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.

Back Page: My Favorite Lesson: Arctic Search and Destroy

2014 ◽  
Vol 108 (1) ◽  
pp. 80
Author(s):  
Cindy Kroon
Keyword(s):  
Common Core ◽  
Common Core State Standards ◽  
State Standards ◽  
Polar Coordinate System ◽  
Complex Numbers ◽  
Core State ◽  
Rectangular Coordinates ◽  
Polar Coordinate ◽  
The Common ◽  
Polar Forms

“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).

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.

scholarly journals The Lagrangian and Hamiltonian for the Two-Dimensional Mathews-Lakshmanan Oscillator

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Wang Guangbao ◽  
Ding Guangtao
Keyword(s):  
Lagrange Function ◽  
Three Dimensional ◽  
Hamiltonian Function ◽  
First Integral ◽  
Dimensional Case ◽  
Analytical Mechanics ◽  
Two Dimensional ◽  
Rectangular Coordinates ◽  
Polar Coordinate ◽  
The One

The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
I. H. Stampouloglou ◽  
E. E. Theotokoglou
Keyword(s):  
Fourth Order ◽  
Biharmonic Equation ◽  
Direct Determination ◽  
Additional Term ◽  
Elastic Solid ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Radial Coordinate ◽  
Displacement Fields ◽  
Polar Coordinate

From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate r. According to the proposed investigation additional terms, φ¯−m(r,θ)(1≤m≤n) other than the usually tabulated in the Michell solution (1899, “On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates,” Proc. Lond. Math. Soc., 31, pp. 100–124) are found. Finally the stress and the displacement fields due to each one additional term of φ¯−m(r,θ) are determined.

scholarly journals Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Iterative Method ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Poisson’S Equation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Poisson's Equation ◽  
Polar Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Diffusion Convection

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.

scholarly journals Determination of Ship Approach Parameters in the Polar Coordinates System

2014 ◽  
Vol 96 (1) ◽  
pp. 1-8
Author(s):  
Andrzej Banachowicz ◽  
Adam Wolski
Keyword(s):  
Coordinate System ◽  
Target Position ◽  
Geometric Interpretation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Polar Coordinate ◽  
Radar Measurements ◽  
Cartesian Coordinate ◽  
Radar Antenna

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

scholarly journals Modelling primary and secondary growth processes in plants: a summary of the methodology and new data from an early lignophyte

2003 ◽  
Vol 358 (1437) ◽  
pp. 1473-1485 ◽  
Author(s):  
Thomas Speck ◽  
Nick P. Rowe
Keyword(s):  
Tissue Distribution ◽  
Secondary Growth ◽  
Polar Coordinates ◽  
Ontogenetic Stage ◽  
Fossil Plants ◽  
Growth Processes ◽  
Bending Modulus ◽  
Equisetum Hyemale ◽  
Evolutionary Context ◽  
Polar Coordinate

A mathematical method, based on polar coordinates that allow modelling of primary and secondary growth processes in stems of extant and fossil plants, is summarized and its potential is discussed in comparison with numerical methods using digitizing tablets or electronic image analysing systems. As an example, the modelling of tissue distribution in the internode of an extant sphenopsid ( Equisetum hyemale ) is presented. In the second half of the paper we present new data of a functional analysis of stem structure and biomechanics of the early lignophyte Tetraxylopteris schmidtii (Middle Devonian) using the polar coordinate method for modelling the tissue distribution in stems of different ontogenetic age. Calculations of the mechanical properties of the stems, based on the modelling of the tissue arrangement, indicate that there is no increase in structural bending modulus throughout the entire development of the plant. The oldest ontogenetic stage has a significantly smaller bending elastic modulus than the intermediate ontogenetic stage, a ‘mechanical signal’, which is not consistent with a self–supporting growth form. These results, and the ontogenetic variations of the contributions of different stem tissues to the flexural stiffness of the entire stem, are discussed in the evolutionary context of cambial secondary growth.

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