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.

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 Calculating the near field of a line of sources using Mellin transforms

2007 ◽  
Vol 49 (1) ◽  
pp. 99-109
Author(s):  
P. A. Martin
Keyword(s):  
Line Segment ◽  
Near Field ◽  
Slender Body ◽  
Polar Coordinates ◽  
Asymptotic Approximations ◽  
Systematic Method ◽  
Mellin Transforms ◽  
Cylindrical Polar Coordinates

In slender-body theories, one cften has to find asymptotic approximations for certain integrals, representing distribution:; of sources along a line segment. Here, such approximations are obtained by a systematic method that uses Mellin transforms. Results are given near the line (using cylindrical polar coordinates) and near the ends of the line segment (using spherical polar coordinates).

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.

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.

Limits in 𝒫ℳℱ of Teichmüller geodesics

2019 ◽  
Vol 2019 (747) ◽  
pp. 1-44 ◽  
Author(s):  
Jon Chaika ◽  
Howard Masur ◽  
Michael Wolf
Keyword(s):  
Line Segment ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Quadratic Differentials ◽  
Limit Set ◽  
Unique Point ◽  
Time Intervals ◽  
Limit Sets ◽  
Bounded Distance ◽  
Uniquely Ergodic

AbstractIn this paper we consider the limit set in Thurston’s compactification{\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}}of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.

scholarly journals QUADRISECANT APPROXIMATION OF HEXAGONAL TREFOIL KNOT

2011 ◽  
Vol 20 (12) ◽  
pp. 1685-1693 ◽  
Author(s):  
GYO TAEK JIN ◽  
SEOJUNG PARK
Keyword(s):  
Line Segment ◽  
Intersection Point ◽  
Straight Line Segment ◽  
Closed Curve ◽  
Straight Line ◽  
Trefoil Knot ◽  
The Given ◽  
Marked Points

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

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