Some Comments on Sherratt's Principle

1977 ◽  
Vol 9 (12) ◽  
pp. 1417-1419
Author(s):  
R Vaughan
Keyword(s):  
Probability Density ◽  
Polar Coordinates ◽  
Cartesian Coordinates

The term ‘density of dwellings' used by geographers is conceptually different to the term ‘probability density of dwellings' used by statisticians. In Cartesian coordinates the numerical values only differ by a scaling constant, but in polar coordinates this is not the case. To help clear up the resulting confusion, this paper attempts to show the relation between the two concepts.

scholarly journals The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170652 ◽  
Author(s):  
J. Gratus ◽  
T. Banaszek
Keyword(s):  
Coordinate Transformation ◽  
Equations Of Motion ◽  
Flat Space ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Transformation Rules ◽  
Integral Transformations ◽  
Second Derivatives ◽  
Electric Dipoles ◽  
Derivatives Of

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.

Coherent states for Smorodinsky-Winternitz potentials

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Nuri Ünal
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Wave Packets ◽  
Harmonic Oscillators ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Physical Time ◽  
First Case

AbstractIn this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem. In the first case, we find the non-spreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three non-isotropic harmonic oscillators. All of these states evolve in physical-time. We also show that in parametric-time, the second case can be transformed to the first one with vanishing eigenvalues.

scholarly journals Lane Detection for Autonomous Vehicles using Open CV Library

2020 ◽  
Vol 6 (12) ◽  
pp. 176-180
Author(s):  
Aarushi Mittal and Narinder Kaur
Keyword(s):  
Hough Transform ◽  
Autonomous Vehicles ◽  
Region Of Interest ◽  
Traffic Signals ◽  
Lane Detection ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Path Discovery

For vehicles to have the option to drive without anyone else, they have to comprehend their encompassing world like human drivers, so they can explore their way in roads, pause at stop signs and traffic signals, and try not to hit impediments, for example, different vehicles and pedestrians. In view of the issues experienced in identifying objects via self-governing vehicles an exertion has been made to show path discovery utilizing OpenCV library. The explanation and method for picking grayscale rather than shading, distinguishing and detecting edges in an image, selecting region of interest, applying Hough Transform and choosing polar coordinates over Cartesian coordinates has been discussed.

Exploroing Polar Curves with GeoGebra

2012 ◽  
Vol 106 (3) ◽  
pp. 228-233
Author(s):  
Tuyetdong Phan-Yamada ◽  
Walter M. Yamada
Keyword(s):  
Trigonometric Function ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Step Process ◽  
Cartesian Plane

Most trigonometry textbooks teach the graphing of polar equations as a two-step process: (1) plot the points corresponding to values of θ such as π, π/2, π/3, π/4, π/6, and so on; and then (2) connect these points with a curve that follows the behavior of the trigonometric function in the Cartesian plane. Many students have difficulty using this method to graph general polar curves. The difficulty seems to stem from an inability to convert changes in the value of the trigonometric equation as a function of angle (abscissa vs. ordinate in Cartesian coordinates) to changes of the radius as a function of angle (r[θ] in polar coordinates). GeoGebra provides a tool to help students visualize this relationship, thus significantly improving students' ability

Iteration Logistic Map Dynamics in the Polar Coordinates

Fractals ◽  
1998 ◽  
Vol 06 (01) ◽  
pp. 11-22
Author(s):  
Haijian Sun ◽  
Lin Liu ◽  
Aike Guo
Keyword(s):  
Logistic Map ◽  
Polar Coordinates ◽  
Ancient China ◽  
Cartesian Coordinates ◽  
Logistic Equations ◽  
The World ◽  
The Law

The complicated dynamics of logistic maps with four fundamental operations in the polar coordinates are analyzed in this paper. Iterations of single logistic map in polar coordinates exhibit similar characteristics as in the Cartesian coordinates. Iterations of double logistic maps, however, are more interesting in polar coordinates than those in Cartesian coordinates. Under different iterative rules, the dynamics of double logistic maps show some funny artistic patterns. Some of them look like the formation of an embryo, and some even look like Taiji diagram, which is the representative symbol of Taoism in ancient China. It is interesting to see that Taiji, which is believed in Taoism to be the mysterious origination of the world, is an interim phenomenon within the evolution of the systems governed by the law of the logistic equations.

Coping with Curvilinear Coordinates

1998 ◽  
Vol 26 (4) ◽  
pp. 309-317 ◽  
Author(s):  
J. Paavola ◽  
E.-M. Salonen
Keyword(s):  
Main Idea ◽  
Curvilinear Coordinates ◽  
Polar Coordinates ◽  
Systematic Method ◽  
Cartesian Coordinates ◽  
Orthogonal Coordinates

A systematic method to generate expressions appearing in physics and engineering and valid in curvilinear orthogonal coordinates is presented. The method, which is called ‘the method of local cartesian coordinates’ achieves the same as ‘the method of moving axes’ presented in Love [1] but with a smaller effort and with more familiar mathematical tools. The main idea is: if we have an expression valid in rectangular cartesian coordinates, a corresponding expression for curvilinear orthogonal coordinates can be formed with simple steps. Some general expressions are generated, but the method can be used equally well to produce the desired formulas directly in specific cases. For this purpose, polar coordinates are employed extensively as a demonstration example.

Free Vibration Analysis of Parabolic Arches in Cartesian Coordinates

2003 ◽  
Vol 03 (03) ◽  
pp. 377-390 ◽  
Author(s):  
Byoung Koo Lee ◽  
Sang Jin Oh ◽  
Guangfan Li ◽  
Kou Moon Choi
Keyword(s):  
Differential Equations ◽  
Slenderness Ratio ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Chord Length ◽  
Polar Coordinates ◽  
Length Ratio ◽  
Mode Shapes ◽  
Dimensional System ◽  
Cartesian Coordinates

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.

scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

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