Arc Of Circle Research Articles

Multiscale Detection of Circles, Ellipses and Line Segments, Robust to Noise and Blur

This paper proposes a basic taxonomy of image contours. Our goal is to classify smooth curves into five categories, namely, circles, ellipses, line segments, arcs of circles and arcs of ellipses. These geometrical structures have been chosen as they serve as input of many computer vision tasks. The proposed strategy is applied on a set of initial disjoint contours, which are grouped together to form the aforementioned structures. These, in turn, are validated using an a contrario approach that guarantees a reduced number of false detections. The use of a multiscale strategy permits the detection at different resolution levels, which makes the method robust to noise and blur.

The principle of dividing feasible trajectories in a robot control problem

The problem of control of mobile robot is considered. It is assumed that the start and end points of the trajectory are set, and one or more obstacles in the form of circles are present on the line connecting these points. Circles are defined by the coordinates of their centers and the lengths of their radius. The feasible trajectory must not share points with the interiors of these circles. The proposed principle is to represent any feasible trajectory as a sequence of straight sections and arcs of circles that limit circular obstacles. The robot’s movement along straight sections is described by a second-order differential equation, and the movement along arcs is uniform with a given minimum speed.

Infinite plate loaded with normal force moving along a complex path

Introduction. A technique of solving the problem on an infinite plate lying on an elastic base and periodically loaded with a force that moves along an arbitrary closed trajectory and according to an arbitrary law; is considered. Materials and Methods. An original method for solving problems on the elasticity theory for plates loaded with a force moving arbitrarily along a closed trajectory of arbitrary shape is considered. The problem on an infinite plate lying on an elastic foundation is investigated. The plate is loaded with a normal force moving at a variable speed. The load is decomposed into a Fourier series on a time interval whose length is equal to the time of its passage along the trajectory. The solution to this problem is realized through a superposition of solutions to the problems corresponding to the load defined by the summands of the specified Fourier series. The final problem solution is presented in the form of a segment of the Fourier series, each summand of which corresponds to the solution to the problem on the action on an infinite plate of the load distributed along a closed trajectory of the force motion. The fundamental solution to the vibration equation of an infinite plate lying on an elastic foundation is used to construct these solutions. Results. A solution to the problem of an infinite plane, along which a concentrated force moves at a variable speed, is presented. A smooth closed curve consisting of arcs of circles was considered as a trajectory. The behavior of displacements and stresses near the moving force is investigated; and the process of the elastic wave energy propagation is also studied. For this purpose, a change in the Umov-Poynting vector is considered. Discussion and Conclusions. The results obtained can be used in calculations for road design. The study of the propagation of the energy of elastic waves from moving vehicles will provide the assessment of the impact of these waves on buildings located near the road. Analysis of the behavior of displacements and stresses near the moving force will allow assessing the wear of the road surface.

The effect of replacing non-analytic trajectories with break points on smooth paths to the complexity of deformation and loading processes of materials

This article is devoted to an experimental study of the effect of rounding off corner points of two-link strain trajectories on complex loading processes during elastoplastic deformation of materials. Replacing corner points in their vicinity with local sections of circles allows a nonanalytic trajectory to be replaced with a smooth trajectory. Experimental studies were performed on thin-walled tubular specimens of the low-carbon steel St3 on an SN-EVM automated testing system. The loading programs for tubular specimens were set in the Ilyushin's deviatoric strain space. The rounding of the corner point of a two-link strain trajectory with an angle of 90° between the branches by arcs of circles with curvatures of 200, 400, as well as the rounding of the corner point of a two-link strain trajectory with an angle of 135° between the branches by arcs with curvatures of 400, 800 are considered. The experimental data characterizing the vector and scalar properties of the material are presented. The experimental data show that the effect of complex loading on the relationship between stresses and strains in a curved section is not immediately apparent. In the curved section, the magnitude of the stress vector modulus first increases, and then decreases with the formation of stress dives. The minimum point of the stress dive is located on the next straight branch of the strain trajectory. In the curvilinear section, the angle of delay increases, and in the next straight branch it decreases, and with the increase of the strain it tends to be zero. The rate of decrease of the angle of delay depends little on the differences in the geometry of the previous history of strain trajectory. In the second straight branch, the experimental results for a smooth and original two-link strain trajectories become little distinguishable from each other. Thus, replacing the original non-analytical strain trajectory to a smooth trajectory affects the complexity of the process of deformation and loading of the materials only in the vicinity of the corner point. This circumstance can be taken into account when numerically modeling the processes of elastoplastic deformation of materials and integrating the defining relations, replacing nonanalytic trajectories with smooth ones. This can be taken into account in the numerical calculation of elastic-plastic deformation and integration of constitutive relations, replacing non-analytical strain trajectories by smooth ones.

Uniform point variance bounds in classical beta ensembles

In this paper, we give bounds on the variance of the number of points of the Circular and the Gaussian [Formula: see text] Ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on [Formula: see text].

New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics

The circle and sphere have been studied since the ancient Egyptians and Greeks before the Common Era (BCE). The recent scientific renaissance has also used them in different fields. It is also mentioned in the Prophet Mohamed`s Holy Quran. This article introduces a new Theorem (S. El-Mongy`s Theorem) as an empirical formula to correlate the constant (e) with circle and sphere. It states that “the arc length is correlated as a direct function in {(e π r sA)}, whatever the central angle (ϴ) and radius (r). The factor sA is (ϴ/10ϕ). The formula can also be written as; AL = {(0.0174533185 r ϴ)}. Where the value 0.0174533185 is a constant called Sayed`s number (Is) and equals (eπ/10ϕ). This factor is very close to value (π/180 = 0.0174532925) with ~1.5x10-4 % difference. The formula was applied for calculation the arc length (AL) of circles of different radii and angles. The results of this formula were validated and verified for very wide range; from 0.5 cm to 4.4x1023 km (46.5x109 light-years; ly) and compared with the old published arc length formula results. The difference is from 0.000% to 0.002% only. The formula was also used as trigonometric functions of circular orbits for calculation the distances between the Earth and Sun, Moon, planets, stars, and EH-M87 Black hole with relatively small error; the difference is from 0.26% to maximum ~ 2.27%. The error was 0.29% for ~54 x 106 ly distance to the M87 black hole. The S. El-Mongy formula may open the door for further scientific and engineering applications.

Mindset matters: how mindset affects the ability of staff to anticipate and adapt to Artificial Intelligence (AI) future scenarios in organisational settings.

Any first step in organisational adaptation starts with individuals’ responses and willingness (or otherwise) to change an aspect of themselves given the transcontextual settings in which they are operating (Bateson in Small arcs of larger circles: framing through other patterns, Triarchy Press, Axminster, 2018). This research explores the implications for organisational adaptation strategies when Artificial Intelligence (AI) is being embedded into the ecology of the organisation, and when employees have a dominant fixed or growth mindset (Dweck in Mindset: changing the way you think to fulfil your potential. Robinson, London, 2017). Research participants were supplied with a single scenario based in 2030, where—as a result of Artificial Intelligence technology implementation—employees were going to be displaced. Using Torbert’s (Organizational wisdom and executive courage, New Lexington Press, San Francisco, 1998) ‘first, second and third person’ research theory, participants were asked to independently review their thoughts, sense, and image of the future from a fixed mindset position (considered to be the worst case), then from a growth mindset perspective (best case), and then do the same collectively. Five key findings are outlined which support the principle that having a growth mindset is a key component of adaptive capacity and futures literacy. The five key findings conclude that AI adaptation processes need to include compassion and authenticity, embodiment, fundamental needs and motivations, mutual learning and considering what lies beyond the edges of the organisation (Bateson in Small arcs of larger circles: framing through other patterns, Triarchy Press, Axminster, 2018).

Algorithm for screws profiling of a conical trilobed compressor - analytical solution

Screw compressors, as rotary machinery, are capable of working with high compression ratios for gas compression. They transform the mechanical work done by an electric motor or turbine into the potential energy of a gas, by reducing its volume and increasing the pressure. The essential elements of these compressors are the two helical rotors with lobes, mounted in a housing. Existing rotors in the construction of compressors are working bodies whose composite profiles include circle arcs, line segments and cycloidal curves, while also often including non-analytical curves. Such geometry of profiles gives the gear processes superior characteristics, in the sense of reducing friction and improving sealing conditions. This paper analyses, by analytical approach, the possibility of constructing a multilobated conical screw compressor with a number of lobes Z, Z + 1, of the male and female rotor. The parametric equations of the contained worm and the kinematics of the working process of the conical screw compressor shall be defined. The theory of surface wrapping comprises analytical methods of study, which were developed on the basis of the fundamental theorems on wrapping of known surfaces in differential geometry and the kinematics laws on rotor movements. Thus, this paper proposes an algorithm based on the approach to the problems analyzed under the “virtual focal points theorem” between the complex surfaces of the conical surface lobes, which constitute the active lobes of the male and female rotors. Normal to active surfaces are defined, and also the speed in relative movement, in the “gear” process of the conical screw compressor. Regarding rotors in conical screw compressors, the circular pitch of the teeth is constant. Also, the screw pitch of the conical compressor rotors is a constant value. In these conditions, the rotors of the screw compressors are acknowledged as active surfaces, conical screw surfaces with constant axial screw pitch. The cross section (in the normal plane to the driving rotor axis) determines a plane gear. This defines the specific winding condition, and the shape of the conical screw surface of the female rotor.

Geometry of Ancient Mazes, Part I: ‘Cretan’ Mazes

The aim of the present survey is to investigate the processes implemented for conceiving and setting up two major types of mazes which have come down to us on various media (stone, mosaic, painting, ceramic, coins…) from Prehistoric times to Roman Empire (and, for some, even later). Based on a corpus of 83 mazes, the two sets usually known as ‘Cretan’ and ‘Roman’ could be characterized and differentiated. Most of them show the walls of the Labyrinth and the path exists only in negative. And, at variance with the original myth (Theseus and the Minotaur) there is only one possible path, which shows no dead ends or crossings. This first part of the study is devoted to Cretan mazes, the apparently complex structure of which is usually composed of two pairs of imbricated laces, although extensions and variants exist. This structure can be modeled by a square pattern to which several sets of concentric equidistant arcs of circles are connected. Based on this model, a set up process can be proposed, enabling to memorize the design and reproduce it by heart. This is doubtless why Cretan mazes could persist along centuries and be found as far as America and India

Trajectory planning and tracking control for autonomous parallel parking of a non-holonomic vehicle

This paper proposes autonomous parallel parking for a front-wheel steering vehicle, with highlights on a trajectory planning method and on a trajectory tracking control method. The trajectory planning problem is decoupled into the path planning problem and the longitudinal velocity planning problem to reduce the difficulty of the trajectory planning problem. First, a collision-free path by combining circle arcs with straight line is created to park the vehicle in one or more maneuvers on the premise of meeting the kinematic constraint of vehicle, and then the path is transformed into a continuous-curvature path using B-spline curve. Second, the longitudinal velocity is created using B-spline curve on the premise of meeting the performance constraints of driving and braking system. To execute the generated trajectory, a non-time reference path tracking sliding mode control strategy is deduced by Lyapunov stability theory, and a longitudinal velocity tracking proportional–integral control strategy is proposed based on smooth handoff method. Finally, the parking performance is verified based on model-in-the-loop simulation system.

Observation and modeling of droplet shape on metal fiber with gravity effect

Droplet retention on metal fibers is a common physical phenomenon, and its drainage is significantly affected by the droplet shape. The purpose of this study is to experimentally investigate the effect of gravity on the shape of droplet on a metal fiber, and to present a method to quantitatively describe the droplet shape. It is found that droplets on a metal fiber are in the shape of clam-shell, and the droplet profile curves from the front view and the side view are both approximate to ellipse arcs or circle arcs. This finding inspires the description method of the droplet shape, covering the solid-liquid interface and the gas-liquid interface. In this method, the solid-liquid interface is described by a cylindrical function with the triple contact line as the boundary, and the triple contact line is expressed as the intersection line of two cylindrical surfaces; the gas-liquid interface is simplified as the curved surface formed by the profile curve revolving around the vertical axis, and the profile curve at an arbitrary azimuthal angle is described by the ellipse function with the variable semi-axis. The proposed model is validated by the experiments and the results show that, the goodness of fit of the predicted profiles is larger than 88%, and fthe deviations between the predicted volume and the experimental volume are within 10%.

New exotic capillary free-surface flows

Abstract

Spherical great circle arcs based indicators for evaluating distortions of map projections

Spherical great circle arcs based indicators for evaluating distortions of map projections

Comparing Performance of Different Equating Methods in Presence and Absence of DIF Items in Anchor Test

This study aims to compare the performance of different small sample equating methods in the presence and absence of differential item functioning (DIF) in common items. In this research, Tucker linear equating, Levine linear equating, unsmoothed and presmoothed (C=4) chained equipercentile equating, and simplified circle arc equating methods were considered. The data used in this study is 8th-grade mathematics test item responses which obtained from Trends in International Mathematics and Science Study (TIMSS) 2015 Turkey sample. Item responses from Booklet-1 (N=199) and Booklet-14 (N=224) are chosen for this study. Data analyses were completed in four steps. In the first step, assumptions for DIF detection and test equating methods were checked. In the second step, DIF analyses were conducted with Mantel Haenszel and logistic regression methods. In the third step, Booklet 1 was chosen as base form and Booklet 14 chosen as a new form, then test equating was conducted under common item nonequivalent groups design. Test equating was done in two phases: the presence and absence of DIF items in the common items. Equating results were evaluated based on standard error of equating (se), bias and RMSE indexes. DIF analyses showed that there were two sizeable DIF items in anchor test. Equating results showed that performances of equating methods are similar in presence and absence of DIF items from anchor test and there is no notable change in se, bias and RMSE values. While the circle arc equating method outperformed other equating methods based on se, 4-moment presmoothed chained equipercentile equating method outperformed other methods based on bias and RMSE evaluation criteria.

Attitude kinematics using slewing transformation of a single axis

Attitude kinematics using a single axis was developed to establish the relationships among the axis trajectory, enclosed solid angle and the associated attitude transformation. An arbitrary transformation due to a sequence of finite rotations or due to propagation consisting of an infinite number of infinitesimal rotations is equivalent to the product of a slewing transformation along a great circle arc linking the initial and final orientations of the axis followed by a rotational transformation about the axis. A key finding of this work is that the slewing transformation commutes with the rotational transformation, if the trajectory of the slewing transformation is fixed in space and the axis orientation is accounted for in the rotational transformation. The results were used to establish two new kinematic theorems. The new theorems provided simple derivations of Ishlinskii’s theorem, the Goodman-Roberson theorem and a result found by Montgomery. The findings in this work were applied to attitude propagation by separating attitude rotational increments that are normal to the axis with those that are parallel to the axis. The normal and parallel angular increments were integrated separately to form the slewing transformation and rotational transformation, respectively. The product of the two commuting transformations formed the total propagation transformation. The findings were validated by using them to create and implement a propagation algorithm. Numerical testing showed that the new algorithm has accuracy comparable to traditional attitude propagation algorithms when stress tested with pure coning motion.

Convergence of a Limit Periodic Schur Continued Fraction

In this paper, we show that if the parameters of a Schur continued fraction tend to zero, then the functions to which the even convergents converge inside the unit disk and the functions to which the odd convergents converge outside the unit disk cannot have a meromorphic continuation to each other through any arc of the unit circle. This result is obtained as a consequence of the convergence theorem for limit periodic Schur continued fractions.

SPATIAL TECHNICAL CURVES AS SMOOTH LINES OF THE ARCS NORMCURVES

In solving applied problems one often encounters with the construction of spatial curves for a number of pre-set conditions. As a rule, the constructed curve is set by a set of pre-calculated or experimentally obtained conditions (points, tangents, values of curvature and torsion at these points). The horizontal and frontal projections (plan and profile) are calculated independently. As a result, the outline will consist of arcs of spatial curves no lower than the fourth order with unpredictable differential properties. The arcs of circular norm-curved spaces are devoid of this drawback.The article discusses some theoretical questions of curve theory for a rational choice of smooth contour components. It is proposed to use arcs of cubic circles as components of a spatial smooth one-dimensional contour. A cubic circle, being a special case of a cubic ellipse, intersects an improper plane at a real point and at two cyclic points. Such curves have better differential properties from the standpoint of the monotonicity of changes in the values of the angles of inclination of the tangents, curvature, and torsion, which significantly affects the dynamic qualities of the contour being constructed.

Composites of nanodimensional glass in the system Na2O–SiO2/Mesoporous silica and their high ionic conductivity

Nanodimensional glass samples of compositions xNa2O-(100–x)SiO2, with x having values of 25, 35, and 45, have been synthesized within the nanochannels of diameter ca. 5 nm of mesoporous silica SBA-15 (Santa Barbara Amorphous 15) by a sol–gel technique. The dc electrical conductivities of the nanocomposites, as well as of bulk glasses, have been determined in the temperature range 300–500 K by carrying out ac impedance measurements over the frequency range 20 Hz to 1 MHz on samples of diameter ca. 10 mm and thickness ca. 0.7 mm, and determining the intersection points of the arcs of circles in the complex impedance plots at high frequency with the real axis. The conductivities of the nanocomposites were found to be 4–5 orders of magnitude higher than those of the corresponding bulk glasses at room temperature. This may be ascribed to the presence of greater free volume in the nanoglasses as compared to their bulk counterparts, thus providing faster pathways for ion migration therein, and hence drastically lowering the activation energy of sodium ion migration. The nanocomposite with 45 mol% Na2O exhibited an electrical conductivity of the order of 10−3 S cm−1 at room temperature (ca. 300 K). The nanocomposites also showed outstanding electrochemical performances. The maximum specific capacitance was seen for the nanocomposite with nanoglass composition 45Na2O–55SiO2, amounting to 1344 F/g at a scan rate of 10 mV/s. Around 88.8% retention of capacitance over 100 successive cycles indicates good sustainability of the samples. These nanocomposites are thus highly promising for the fabrication of solid-state sodium-ion batteries.

Torsion with Circular Shear of a Mooney–Rivlin Solid

The axisymmetric torsion of a layer of incompressible Mooney–Rivlin material placed between two rigid coaxial cylindrical surfaces is considered. It is assumed that the ends of the cylinder are fixed to avoid axial displacement and the cross-sections orthogonal to the axis of symmetry do not distort during deformation. In this case, the material points move along arcs of circles with an angular velocity which in the general case may depend on both the axial and the radial coordinates. Closed-form solution of the equilibrium equation is obtained. The well-known Rivlin solution is a particular case of this solution. Like in the classical solution, the stress tensor is a function of only the radial coordinate and the twist angle of the material points is a linear function of the axial coordinate. It is shown that torsion that is symmetric about a plane orthogonal to the axis of rotation can be realized only within the framework of the Rivlin solution. The obtained exact solution can be realized in the presence of friction on cylindrical surfaces. The initial-boundary-value problem described by this solution is considered.

An Iterative Dynamic Programming Approach to the Multipoint Markov-Dubins Problem

A new solution to the multipoint Markov-Dubins problem via iterative dynamic programming is herein presented. The shortest path problem connecting a sequence of given points in the plane while maintaining angle continuity and bounded curvature is presented. As in the classic two points Dubins problem, the solution is a juxtaposition of line segments and circle arcs. This problem is relevant for the path planning of a non-holonomic robot, such as a wheeled vehicle. The proposed method is robust and computationally inexpensive with respect to existing solutions and is therefore suitable to be integrated as motion primitive into Dubins-based applications, e.g. orienteering problems or waypoint following robotics.