Some Remarks on Higher Curvature Theory

1967 ◽  
Vol 89 (1) ◽  
pp. 84-86 ◽  
Author(s):  
G. R. Veldkamp
Keyword(s):  
Polar Coordinates ◽  
Rectangular Coordinates ◽  
Curvature Theory

Freudenstein introduced the concept of generalized Burmester points using proofs involving polar coordinates. In the present paper rectangular coordinates are used; this leads to an extension and a slightly modified interpretation of the theory.

Microcomputer-Assisted Mathematics: Computer-Assisted Polar Graphing

1987 ◽  
Vol 80 (3) ◽  
pp. 246-250
Author(s):  
Karen Doyle Walton ◽  
J. Doyle Walton
Keyword(s):  
Secondary School ◽  
Coordinate System ◽  
Polar Coordinates ◽  
Computer Assisted ◽  
College Level ◽  
Rectangular Coordinates ◽  
The Subject

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.

Finding Points of Intersection of Polar-Coordinate Graphs

1991 ◽  
Vol 84 (6) ◽  
pp. 472-477
Author(s):  
Warren W. Esty
Keyword(s):  
Polar Coordinates ◽  
Rectangular Coordinates ◽  
Polar Coordinate

Students studying polar coordinates may be required to find the points of intersection of the polar-coordinate graphs of two functions, f and g. Their experiences with rectangular coordinates may lead them to expect that all the points of intersection can be found by solving the equation f((J) = g (O). They may be distressed to discover that points of intersection of the graphs can occur that do not correspond to solutions of that equation.

Axes for symmetric convex curves

2018 ◽  
Vol 102 (553) ◽  
pp. 23-30
Author(s):  
David L. Farnsworth
Keyword(s):  
Line Segment ◽  
Closed Curve ◽  
Polar Coordinates ◽  
Symmetric Convex ◽  
Minkowski Geometry ◽  
Convex Curves ◽  
Rectangular Coordinates

Curves are given in polar coordinates (r, θ)by equations of the form r = f (θ), where for f (θ) > 0 all θ. Consider curves which are symmetric about the origin O, so that, f(θ + π) = f (θ) for all θ. For such a curve, its interior is the set {(r, θ) : 0 ≤ r ≤ f (θ)}. Further, assume that the curve is convex. Recall that a closed curve is convex if a line segment between any two of its points has no points exterior to the curve [1], [2, pp. 198-203]. We call these curves M-curves, because the curves are fundamental objects in Minkowski geometry, where they are called Minkowski circles or simply circles [3, 4]. That application is briefly discussed in the Section 4 but is not required for our purposes.Examples of M-curves are displayed in Figures 1 to 6. In order to express these curves as functions in rectangular coordinates, we need axes.

Scattering by a horizontal subsurface penny-shaped crack

1986 ◽  
Vol 408 (1835) ◽  
pp. 277-294 ◽  
Keyword(s):  
Crack Opening ◽  
Resonance Phenomenon ◽  
Polar Coordinates ◽  
Axially Symmetric ◽  
Opening Displacement ◽  
Time Harmonic ◽  
Wave Resonance ◽  
Penny Shaped Crack ◽  
Rectangular Coordinates ◽  
Shaped Crack

The scattering by a horizontal subsurface penny-shaped crack subjected to axially symmetric loading is investigated. The formulation begins with deriving the response of a time harmonic point force in rectangular coordinates. Then, the integral representation and integral equations are converted into polar coordinates by applying the condition of axial symmetry. The results contain crack opening displacement (COD), stress intensity factors, scattered pattern and the frequency spectrum of the Rayleigh wave and the back-scattered longitudinal wave. Resonance phenomenon is compared with the plane strain case solved in an earlier paper.

Parabolic coordinates

2021 ◽  
Vol 105 (563) ◽  
pp. 226-236
Author(s):  
Steven J. Kilner ◽  
David L. Farnsworth
Keyword(s):  
Distance Function ◽  
Euclidean Distance ◽  
Rigid Motion ◽  
Polar Coordinates ◽  
Coordinate Systems ◽  
Rectangular Coordinates ◽  
Selection Of

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.

scholarly journals Monte Carlo Simulations of Two and Three Dimensional Fluids Revisited: A Successful Demonstration of the Use of Polar Coordinates

1991 ◽  
Author(s):  
Francis J. Conlan
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Three Dimensional ◽  
Polar Coordinates ◽  
Sampling Area ◽  
Simulation Techniques ◽  
Virtual Imaging ◽  
Virtual Images ◽  
Rectangular Coordinates ◽  
Mc Simulation

Abstract Standard Monte Carlo (MC) simulation techniques in current use (Valleau and Whittington, Valleau and Torrie, 1977) in the study of Statistical properties of fluids are limited to being performed in rectangular coordinates and rely on a non-physical construct called virtual images. Virtual imaging was a technique used in the early days of computer simulations (1950–1960) to reduce extraneous boundary effects in Monte Carlo simulations of fluids (Wood, 1968). Simulations were restricted to rectangular sampling regions providing the central sampling area with virtual images of itself. All particles, including those near the boundary, by other particles.

Symmetry’s Applications to Bihamonic Equation and the Generalized Solution of Elasticity Problem

2011 ◽  
Vol 52-54 ◽  
pp. 899-904
Author(s):  
Jian Qin ◽  
Ke Fu Huang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Eigenfunction Expansion ◽  
Generalized Solution ◽  
Elasticity Problem ◽  
Polar Coordinates ◽  
Invariant Solutions ◽  
Rectangular Coordinates ◽  
Elasticity Solutions ◽  
Lie Transformations

The generalized elasticity solutions are obtained in this paper by symmetries from Lie transformations. The symmetries of the bihamonic equation are obtained by mathematica package SYM. The several invariant solutions are found by solving the corresponding ordinary differential equations from Lie transformations. The superposition of invariant solutions, corresponding to an eigenfunction expansion, could yield the generalized elasticity solutions in rectangular coordinates and polar coordinates, where the eigenvalues arises from the invariance of bihamonic equation.

scholarly journals Transformation between polar and rectangular coordinates of stiffness and dampness parameters in hydrodynamic journal bearings

Friction ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 201-206 ◽  
Author(s):  
Zhuxin Tian ◽  
Yu Huang
Keyword(s):  
Reynolds Equation ◽  
Journal Bearing ◽  
Stability Boundary ◽  
Journal Bearings ◽  
Polar Coordinates ◽  
Oil Film ◽  
Rectangular Coordinates ◽  
Bearing Stiffness ◽  
The Stability

Abstract The stiffness and dampness parameters of journal bearings are required in rectangular coordinates for analyzing the stability boundary and threshold speed of oil film bearings. On solving the Reynolds equation, the oil film force is always obtained in polar coordinates; thus, the stiffness and dampness parameters can be easily obtained in polar coordinates. Therefore, the transformation between the polar and rectangular coordinates of journal bearing stiffness and dampness parameters is discussed in this study.

scholarly journals Encryption using circular harmonic key

DYNA ◽  
2015 ◽  
Vol 82 (190) ◽  
pp. 70-73
Author(s):  
Jorge Enrique Rueda-Parada
Keyword(s):  
Fourier Transform ◽  
Added Value ◽  
Polar Coordinates ◽  
Computational Tool ◽  
Optical Encryption ◽  
Tolerance Level ◽  
Rectangular Coordinates ◽  
The Fourier Transform

This work presents a study of variance to rotation key encryption processors based on the Fourier transform. It was determined that the key in rectangular coordinates allows a tolerance level of less than 0.2 degrees of rotation of the key in the decryption process. Thus, the solution is to build the key in polar coordinates, by means of circular harmonics expansion; in this way, the tolerance threshold rises to about 40 degrees of rotation of the key in the decryption process. This solution is an added value for optical encryption processors. I have developed a computational tool for simulations and results obtained in this study.

Non-Polar Variation of Latitude

1972 ◽  
Vol 1 ◽  
pp. 93-101 ◽  
Author(s):  
S. Yumi
Keyword(s):  
Polar Motion ◽  
Polar Coordinates ◽  
Normal Variation ◽  
Latitude Variation ◽  
Local Trend

ABSTRACTAnalysing the residual latitude of the station, local trend in latitude variation other than by the polar motion was found.Residual latitude was calculated for each of 26 stations which gave the continuous records of observation during 6 years comprising — 1962 — 1967 as a difference between observed variation of latitude and – normal variation calculated by the polar coordinates Iderived from all the results of 26 stations.As far as the results during these six years are concerned, local trend at any station it seemed to be expressed in terms of 3λ.Assumed effect of local trend on the coordinates values of the instantaneous pole is also discussed.

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