Cartesian Coordinate Systems

2011 ◽  
pp. 1-30
Author(s):  
Fletcher Dunn ◽  
Ian Parberry
Keyword(s):  
Coordinate Systems ◽  
Cartesian Coordinate

Experimental and numerical investigation of railroad vehicle braking dynamics

2009 ◽  
Vol 223 (3) ◽  
pp. 255-267
Author(s):  
C Mellace ◽  
A P Lai ◽  
A Gugliotta ◽  
N Bosso ◽  
T Sinokrot ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
The Body ◽  
Computer Algorithms ◽  
Coordinate Systems ◽  
State Equations ◽  
Relative Coordinates ◽  
Linear State ◽  
Cartesian Coordinate ◽  
Railroad Vehicle ◽  
Braking Dynamics

One of the important issues associated with the use of trajectory coordinates in railroad vehicle dynamic algorithms is the ability of such coordinates to deal with braking and traction scenarios. In these algorithms, track coordinate systems that travel with constant speeds are introduced. As a result of using a prescribed motion for these track coordinate systems, the simulation of braking and/or traction scenarios becomes difficult or even impossible. The assumption of the prescribed motion of the track coordinate systems can be relaxed, thereby allowing the trajectory coordinates to be effectively used in modelling braking and traction dynamics. One of the objectives of this investigation is to demonstrate that by using track coordinate systems that can have an arbitrary motion, the trajectory coordinates can be used as the basis for developing computer algorithms for modelling braking and traction conditions. To this end, a set of six generalized trajectory coordinates is used to define the configuration of each rigid body in the railroad vehicle system. This set of coordinates consists of an arc length that represents the distance travelled by the body, and five relative coordinates that define the configuration of the body with respect to its track coordinate system. The independent non-linear state equations of motion associated with the trajectory coordinates are identified and integrated forward in time in order to determine the trajectory coordinates and velocities. The results obtained in this study show that when the track coordinate systems are allowed to have an arbitrary motion, the resulting set of trajectory coordinates can be used effectively in the study of braking and traction conditions. The results obtained using the trajectory coordinates are compared with the results obtained using the absolute Cartesian-coordinate-based formulations, which allow modelling braking and traction dynamics. In addition to this numerical validation of the trajectory coordinate formulation in braking scenarios, an experimental validation is also conducted using a roller test rig. The comparison presented in this study shows a good agreement between the obtained experimental and numerical results.

scholarly journals New upper bounds for noncentral chi-square cdf

2020 ◽  
pp. 70-74
Author(s):  
Valeriy A. Voloshko ◽  
Egor V. Vecherko
Keyword(s):  
Density Function ◽  
Upper Bounds ◽  
Cumulative Density Function ◽  
Coordinate Systems ◽  
Chi Square ◽  
One Dimensional ◽  
Chi Squared ◽  
Cartesian Coordinate ◽  
The One ◽  
Inverse Trigonometric Functions

Some new upper bounds for noncentral chi-square cumulative density function are derived from the basic symmetries of the multidimensional standard Gaussian distribution: unitary invariance, components independence in both polar and Cartesian coordinate systems. The proposed new bounds have analytically simple form compared to analogues available in the literature: they are based on combination of exponents, direct and inverse trigonometric functions, including hyperbolic ones, and the cdf of the one dimensional standard Gaussian law. These new bounds may be useful both in theory and in applications: for proving inequalities related to noncentral chi-square cumulative density function, and for bounding powers of Pearson’s chi-squared tests.

Mathematical Model for Kinematic Analysis of Working Coordinates of General Mechanisms

1995 ◽  
Author(s):  
J. Y. Wang ◽  
H. J. Yeh ◽  
T. C. Lin ◽  
J. K. Wu
Keyword(s):  
Kinematic Analysis ◽  
Inverse Kinematic ◽  
Coordinate Space ◽  
Coordinate Systems ◽  
Constraint Equations ◽  
Path Design ◽  
Joint Coordinates ◽  
Solid Foundation ◽  
Inverse Kinematic Analysis ◽  
Cartesian Coordinate

Abstract Mathematical models have been derived for the kinematic analysis of working coordinates, which contain the formulation of the working coordinates constraint equations in terms of the relative joint coordinates, the transformation of the Jacobian matrix of the associated constraint equations from the Cartesian coordinate space to the relative joint coordinate space, and formulation for velocity and acceleration calculation. Such models lay out a solid foundation for the computational inverse kinematic analysis. In addition, moveable working coordinates are derived for both local and global coordinate systems. Application of this can be the working path design of general manipulators.

Study on the Diffusion Equations of Water Pollutants in Various Water Areas with Different Waterflow States

2011 ◽  
Vol 183-185 ◽  
pp. 1030-1034
Author(s):  
Xiao Ling Lei ◽  
Bo Tao
Keyword(s):  
Coordinate System ◽  
Diffusion Equations ◽  
Cylindrical Coordinate System ◽  
Cartesian Coordinate System ◽  
Coordinate Systems ◽  
Spherical Coordinate ◽  
Water Pollutants ◽  
Cylindrical Coordinate ◽  
Cartesian Coordinate ◽  
Further Development

The development and application of the diffusion equations of water pollutants are synthetically discussed. Depending on Cartesian Coordinate system, the water pollutants diffusion equations in different waterflow states are reviewed. And further development of the water pollutants diffusion equations in different waterflow states is extended to Cylindrical Coordinate system and Spherical Coordinate system respectively. This makes the simulating and modeling of water pollutants diffusion much more accurate and convenient in various water areas with different waterflow states by using different coordinate systems.

Representation of Reservoir Geometry for Numerical Simulation

1970 ◽  
Vol 10 (04) ◽  
pp. 393-404 ◽  
Author(s):  
G.J. Hirasaki ◽  
P.M. O'Dell
Keyword(s):  
Coordinate System ◽  
Curvilinear Coordinates ◽  
Reference Plane ◽  
Cartesian Coordinate System ◽  
Curvilinear Coordinate System ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Curvilinear Coordinate ◽  
Reservoir Geometry ◽  
Cartesian Coordinate

Abstract For most reservoirs the reservoir thickness and dip vary with position. For such reservoirs, the use of a Cartesian coordinate system is awkward as the coordinate surfaces are planes and the finite-difference grid elements are rectangular parallepipeds. However, these reservoirs may be efficiently parallepipeds. However, these reservoirs may be efficiently modeled with a curvilinear coordinate system that has coordinate surfaces that coincide with the reservoir surfaces. A procedure is presented that may be used to determine a curvilinear coordinate system that will conform with the geometry of the reservoir. The reservoir geometry is described by the depth of the top of the reservoir and the thickness. The mass conservation equations are presented in curvilinear coordinates. The finite-difference equations differ from the usual Cartesian coordinate formulation by a factor multiplying the pore volume and transmissibilities. A numerical example is presented to illustrate the magnitude of the error that may occur in the computed oil recovery if the Cartesian coordinate system is simply modified to yield the correct depth and pore volumes. Introduction Many reservoirs have a shape that is inconvenient and possibly inaccurate to model with Cartesian coordinates. The use of a curvilinear coordinate system that follows the shape of the reservoir can be advantageous for such reservoirs. The formulation discussed here will have the greatest advantage in modeling thin reservoirs but will have little advantage in modeling a reservoir whose thickness is greater than its radius of curvature, such as a pinnacle reef. pinnacle reef. In this paper the reader is introduced to various grid systems used to model reservoirs. A brief discussion of some concepts of differential geometry contrasts differences between Cartesian coordinates and curvilinear coordinates. A curvilinear coordinate system for modeling reservoir geometry is presented. Formulation of the conservation equations in curvilinear coordinates and the necessary modifications to pore volume and transmissibility are discussed. A numerical example illustrates the magnitude of the error that may result from some coordinate systems. COORDINATE SYSTEMS AND RESERVOIR GRID NETWORKS A reservoir is usually described with the depth, thickness, boundaries, etc., shown on a structure map with sea level as a reference plane. For example, the subsea depth may be shown as a contour map on the reference plane with a Cartesian coordinate grid superimposed on the reference plane as shown on Fig. 1. The Cartesian coordinates, plane as shown on Fig. 1. The Cartesian coordinates, (y1, y2), have been defined as the coordinates for the reference plane. If the reservoir surfaces are parallel planes, Cartesian coordinates may be used. The Cartesian coordinate may be rotated such that the coordinate surfaces coincide with the reservoir surfaces. SPEJ P. 393

scholarly journals A Continuous Coordinate System for the Plane by Triangular Symmetry

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 191 ◽  
Author(s):  
Benedek Nagy ◽  
Khaled Abuhmaidan
Keyword(s):  
Image Processing ◽  
Coordinate System ◽  
Computer Science ◽  
Closed Interval ◽  
Triangular Grid ◽  
Digital Geometry ◽  
Cartesian Coordinate System ◽  
Coordinate Systems ◽  
Cartesian Coordinate ◽  
New System

The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems. The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system.

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