzzy systems, called FPA-fuzzy. In this hybrid computation, both the fuzzy structure and the number of IF–THEN rules are optimized through the FPA evolutionary process. This approach overcomes the drawback of the structure tuning problem in conventional fuzzy systems. After deriving the polar-space kinematics model of Mecanum vehicles, an optimal EML FPA-fuzzy online control scheme is synthesized, and the global stability is proven via Lyapunov theory. An embedded cyber-physical robotic system is then constructed using the typical 5C strategy. The proposed FPA-fuzzy computation collaborates with the advanced sensors and actuators of Mecanum vehicles to design a cyber-physical robotic system. Compared with conventional Cartesian-space control methods, the proposed EML FPA-fuzzy has the advantages of metaheuristics, fuzzy online control, and cyber-physical system design in polar coordinates. Finally, the mechatronic design and experimental setup of a Mecanum vehicle cyber-physical system is constructed. Through experimental results and comparative works, the effectiveness and merit of the proposed methods are validated. The proposed EML FPA-fuzzy control approach has theoretical and practice significance in terms of its real-time capability, online parameter tuning, convergent behavior, and hybrid artificial intelligence.
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45

Ishida, Y., T. Ikeda, and T. Yamamoto. "Nonlinear Forced Oscillations Caused by Quartic Nonlinearity in a Rotating Shaft System." Journal of Vibration and Acoustics 112, no. 3 (July 1, 1990): 288–97. http://dx.doi.org/10.1115/1.2930507.

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This paper deals with nonlinear forced oscillations in a rotating shaft system which are caused by quartic nonlinearity in a restoring force. These oscillations are theoretically analyzed by paying attention to the nonlinear components represented by the polar coordinates. It is clarified which kind of nonlinear component has an influence on each oscillation. In experiments it was shown that, when the shaft was supported by double-row angular contact ball bearings, the restoring force had nonlinear spring characteristics involving quartic nonlinearity in addition to quadratic and cubic ones. Experimental results were compared with the theoretical results regarding the probability of occurrence and the shapes of the resonance curves.
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46

Huang, Lve, Hua Biao Yan, and Lu Min Tan. "A Novel Method of Motion People Detection and Tracking in Moving Background." Applied Mechanics and Materials 373-375 (August 2013): 547–51. http://dx.doi.org/10.4028/www.scientific.net/amm.373-375.547.

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The Surendra background update and novel fast model matching were mixed which can reduce the matching region. A Yuntai tracking system was present for people tracking, the fuzzy control tracking based on polar coordinates was also present, which makes the tracking of people object always in the video range and the Yuntai neednt move frequency. Results indicate that the algorithm is superior to the previously published variants of the model matching and the Yuntai system track people in real time.
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47

Mao, Yi, Yun-ru Li, and Jin-fu Zhu. "Research on Key Technologies of Network Centric System Distributed Target Track Fusion." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/6137015.

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To realize common tactical picture in network-centered system, this paper proposes a layered architecture for distributed information processing and a method for distributed track fusion on the basis of analyzing the characteristics of network-centered systems. Basing on the noncorrelation of three-dimensional measurement of surveillance and reconnaissance sensors under polar coordinates, it also puts forward an algorithm for evaluating track quality (TQ) using statistical decision theory. According to simulation results, the TQ value is associated with the measurement accuracy of sensors and the motion state of targets, which is well matched with the convergence process of tracking filters. Besides, the proposed algorithm has good reliability and timeliness in track quality evaluation.
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48

Lie, S. T. "Evaluation of singular boundary element kernels using polar coordinates system over a flat quadratic triangle in elastostatics." Engineering Analysis with Boundary Elements 10, no. 2 (1992): 173–77. http://dx.doi.org/10.1016/0955-7997(92)90048-c.

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49

Russo, G. S., and C. J. Bruce. "Neurons in the supplementary eye field of rhesus monkeys code visual targets and saccadic eye movements in an oculocentric coordinate system." Journal of Neurophysiology 76, no. 2 (August 1, 1996): 825–48. http://dx.doi.org/10.1152/jn.1996.76.2.825.

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1. We investigated whether neurons in the supplementary eye field (SEF) of macaque monkeys code saccadic eye movements in oculocentric coordinates (relative to the current direction of fixation) or in craniocentric coordinates (relative to the head). Craniocentric coding in SEF had been previously suggested by the convergent appearance of electrically elicited saccades originating at different orbital positions. 2. We primarily studied SEF neurons that started responding before the beginning of saccades because such presaccadic activity is likely related to saccade generation and metrics. Using a memory-saccade task, we classified the presaccadic activity of each neuron as either purely visual related, purely movement related, or both visual and movement related. 3. We then mapped the response fields (receptive fields and movement fields) of SEF neurons from different orbital positions. When mapped relative to a central fixation point, the strongest responses for a given SEF neuron invariably occurred for a particular polar direction with fairly symmetrical declines for departures from that direction. When tested using other fixation point locations, their strongest responses almost always continued to occur for stimuli having the same polar direction relative to each fixation point tested, and thus they appeared to code both stimulus direction and saccade direction in an oculocentric coordinate system. 4. The effect of eye position on SEF presaccadic activity was quantified in two ways by computing, for each neuron, 1) an "intersection distance," the eccentricity of the point where extensions of the neuron's optimal polar directions measured at two eccentric orbital positions converged, and 2) an "orbital perturbation index" such that an index of 0 corresponded to no change in the neuron's optimal polar direction across different orbital positions (i.e., perfectly oculocentric response fields) and an index of 1 corresponded to optimal polar directions that converged to the same craniocentric goal regardless of initial eye position (i.e., perfectly craniocentric response fields). For neurons with both visual and movement responses, these measures were calculated separately for each type of activity using tasks that temporally separated the visual cue presentation and the saccade to it. 5. Almost all of the intersection distances were well beyond the oculomotor range (+/- 50 degrees) of the monkey (38/39 for movement activity and 62/66 for visual activity). The median intersection distance for visual activity was very large (274 degrees), and the median for movement activity was slightly divergent (beyond infinity). Thus SEF neurons rarely showed a conspicuous convergence of response field direction. 6. Likewise, the mean orbital perturbation indexes were very small (-0.04 +/- 0.21, mean +/- SD, for movement activity and 0.09 +/- 0.15 for visual activity), also indicating that SEF neurons code stimuli and saccades in an oculocentric manner. 7. For neurons with both visual and movement activities, the orbital perturbation indexes of the two activities were not significantly correlated (r = 0.16), even though their characteristic directions (optimal polar direction estimated from the center of the screen) were almost the same (circular correlation, r+ = 0.97). The lack of a significant correlation between the visual and movement activity orbital perturbation indexes is consistent with the hypothesis that most of the variation in this index represents statistically independent errors of measurement. Conversely, the strong covariation of visual and movement activity characteristic directions indicates that directional preference is a fundamental functional property of SEF presaccadic activity.(ABSTRACT TRUNCATED)
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Goodall, Colin R., and Kanti V. Mardia. "A geometrical derivation of the shape density." Advances in Applied Probability 23, no. 03 (September 1991): 496–514. http://dx.doi.org/10.1017/s0001867800023703.

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The density for the shapes of random configurations of N independent Gaussian-distributed landmarks in the plane with unequal means was first derived by Mardia and Dryden (1989a). Kendall (1984), (1989) describes a hierarchy of spaces for landmarks, including Euclidean figure space containing the original configuration, preform space (with location removed), preshape space (with location and scale removed), and shape space. We derive the joint density of the landmark points in each of these intermediate spaces, culminating in confirmation of the Mardia–Dryden result in shape space. This three-step derivation is an appealing alternative to the single-step original derivation, and also provides strong geometrical motivation and insight into Kendall's hierarchy. Preform space and preshape space are respectively Euclidean space with dimension 2(N–1) and the sphere in that space, and thus the first two steps are reasonably familiar. The third step, from preshape space to shape space, is more interesting. The quotient by the rotation group partitions the preshape sphere into equivalence classes of preshapes with the same shape. We introduce a canonical system of preshape coordinates that include 2(N–2) polar coordinates for shape and one coordinate for rotation. Integration over the rotation coordinate gives the Mardia–Dryden result. However, the usual geometrical intuition fails because the set of preshapes keeping the rotation coordinate (however chosen) fixed is not an integrable manifold. We characterize the geometry of the quotient operation through the relationships between distances in preshape space and distances among the corresponding shapes.
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