Showing papers on "Parabola published in 2013"
TL;DR: In this paper, the authors consider the problem of finding a Heisenberg uniqueness pair for a simple set Γ, Λ in the plane, where Γ is a straight line or a union of two straight lines.
Abstract: Let Γ denote the parabola y=x2 in the plane. For some simple sets Λ in the plane we study the question whether (Γ,Λ) is a Heisenberg uniqueness pair. For example we shall consider the cases where Λ is a straight line or a union of two straight lines.
30 citations
TL;DR: In this article, a method for the design of gear tooth profiles using parabolic curve as its line of action is proposed, which is based on the meshing theory, and the effect of the two design parameters that present the size (or shape) of the parabola curve relative to the gear size, on the shape of tooth profiles and on the contact ratio are also studied through the design an example drive.
26 citations
TL;DR: In this paper, the authors considered the problem of characterizing a parabola to be a strictly convex curve in the plane and showed that it is possible to characterise a paraboloid with respect to the tangent of a chord.
Abstract: . Archimedes knew that the area between a parabola and anychord AB on the parabola is four thirds of the area of triangle ∆ABPwhere P is the point on the parabola at which the tangent is parallel toAB. We consider whether this property (and similar ones) characterizesparabolas. We present five conditions which are necessary and sufficientfor a strictly convex curve in the plane to be a parabola. 1. IntroductionA parabola is the set of points in the plane which are equidistant from apoint F called the focus and a line l called the directrix. Archimedes foundsome interesting area properties of parabolas.Consider the region bounded by a parabola and a chord AB. Let P be thepoint on the parabola where the tangent is parallel to the chord AB. The linethrough P parallel to the axis of the parabola meets chord AB at a point V .Then, he showed that the area of the parabolic region is a|PV | 3/2 for someconstant a, which depends only on the parabola.Furthermore, he proved that the area of the parabolic region is 4/3 timesthe area of triangle △ABP whose base is the chord and whose third vertex isP. For the proofs of Archimedes, see Chapter 7 of [8].In this paper, we consider whether this property (and similar ones) charac-terizes parabolas. As a result, we present five conditions which are necessaryand sufficient for a strictly convex curve in the plane to be a parabola.Usually, a curve X in the plane R
21 citations
TL;DR: In this paper, the authors focused on distributed modal neural sensing signals on a flexible simply-supported parabolic cylindrical shell panel, which is fully laminated with a piezoelectric layer on its outer surface and segmented into infinitesimal elements (neurons).
16 citations
TL;DR: In this article, a detailed mathematical model for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation, is presented.
Abstract: The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. This article expands on our previously published work concerning a real-world billiard. Here we provide a detailed mathematical model for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge, and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola, i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The representation of the coefficient of restitution employed in the model resulted in approximate agreement with the experimental data for all boundary shapes investigated. We show how the coefficient of restitution varies under different model assumptions. It is shown that the data can be successfully modeled with a simple set of parameters.
13 citations
TL;DR: In this article, the authors mainly focused on the design of transition curves of the cubic parabola type in track alignment design, and derived the transition curves from the theory of cubic parabolos using calculus techniques.
Abstract: This paper mainly focuses on the design of transition curves of the cubic parabola type in track alignment design. Equations for the transition curves are first derived from the theory of cubic parabolas using calculus techniques. They are then analyzed using numerical analysis methods. The proposed formulation is evaluated by comparison of its calculated results with data in 684 actual cases of transition curves. The accuracy of the proposed method is verified in this study by comparing the estimated results with collected data from actual engineering projects, and the applicability of the new method to the engineering practice of track alignment design is justified. The chainage and coordinates of the control points as well as other curve points and the tangential angle can be easily calculated using the transition curve equations.
12 citations
TL;DR: In this article, the nonlinear dynamics of a particle on a rotating parabola are analyzed by means of the analytic and semi-analytic approaches and the effects of different parameters on the governing equation are evaluated.
10 citations
TL;DR: In this paper, the theory of Diophantine approximation on planar curves has been studied in the case of parabola and it has been shown that the theory can be extended to manifolds.
Abstract: The well known theorems of Khintchine and Jarn\'ik in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.
10 citations
TL;DR: A parabolic analog of the arbelos is introduced, using theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties ofThe arbelo, some of which may be new.
Abstract: The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we ...
8 citations
TL;DR: In this paper, the authors show that although the wording parabolic is stricto sensu incorrect, it is though not far from reality and they show how to calculate simply the approximation committed by calling this trajectory parabolic.
7 citations
TL;DR: In this paper, it was shown that the noncolliding orbits in the "external configuration space" R2k+1 backslash{0} are all conics, and that a conic orbit is an ellipse, a parabola, or a branch of a hyperbola according to the total energy.
TL;DR: A novel method is proposed for pose estimation based on parabolic motion by employing the projective geometry properties of vanishing point and vanishing line implicated in the projected parabola, provided that the intrinsic parameters of camera are specified.
TL;DR: In this article, the authors give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Žadnik for all parabolic geometry, which is based on the definition of a natural connection on the space of Weyl structures.
Abstract: We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Žadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.
Patent•
08 May 2013
TL;DR: In this article, a tracing method of the maximum power point of a solar battery was proposed. But the method is restricted to a part of the power/voltage curve which is close to the maximum point of the battery output.
Abstract: The invention discloses a tracing method of a solar battery maximum power point. A part of curve which is nearby the maximum power point of a solar battery output characteristic power/ voltage curve 1 is approximated to a parabola 2, a parabolic equation is solved by using three groups of power/ voltage data 3, 4 and 5 on the part of the curve, and an extreme point 6 of the parabolic equation is the maximum power point 7 output by the solar battery.
TL;DR: The Crank-Nicolson scheme, which is generally accepted as an improvement of the Schmidt scheme, is subjected not only to stability analysis, but also absolute relative error analysis to guide Mathematicians and Engineers alike to know the true performance of these numerical solution methods.
Abstract: numerical algorithms employed in the solution of Parabolic Partial Differential Equations are the subject of this paper. In particular, the Crank-Nicolson scheme, which is generally accepted as an improvement of the Schmidt scheme, is subjected not only to stability analysis, but also absolute relative error analysis to guide Mathematicians and Engineers alike to know the true performance of these numerical solution methods. The Heat Equation with Dirichlet conditions conducting heat is analysed by employing the analytical method of solution where the method of Separation of Variables is used. The same equation is then solved with the Schmidt scheme as well as the Crank-Nicolson scheme and the results compared to the analytical solution. It is shown that provided stability conditions for both numerical schemes are not compromised, the Schmidt scheme is better than the Crank-Nicolson scheme at the particular point 80% from the conducting end of the rod. With the rod discretized into six points, both ends of the rod produce the same results for both numerical schemes. With the remaining four points, it is shown that three points produced values which showed that the Crank-Nicolson scheme is better than the Schmidt scheme at those three points, but not the fourth.
TL;DR: In this article, the authors generalized Trapezon's result on bending oscillation of beams of constant thickness whose width varies in accordance with a fourth order parabola and showed that the existence of a closed formula is possible for any power function describing the width of the beam.
Abstract: In this study Trapezon’s result is generalized on bending oscillation of beams of constant thickness whose width varies in accordance with a fourth order parabola. The existence of a closed formula is shown, which offers solution not only for a fourth order parabola but for every power function describing the width of the beam. In particular, if the exponent is a power of 2 then also the closed form of the general solution can be given.
01 Jan 2013
TL;DR: In this article, the authors introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features.
Abstract: We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y 2 . Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected concurrences and collinearities, a construction for the evolute, and the determination of (up to) four points on the parabola whose normals meet.
TL;DR: For certain kinds of parabolic Calderon-Zygmund operators, this article showed that the L2(E,dσ)-boundedness of T is equivalent to the parabolic uniform rectifiability of E. This is a parabolic version of a well-known result of G. David and S. Semmes.
Abstract: Let E be a subset in (n+1)-dimensional Euclidean space with parabolic homogeneity, codimension 1, and with an appropriate surface measure σ associated with it. For certain kinds of parabolic Calderon–Zygmund operators T we prove that the L2(E,dσ)-boundedness of T is equivalent to the parabolic uniform rectifiability of E. This is a parabolic version of a well-known result of G. David and S. Semmes.
TL;DR: In this paper, the truncated parabolic function has been shown to be a representation of the Dirac function and has the same bound state spectrum, tunneling and reflection amplitudes as Dirac potential as the width of the parabola approximates to zero.
Abstract: We study the truncated parabolic function and demonstrate that it is a representation of the Dirac function. We also show that the truncated parabolic function, used as a potential in the Schr¨ odinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac potential, as the width of the parabola approximates to zero. Dirac potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose‐Einstein condensate in a harmonic trap. We show that a harmonic trap with a function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.
Journal Article•
TL;DR: In this article, a model of atomic structure is presented, where a series of parabolas represent the atom shells (vertical sections along paraboloids) from outer to inner part of the atom and a pair of electrons exists in a circular motion in each orbit.
Abstract: A model of atomic structure is presented. Many questions that have baffled scientists for the past decades may be answered by this visualized mathematical model. In particular, the model mathematically relates the properties of the parabola to the atoms' shells and orbits. A series of parabolas represents the atom shells (vertical sections along paraboloids) from outer to inner part of the atom. A pair of electrons exists in a circular motion in each orbit. The orbits are contained in the parabolic shells. Orbits in a shell equal to the square of shell number. Furthermore, hydrogen atomic spectrum was simulated and a new relation was verified. The frequencies of the visible spectra of the hydrogen atom were found to be related to the change in the area enclosed by the first orbit in the second shell, 2s, when the electron is excited into the inner concentric orbits. A proportionality constant equals 0.238725 femto-sec has been found to satisfy that relation.
Journal Article•
TL;DR: The experimental results of emulation data and real image show that the algorithm can efficiently detect the parabola object feature and compute the feature parameters in the optical image, and the calculation accuracy of each parameter is above 98%.
Abstract: The parabola object feature detection of optical image is a difficult research topic in computer vision fields.A parabola feature detection algorithm based on the least square method is presented.Using Canny algorithm to detect edge,then the least square method is used to compute the coefficients of the parabola equation.The conic is constrained by parabola and the algorithm,so it can get the optimal solution without iteration computation.Finally the coordinate transformation of the parabola equation is computed and the parameters of parabola can be achieved.The experimental results of emulation data and real image show that our algorithm can efficiently detect the parabola object feature and compute the feature parameters in the optical image,and the calculation accuracy of each parameter is above 98%.
TL;DR: In this article, a parabola in the standard form corresponding to three points on the parabolas is given, such that the normals at these three points concur at a point.
Abstract: Given a parabola in the standard form corresponding to three points on the parabola, such that the normals at these three points concur at a point the equation of the circumscribing circle through the three points provides a tremendous opportunity to illustrate ‘The Art of Algebraic Manipulations’. The equation of the circle exists as a separate entity depending on the values of the parameters The graphical illustrations are based on Scientific WorkPlace 5.5 and they provide interesting interpretations of the roots of cubic equations. Geometry, algebra and the theory of cubic equations are amply illustrated in this article.
Proceedings Article•
04 Nov 2013
TL;DR: In this paper, the scalar radiation problem using parabolic equation is considered and the scattered field calculation technique based on two parabolic equations for waves propagating in two opposite directions is presented.
Abstract: In this paper the scalar radiation problem using parabolic equation is considered. The scattered field calculation technique based on two parabolic equations for waves propagating in two opposite directions is presented. The near- and far-field calculation results for parabolic cylinder reflector and various excitation conditions are shown. The near-field amplitude for cylindrical Luneburg lens and line current excitation are produced.
Posted Content•
TL;DR: In this paper, five conditions are presented which are necessary and sufficient for a strictly convex curve in the plane to be a parabola, where the tangent is parallel to the chord.
Abstract: Archimedes knew that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $\Delta ABP$ where P is the point on the parabola at which the tangent is parallel to $AB$. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.
Journal Article•
TL;DR: A new kind of cubic algebraic trigonometric interpolation splines with a shape parameter over spaceΩ=span { 1, t, sint, cost , sint2t, cos2t } was presented.
Abstract: A new kind of cubic algebraic trigonometric interpolation splines with a shape parameter over spaceΩ=span { 1 , t , sint , cost , sint2t , cos2t } was presentedThe interpolation splines have many similar properties with cubic B-splinesThe curves and surfaces can interpolate directly some control points without solving system of equations or inserting some additional control pointsThe curves can be used to exactly represent straight line segment , circular arc , elliptic arc , parabola and some transcendental curves such as circular helixThe corresponding tensor product surfaces can also precisely represent some quadratic surfaces and transcendental surfaces , such as sphere , cylindrical surfaces and helix tubeThe shape of the curves and surfaces can be modified globally through changing the values of the parametersFurthermore , the local parameters are introduced in the splines using the singular blending techniqueExamples are given to illustrate that the splines can be used as a novel efficient model for geometric design in the fields of CAGD
TL;DR: Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line as discussed by the authors, and every cubic has rotational symmetrized symmetrization.
Abstract: Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cub...
TL;DR: In this article, a method of parabola error evaluation using Geometry Ergodic Searching Algorithm (GESA) was proposed according to geometric features and fitting characteristics of Parabola errors.
Abstract: A method of parabola error evaluation using Geometry Ergodic Searching Algorithm (GESA) was proposed according to geometric features and fitting characteristics of parabola error. First , the feature points of least-squared parabola are set as reference feature points to layout a group of auxiliary feature grid points. After that, a series of auxiliary parabolas as assumed ideal parabolas are reversed with the auxiliary feature points.The range distance from given points to these assumptions ideal parabolas are calculated successively.The minimum one is parabola profile error.The process of GESA was detailed discribed including the algorithm formula and contrastive results in this paper.Simulation experiment results show that the geometry ergodic searching algorithm is more accurate than the least-square method. The parabola profile error can be evaluated steadily and precisely with this algorithm based on the minimum zone.
01 Oct 2013
TL;DR: In this paper, it was shown that an inverse quartic function can give a much better fit than an inverse parabola in almost all FLC test samples showing asymmetric strain distributions.
Abstract: The current study aims to determine the limit strains more accurately and reasonably when producing a forming limit curve (FLC) from experiments. The international standard ISO 12004-2 in its recent version (2008) states that the limit major strain should be determined by using the best-fit inverse second-order parabola through the experimental strain distribution. However, in cases where fracture does not occur at the center of the specimen, due to insufficient lubrication, the inverse parabola does not give a realistic fit because of its intrinsic symmetry in shape. In this study it is demonstrated that an inverse quartic function can give a much better fit than an inverse parabola in almost all FLC test samples showing asymmetric strain distributions. Using a quartic fit creates more reliable FLCs.
TL;DR: In this article, all self-maps of the unit disk in the complex plane can be classified as elliptic, hyperbolic or parabolic, and the parabolic case is the most complicated one and branches into zero-step and non-zero-step cases.
Abstract: Based on dynamical behavior, all self-maps of the unit disk in the complex plane can be classified as elliptic, hyperbolic or parabolic. The parabolic case is the most complicated one and branches into two subcases - zero-step and non-zero-step cases. In several dimensions, zero-step and non-zero step cases can be defined for sequences of forward iterates, but it is not known yet if the classification can be extended to parabolic maps of the ball. However, some geometric properties of the forward iterates can be generalized to higher-dimensional case.
01 Jan 2013
TL;DR: In this paper, an approach that takes onto consideration all nonlinear terms of Navier-Stokes equations is presented and new solutions of the Poiseuille problem are obtained and their nonlinear properties are identified.
Abstract: Poiseuille problem is the first problem in theoretical hydromechanics for which the exact solution has been found. The solution is a steady state solution of Navier–Stokes equations and it gives the velocity profile known as "Poiseuille parabola". Experimental studies show that parabolic profile occurs very seldom in fluid flows. Usually more complex structures are observed. This fact makes us again focus attention on the problem to obtain other solutions. This paper presents an approach that takes onto consideration all nonlinear terms of Navier–Stokes equations. New solutions of the Poiseuille problem are obtained and their nonlinear properties are identified.