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Geographic routing in wireless ad hoc networks is characterized by routing decisions made from locally available position information, which entails network scalability. However, it requires an effective recovery approach to sending a packet bypassing network holes whenever the simple greedy forwarding fails. Among well-known approaches, rotational sweep routing algorithms based on a circular arc are able to achieve packet delivery guarantee as well as low routing path stretch under the impractical assumption that a wireless link exists between two nodes if and only if their distance is less than one unit. Instead, we propose a cooperative rotational sweep algorithm taking into account practically imperfect wireless connections. The algorithm involves a regular rotational sweep procedure and a cooperative one both making use of iterative sweeps with circular arcs of decreasing size subjective to a minimum size constraint. Essentially, the cooperative rotational sweep procedure resolves hidden node issues through exploiting packet header overheads for memory of the latest routing path while iterative sweeps reduce the possibility of missing pivotal relays. Simulation results demonstrate that the proposed scheme presents the benefit of using packet header overheads to support high end-to-end routing success probabilities without sacrificing the inherent feature of localized routing. [ABSTRACT FROM AUTHOR]

A "circular arc–involute–circular arc" circular arc gear pump was developed based on a gear meshing principle and coordinate transformation as well as an accurate calculation model of the radial force. The dependence of the radial force on the meshing angle was investigated. The temperature rise of journal bearings in the pump was evaluated for bearings with and without herringbone grooves. Furthermore, the influence of the rotational speed and outlet pressure on this rise was assessed. The results revealed using herringbone groove on the inner wall of bearing was effective in reducing the temperature increase. Therefore, the use of grooves represents a suitable method of reducing the temperature rise in the journal bearings of a high-speed gear pump. [ABSTRACT FROM AUTHOR]

In the field of visual navigation, crater is an ideal navigation landmark on planet surface, because of its universality and significance. During the descent phase of lander, crater needs to be extracted and matched. However, as the landing proceeds, images taken by the on-board camera will change, such as the variation of scale, translation, rotation, illumination which will increase the difficulty of the matching algorithm. This paper proposes a more accurate and faster crater matching algorithm based on feature descriptor. This algorithm is designed for navigation during the descending phase. It has the invariance of scaling, rotation, illumination. First, this algorithm extracts circular arc by Gaussian Pyramid and ELSD algorithm, so that this algorithm has the invariance against scale variation. Then, in order to describe the circular arc, this algorithm determine the direction of the circular arc by the direction histogram. And this algorithm constructs the circular arc band descriptors in the support region to achieve translation, rotation and illumination invariance. At last, matching criteria is the combination of the Nearest Neighbor Distance Ratio (NNDR) and the Euclidean distance constraint. The results are obtained under different image variations. The matching results show that this crater matching algorithm has high crater correct extracting and matching rate and high computationally efficient. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit representation of the complex Chebyshev polynomials on a given arc of the unit circle (in the complex plane) in terms of real Chebyshev polynomials on two symmetric intervals (on the real line). The real Chebyshev polynomials, for their part, can be expressed via a conformal mapping with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's. [ABSTRACT FROM AUTHOR]

The geometric properties of conjugated circular arcs connecting two points on the plane with set directions of tan- gent vectors are studied in the work. It is shown that pairs of conjugated circular arcs with the same conditions in frontier points create one-parameter set of smooth curves tightly filling all the plane. One of the basic properties of this set is the fact that all coupling points of circular arcs are on the circular curve going through the initially given points. The circle radius depends on the direction of tangent vectors. Any point of the circle curve, named auxiliary in this work, determines a pair of conjugated arcs with given boundary conditions. One more condition of the auxiliary circle curve is that it divides the plane into two parts. The arcs going from the initial point are out of the circle limited by this circle curve and the arcs coming to the final point are inside it. These properties are the basis for the method of conjugated circular arcs tracing pro- posed in this article. The algorithm is rather simple and allows to fulfill all the needed plottings using only the divider and ruler. Two concrete examples are considered. The first one is related to the problem of tracing of a pair of conjugated arcs with the minimal curve jump when going through the coupling point. The second one demonstrates the possibility of trac- ing of the smooth curve going through any three points on the plane under condition that in the initial and final points the directions of tangent vectors are given. The proposed methods of conjugated circular arcs tracing can be applied in solving of a wide variety of problems connected with the tracing of cam contours, for example pattern curves in textile industry or in computer-aided-design systems when programming of looms with numeric control.

Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree 10 rather than 6; the Chebyshev error function equioscillates 11 times rather than 7; the approximation order is 10 rather than 6. The method approximates more than the full circle with Chebyshev uniform error of 1=29. The examples show the competence and simplicity of the proposed approximation, and that it can not be improved. [ABSTRACT FROM AUTHOR]

A geometric model of a 1-pair of circular arcs modified wrap (1-CMW), the existing circular arc modified wrap (CMW), was developed. In order to improve the performance and overcome its shortcoming of the CMW, a novel n -pair of circular arcs modified wrap (n -CMW) for scroll compressors was proposed based on the midline design method. A 2-CMW was analyzed, and then a general geometric model of n -CMW with arbitrary pair number n was established. In addition, the generation procedure, general mutual dependence of geometric parameters as well as equations of wrap profiles were obtained, and the influence of geometric parameters on shapes of the n -CMW was analyzed. Flow simulations of the 1-CMW, 2-CMW, 3-CMW and 4-CMW during the working process with the same conditions were conducted. Study results indicate that the proposed n -CMW increases the built-in volume ratio and design flexibility, and is superior to the existing CMW. [ABSTRACT FROM AUTHOR]

Abstract We propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves. The approach is based on a constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the Bézier form is derived and a detailed analysis provided for some low degree cases which were not studied yet. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the simplified radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given. [ABSTRACT FROM AUTHOR]

Highlights • A compact shape representation by piecewise-circular curves suitable for efficient transformation and simplification. • A practical method to compute the coefficients of the Fourier series of the piecewise-circular curve. • A practical method to compute the shape moments of the enclosed region. • As a result, matching and alignment of nonpolygonal shapes based on these standard techniques can be done more efficiently. Abstract Many natural and artificial shapes can be compactly described by sequences of circular arcs and straight line segments. This paper presents practical methods to compute the coefficients of the Fourier series of a piecewise-circular curve and the moments of the enclosed region. These descriptors are the basis of simple and widely used techniques for shape matching and alignment, that can now be efficiently applied to a more expressive data representation. [ABSTRACT FROM AUTHOR]

In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equioscillates five times; the approximation order is four. For θ = π/4 arcs (quarter of a circle), the uniform error is 5.5 × 10−3. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfy the properties of the best uniform approximation and yield the highest possible accuracy.

Circular arcs are a graphical element present in the cadastral systems of many countries. Unfortunately, this type of record of the geometry of parcel borders is a problem described by the directives of Infrastructure for Spatial Information in Europe (INSPIRE) Data Specification on Cadastral Parcels. Because of the difficulties of using such geometric objects, the solution to this problem, as recommended by the European Commission, should be monitored. The target effect should be a cadastral data model based solely on linear segments. Solutions based on a classic approach of converting such data (like arcs), unfortunately, always involves changes of one of the most important attributes of a parcel—its area. The paper presents a proposal for solving this important problem using an algorithm, ensuring the preservation of the area of the parcels after converting the arcs into linear segments. Moreover, attention was paid to the technical aspects of the proposed changes.

Abstract We describe the details of an unconfined insensitive high explosive (PBX 9502) circular arc section experiment, in which, after a transient period, a detonation sweeps around the arc with constant angular speed. The arc section is sufficiently wide that the flow along the centerline of the arc section remains two-dimensional. Data includes time-of-arrival diagnostics of the detonation along the centerline inner and outer arc surfaces, which is used to obtain the angular speed of the steadily rotating detonation. We also obtain the lead shock shape of the detonation as it sweeps around the arc. Reactive burn model simulations of the PBX 9502 arc experiment are then conducted to establish the structure of the detonation driving zone, i.e. the region enclosed between the detonation shock and flow sonic locus (in the frame of the steady rotating detonation). It is only the energy released in this zone which determines the speed at which the steady detonation sweeps around the arc. We show that the sonic flow locus of the detonation driving zone largely lies at the end of, or within, the fast reaction stage of the PBX 9502 detonation, with the largest section of the detonation driving zone lying close to the inner arc surface. We also demonstrate that the reactive burn model provides a good prediction of both the angular speed of the detonation wave and the curved detonation front shape. [ABSTRACT FROM AUTHOR]

The aim of this paper is a construction of parametric polynomial interpolants of a circular arc possessing maximal geometric smoothness. Two boundary points of a circular arc are interpolated together with higher order geometric data. Construction of interpolants is done via a complex factorization of the implicit unit circle equation. The problem is reduced to solving only one nonlinear equation determined by a monotone function and the existence of the solution is proven for any degree of the interpolating polynomial. Precise starting points for the Newton–Raphson type iteration methods are provided and the best solutions are then given in a closed form. Interpolation by parametric polynomials of degree up to six is discussed in detail and numerical examples confirming theoretical results are included. [ABSTRACT FROM AUTHOR]

It has been suggested that a stable Leading Edge Vortex (LEV) can be formed from the sharp leading edge of asymmetric spinnakers, which are high-lift sails used by yachts to sail downwind. If the LEV remains stably attached to the leading edge, it provides an increase in the thrust force. Until now, however, the existence of a stable and attached LEV has only been shown by numerical simulations. In the present work we experimentally verify, for the first time, that a stable LEV can be formed on an asymmetric spinnaker. We tested a 3D printed rigid sail in a water flume at a chord-based Reynolds number of ca. 10 4 . The sail was tested in isolation without hull and rigging. The flow field was measured with Particle Image Velocimetry (PIV) over horizontal cross sections. We found that on the leeward side of the sail (the suction side), the flow separates at the leading edge reattaching further downstream and forming a stable LEV. The LEV grows in diameter from the root to the tip of the sail, where it merges with the tip vortex. We detected the LEV using the γ criterion, and we verified its stability over time. The lift contribution provided by the LEV was computed solving a complex potential model of each sail section. This analysis indicated that the LEV provides more than 20% of the total sail's lift. These findings suggest that the maximum lift of low-aspect-ratio wings with a sharp leading edge, such as spinnakers, can be enhanced by promoting the formation of a stable LEV. [ABSTRACT FROM AUTHOR]

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

This article presents a method of approximating an arc using a polygon. The method uses the condition that the approximated arc describes equal surface areas of the circular sector. The method described in this article ensures that the polygons obtained by approximation have the same area as the original geometric. [ABSTRACT FROM AUTHOR]

In the past 20 years, protected agriculture has been developed rapidly in China, with a total area of 4,000 kha. A large number of mobile platforms are urgently needed to satisfy the need of operation, and research on autonomous vehicles with navigation ability has been highly valued. For structured hard pressed road surface between cultivation beds in modern greenhouse, curb-following navigation technology is most valuable for autonomous vehicle working in greenhouse. However, present curb-following navigation technologies based on machine vision are too complicated and lack of stability, while navigation technologies based on distance sensing cannot reach ideal accuracy of position/orientation detection. The target of curb-following navigation in this paper was to control the vehicle body within a stable position/orientation relative to the curb based on limited simple high/low level signals from arc array of photoelectric switches. Principle of detection and control of position/orientation based on both arc array of photoelectric switches and ideal control area was firstly introduced, and index Nd that represents the number of triggered photoelectric switches and index Nf which represents the center sequence number of triggered photoelectric switches were introduced to establish a position/orientation detection model. The position/orientation was classified by different thresholds of the two indices which can trigger the corresponding control program of trajectory. And then accurate position and orientation which were calculated by the values of two indexes were used to set the parameters of control program to realize the curb line following navigation in real time. This method can realize precise navigation with just limited number of high-low signals from photoelectric switches. The curb-following navigation accuracy relies on number of photoelectric switches, radius and central angle of the arc array, while triggered time interval between adjacent photoelectric switches must satisfy the need of response of sensing, control and mechanical transmission for a vehicle. Differential controlling strategies for different position/orientation states of vehicle based on arc array of photoelectric switches were also put forward. It was found from experiments that as the speed increased, both the transverse error and course angle of the vehicle displacement would rise. While within the designed speed, neither violent shock nor instability was found. Experiment results also indicated that the deviations of vehicle’s position and orientation were kept -35 mm to +15 mm and -5° to +5°, respectively. Relative to the size of the greenhouse vehicle, the error was acceptable which satisfied the curb-following navigation requirement of pesticide spraying, seedling transplanting, transporting etc. in greenhouse. And the control cycle of trajectory was about 2 m along the curb line, which indicated that the vehicle can run smoothly along the curb line under a low control frequency in the greenhouse based on the method above. It was also found that this method could maintain better curb-following navigation accuracy even under interference of sundries if its length was not more than 300 mm, which may meet the actual need of vast majority of the production. In this paper, we proposed new technical ideas for robots running along curb with low cost in the greenhouse. [ABSTRACT FROM AUTHOR]

In this paper, the problem of interpolation of two points, two corresponding tangent directions and curvatures, and the arc length sampled from a circular arc (circular arc data) is considered. Planar Pythagorean–hodograph (PH) curves of degree seven are used since they possess enough free parameters and are capable of interpolating the arc length in an easy way. A general approach using the complex representation of PH curves is presented first and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6 and a detailed analysis of the existence of admissible solutions is provided. In the case of several solutions, some criteria for selecting the most appropriate one are described and an asymptotic analysis is given. Numerical examples are included which confirm theoretical results. • The interpolation of G2 data and an arc lengths is stated in general. • A detailed analysis for circular arc data is provided. • Four real solutions of the problem are confirmed. • An asymptotic analysis together with asymptotic approximation order is given. • A selection of appropriate solution is suggested and everal numerical examples are shown. [ABSTRACT FROM AUTHOR]

Development and wide use of route guidance systems lead to the need for suitable digital maps that can be used for some advanced applications. Sufficient accuracy of road geometry with emphasis on road centerline positions and curvature is crucial. In this paper is presented a method for finding road centerline curvature from raw GPS data. The approach consists of a few processing steps. First it is necessary to fit raw data of each road section using B-splines, and generate equidistant vertices of polyline of the fitted curve. Then follows the appliance of stereographic projection of chosen polyline segments onto the unit sphere. Using the least square method, the plane that best fits the points on the unit sphere is found and the circle that is the intersection of the plane and the unit sphere. Stereographic projection of this circle back to the equatorial plane gives the corresponding circular arc and curvature. The method is also applicable in higher dimensions. The 3D case is numerically presented and results show that the proposed procedure is efficient and yields accurate results.

We show that the nerve and clique complexes of n arcs in the circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time $$O(n\log n)$$ . For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine explicit homology bases and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we show that the Lovász bound on the chromatic number of a circular complete graph is either sharp or off by one. Second, we use the connection to cyclic polytopes to give a novel topological proof of a known upper bound on the distance between successive roots of a homogeneous trigonometric polynomial. Third, we show that the Vietoris-Rips or ambient Čech simplicial complex of n points in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension, and furthermore this homotopy type can be computed in time $$O(n\log n)$$ . [ABSTRACT FROM AUTHOR]

In this paper, we prove a sharp Bernstein-type inequality for algebraic polynomials on circular arcs. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers dimensional synthesis of a six-bar linkage with turning kinematic pairs, in the literature known as Stephenson III Six-bar linkage. The synthesis procedure started from the requirement that it should be a double dwell mechanism and that the coupler point, during dwell, should pass through the given points belonging to the circular arc. The coordinates of those points are not directly given; they lie on the circular arc defined by the corresponding centre of the curve and the crank angle that corresponds to the circular arc of the path during dwell. These values are obtained at the end of the optimization procedure so that in this case the precision points change their positions with the change of the position of the arc on which they lie. As this is the case with double dwell, the coupler point should describe a curve containing two circular arcs which does not have to be symmetric. A new metaheuristic algorithm, known as Cuckoo Search (CS), was used in the procedure of optimum synthesis of mechanism parameters. [Copyright &y& Elsevier]

Abstract: The mathematical modelling of column buckling or beam bending under an axial or transverse load is well established. However, the existent models generally assume a high degree of symmetry in the structure of the column and minor longitudinal and transverse displacements. The situation when the column is made of several components with different mechanical properties asymmetrically distributed in the transverse section, semi-rigid, and subjected to multiple axial loads with significant longitudinal and transverse displacements through compression and bending has not been well characterised. A more comprehensive theoretical model allowing for these possibilities and assuming a circular arc contour for the bend is developed, and used to establish the bending axes, balance between compression and bending, and equivalent stiffness of the column. In certain situations, such as with pull cable catheters commonly used for minimally invasive surgical procedures, the compression loads are applied via cables running through channels inside a semi-rigid column. The model predicts the mathematical relationships between the radius of curvature of the bend and the tension in and normal force exerted by such cables. Conjugate extension with reciprocal compression–bending is a special structural arrangement for a semi-rigid column such that extension of one segment is linked to compression–bending of another by inextensible cables running between them. Leads are cords containing insulated electrical conductor coil and cables between the heart muscle and cardiac implantable electronic devices. Leads can behave like pull cable catheters through differential component pulling, providing a possible mechanism for inside-out abrasion and conductor cable externalisation. Certain design features may predispose to this mode of structural failure. [Copyright &y& Elsevier]

Abstract: Based on the technique of C-shaped Hermite interpolation by a cubic Pythagorean-hodograph (PH) curve, we present a simple method for C-shaped Hermite interpolation by a rational cubic Bézier curve. The method reproduces a circular arc when the input data come from it. Both the Bézier control points, which have a well-understood geometrical meaning, and the weights of the resulting rational cubic Bézier curve are expressed in explicit form. We test our method with many numerical examples, and some of them are presented here to demonstrate the properties of our method. The comparison between our method and a previous method is also included. [Copyright &y& Elsevier]

Abstract: Modern CAD/CAM systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function. Parametric (rational) polynomials are used to approximate the circular arcs. A new approximation method by means of quartic Bézier curves is presented to effectively approximate the circular arcs. It is shown that our method provides the approximation of order eight and yields a smaller error than previous methods. By equally spaced subdivision of circular arcs, the curvature continuous spline approximation algorithm of the circular arcs is given. [Copyright &y& Elsevier]

Abstract: We present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two-point Taylor expansion. We obtain the degree- polynomial for the approximation problem in an efficient way, which is very convenient to increase the degree of polynomial by adding new terms. An upper bound on the approximation error is available, so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user-prescribed error tolerance. [Copyright &y& Elsevier]

Compound toroidal shells whose surface is generated by revolving joined circular arcs are analyzed for stability. The proposed approach makes exact allowance for the geometry of the shell and the way its segments are joined. Shells made of composite materials are analyzed as an example to demonstrate the possibility of optimizing the meridian shape to increase the critical loads for various constructive designs of the joint [ABSTRACT FROM AUTHOR]

Abstract: We obtain cubic and quartic Bézier approximations of circular arcs that respectively satisfy and end-point interpolation conditions. We identify the necessary and sufficient conditions for such approximations to be the best, in the sense that they have the minimum Hausdorff distance to the circular arc. We then establish the existence and uniqueness of these best approximations and present practical methods to calculate them, which are verified by examples. [Copyright &y& Elsevier]

Development and wide use of route guidance systems lead to the need for suitable digital maps that can be used for some advanced applications. Sufficient accuracy of road geometry with emphasis on road centerline positions and curvature is crucial. In this paper is presented a method for finding road centerline curvature from raw GPS data. The approach consists of a few processing steps. First it is necessary to fit raw data of each road section using B-splines, and generate equidistant vertices of polyline of the fitted curve. Then follows the appliance of stereographic projection of chosen polyline segments onto the unit sphere. Using the least square method, the plane that best fits the points on the unit sphere is found and the circle that is the intersection of the plane and the unit sphere. Stereographic projection of this circle back to the equatorial plane gives the corresponding circular arc and curvature. The method is also applicable in higher dimensions. The 3D case is numerically presented and results show that the proposed procedure is efficient and yields accurate results. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the optimum synthesis of a four-bar linkage in which the coupler point performs a path composed of rectilinear segments and a circular arc. The Grashof four-bar linkage whose geometry provides minimum deviations from the given problem for certain parts of the crank cycle is chosen. The motion of the coupler point of the four-bar linkage is controlled within the given values of allowed deviations so that it is always in the prescribed environment of the given point on the observed segment. The synthesis process tends to bring only those path segments that are beyond the boundaries within the prescribed boundary deviations. During the synthesis, allowed deviations change from the initial maximum values to the given minimum ones. Groups of mechanisms realising satisfactory approximation to the desired motion can be obtained by the method of controlled decrease of allowed deviations with the application of the Differential Evolution (DE) algorithm. [Copyright &y& Elsevier]

Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straight-line segments. We demonstrate this by way of three basic and frequent computations on shapes - convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape. [ABSTRACT FROM AUTHOR]

Abstract: Recently some Hermite-type interpolation methods using a rational parametric cubic were proposed; the methods reproduce a circular arc when the input data come from it. A Hermite-type interpolation method is now proposed which reproduces a circular arc when the input data come from it. [Copyright &y& Elsevier]

Abstract: Motivated by the recent work (Farin, G., 2006. Rational quadratic circles are parameterized by chord length, Computer Aided Geometric Design 23, 722–724), we identify a family of curves that can be parameterized by chord length. The α- and -schemes are presented for characterizing planar and spatial curves respectively. Rational chord-length parameterizations are thoroughly investigated. In particular, the low-degree rational curves such as cubics and quartics are studied and applied to geometric Hermite interpolation. The results advise that this new class of curves, subsuming straight lines and circular arcs, have several obvious advantages over general polynomial and rational curves. [Copyright &y& Elsevier]