Arc Of Ellipse Research Articles

Nonlinear error elimination using the fusion of PGC-DCM, geometric fitting, and Atan algorithms.

A phase generated carrier (PGC) demodulation scheme is always accompanied by nonlinear errors. We propose a fusion of PGC differential and cross multiplying (PGC-DCM), geometric fitting, and arctangent (Atan) algorithms for fiber optic interferometric sensors to eliminate nonlinear errors. The output amplitude of the PGC-DCM algorithm is used to judge whether the Lissajous figure of quadrature signals is larger than 1/2 ellipse arc. When the Lissajous figure exceeds 1/2 ellipse arc, the contaminated quadrature signals are corrected by the ellipse correction parameters calculated from the geometric fitting; otherwise, the previous fitting parameters are employed for correction. Geometric fitting is realized by minimizing the Sampson error, and its failure problem under small signals is solved by using the temporary stability of fitting results. Finally, desired signals are extracted from the corrected quadrature signals by the Atan algorithm. Experimental results show that the fusion combines the merits of the three algorithms and expands the application of the geometric fitting in PGC demodulation schemes.

Ameliorated 3 × 3 coupler-based demodulation algorithm using iteratively reweighted ellipse specific fitting.

Passive demodulation scheme using 3 × 3 coupler has been widely used in phase-sensitive optical time-domain reflectometry (φ-OTDR), interrogation of fiber Bragg gratings or fiber optic interferometric sensors, and sensor multiplexing. However, the asymmetry of the 3 × 3 coupler in real applications affects the demodulation performance seriously. We proposed an ameliorated 3 × 3 coupler-based demodulation algorithm using iteratively reweighted ellipse specific fitting (IRESF) to overcome the drawback. IRESF combines iterative reweight technology with ellipse specific fitting, which decreases the weights of high noise points and always outputs ellipse solutions. Any two output signals from the 3 × 3 coupler-based interferometer are fitted by the IRESF and then corrected as a pair of quadrature signals. The stability of the fitting parameters is utilized to resolve the failures of IRESF under small signals. A real-time 1/4 ellipse arc judging module is designed, if the Lissajous figure is larger than 1/4 ellipse arc, IRESF is executed to offer ellipse correction parameters. Otherwise, the fixed parameters preset in the algorithm are used. The fixed parameters are mean values of the fitting parameters of IRESF under a large stimulus. The desired phase signal is finally extracted from the corrected quadrature signals. Experimental results show that the ameliorated algorithm does not require strict symmetry of the 3 × 3 coupler and can work under small signals. The noise floor of the proposed algorithm is -112 dB re rad/√Hz and the demodulated amplitude is 23.15 dB (14.37 rad) at 1 kHz when THD is 0.0488%. Moreover, the response linearity is as high as 99.999%. Compared to the algorithm using direct least squares, the proposed demodulation algorithm is more robust and precise, which has broad application prospects.

High Precision and Stabilization PGC Demodulation Scheme for Fiber Optic Interferometric Sensors

Ellipse fitting algorithm (EFA) has attracted tremendous interest in phase demodulation schemes to eliminate nonlinear errors, however, there is an inherent drawback that the EFA fails to work when the desired phase signal is small. A high precision and stabilization phase generated carrier (PGC) demodulation scheme using hyper least squares (LS) fitting of ellipse is proposed for the first time, to our best knowledge. The contaminated quadrature signals obtained after mixing and low pass filtering are processed by a hyper LS based EFA to eliminate nonlinear errors. Hyper LS relies on algebraic distance minimization with a carefully chosen scale normalization and statistical bias is eliminated up to second order noise terms, which is far superior to the traditional LS. The temporary stability of the fitting result is used to resolve the drawback of the EFA. A module for judging the ellipse arc of quadrature signals is developed. The EFA is executed when the Lissajous figure is larger than 1/4 ellipse arc, otherwise the fitting result of the previous executed EFA is used to correct the ellipse. Experimental results show that the proposed scheme is highly precise and stable regardless of the desired signal amplitude. It has a high response linearity of 99.997%, a minimum detectable phase of 3.16 μrad and a large dynamic range of 123.52 dB @1 kHz & 1% THD.

Stadiums based on curvilinear geometry: Approximation of the ellipsoid offset surface

Abstract This article provides an effective method of determining the coordinates of points and the angles for various surfaces approximating the ellipsoid surface. The stands of the stadium (with a capacity of up to 82,000 seats) with rows shaped by ellipse arcs were designed (in accordance with the European Standard EN 13200-1). An algorithm was proposed to determine the optimal height of the ring steps (with the guarantee of unobstructed vision of the entire playing field from each seat of the stands). The second algorithm ensures a comfortable space in each individual seating place in the stands (deviation of the depth of the row generated by ellipse arcs did not exceed 0.000009 m at any place of the stands). This article encourages readers to program DXF files (read in the AutoCAD system) to streamline design work.

Experiment of swing and deformation of a droplet on a horizontal fiber due to crosswind

The purpose of this paper is to experimentally investigate the oscillate of a clam-shell shaped droplet on a horizontal fiber by the action of crosswind, quantitatively describe the swing and deformation of the droplet, and analyze the damping force by surface tension. The dynamic images of the droplet are recorded by high-speed cameras, the profile curves in the top view and side view of droplet are both approximated to multiple ellipse arcs. Four kinds of liquids and three nylon fibers are used in the experiments. The experimental results show that the droplet oscillates periodically around the fiber by the flow drag force, the swing frequency is inversely proportional to the droplet mass, but does not increase with Re. In addition to the swing around the fiber, the droplet also oscillates periodically in the radial direction, and the frequency of radial oscillation is higher than that of circular swing, which results in the swing amplitude fluctuation. By the flow pressure, the windward side and the leeward side of the droplet deform asymmetrically, the difference of surface tension force on both sides of the droplet forms a moment against the droplet swing around fiber, the resistance moment increases with the liquid surface tension coefficient and fiber diameter.

Wear characteristics of multi-tooth milling cutter in milling CFRP and its impact on machining performance

Multi-tooth milling cutters with segmented right-hand helical cutting edge (SRHCE) and segmented left-hand helical cutting edge (SLHCE) could effectively reduce milling damages for carbon fiber reinforced polymer (CFRP) composites. However, rapid tool wear is still challenging for long-distance milling, and especially wear characteristics of multi-tooth milling cutter and their impacts have not been fully understood. This paper investigated the wear characteristics of the uncoated multi-tooth milling cutter with SLHCE and SRHCE under various feeds per tooth (0.01–0.15 mm/tooth) and a constant spindle speed of 3000 rpm and the effects of tool wear on machining performance by theoretical and experimental analyses. SLHCE and SRHCE experienced abrasive wear, while the wear of SLHCE is slighter than SRHCE because a clearance was formed between SLHCE and nominal tool periphery. Consequently, the profiles of worn SRHCE and SLHCE were different, and they were in ellipse arc and circular arc shape, respectively. The half axis length of ellipse arc worn profile of SRHCE decreased significantly with increasing feed per tooth, while the cutting edge radius of SLHCE changed irregularly. As tool wear aggravated with longer cutting distance, the machining qualities generally worsened, including increased burr damages and surface roughness, and cutting force also increased. Fewer burr damages were found on the top ply, as SLHCE dominated the cutting of the top ply with slighter tool wear because of the clearance, compared to the bottom ply, where SRHCE dominated cutting. In addition, the lowest feed per tooth (0.01 mm/tooth) induced severe tool wear and produced worse machining qualities, especially under longer cutting distances, while the milling force was relatively low. Therefore, more attention should be paid to the SRHCE for multi-tooth milling cutters, and a higher feed rate is preferred under similar conditions.

Construction of conical axoids on the basis of congruent spherical ellipses

Purpose: To carry out the transition from a cylindrical gear in which the centroids are congruent ellipses with centres of rotation in the foci, to a bevel gear on the basic of congruent spherical ellipses. Design/methodology/approach: Congruent ellipses with centres of rotation in the foci serve as centroids for the design of cylindrical gears with non-circular wheels. The article analytically shows that the analogues of ellipses on the plane - congruent spherical ellipses are the basis for the construction of the axoids of the corresponding bevel gears. An analogue of the centre-to-centre distance for ellipses in the plane is the angle between the axes of rotation of conical axoids. Findings: Based on the equality of the arcs of ellipses, the dependence of the angle of rotation of one axoid on the angle of rotation of another is found. Graphs of this dependence for separate cases are given. It is shown under what conditions the axes of axoids intersect at right angle. The parametric equations of spherical ellipses and corresponding axoids are given. They were used to construct spherical ellipses and corresponding conical axoids for different cases. For gears with right angle between the axes, separate positions of the axoids with different angles of their rotation around their axes are constructed. Practical implications: Spherical ellipses are directing curves for the construction of the corresponding conical axoids. Originality/value: The paper shows that congruent spherical ellipses act as centroids for the design of axoids of bevel gears. They roll one by one without sliding, rotating around axes that intersect in the centre of the sphere. To design such gears, it is important to know the interdependence between the geometric parameters, especially for common gears with a right angle between the axes.

Representation of a Monotone Curve by a Contour with Regular Change in Curvature.

The problem of modelling a smooth contour with a regular change in curvature representing a monotone curve with specified accuracy is solved in this article. The contour was formed within the area of the possible location of a convex curve, which can interpolate a point series. The assumption that if a sequence of points can be interpolated by a monotone curve, then the reference curve on which these points have been assigned is monotone, provides the opportunity to implement the proposed approach to estimate the interpolation error of a point series of arbitrary configuration. The proposed methods for forming a convex regular contour by arcs of ellipses and B-spline ensure the interpolation of any point series in parts that can be interpolated by a monotone curve. At the same time, the deflection of the contour from the boundaries of the area of the possible location of the monotone curve can be controlled. The possibilities of the developed methods are tested while solving problems of the interpolation of a point series belonging to monotone curves. The problems are solved in the CAD system of SolidWorks with the use of software application created based on the methods developed in the research work.

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.

A geometric study of different curves for the rotor profiles of a twin-screw compressor

This paper presents the results of comparisons among some patented solutions for profiling the contours of the rotors in twin-screw compressors. Referring to a base case where all the generating curves are circumferences, patents suggesting to replace arcs of circumference with arcs of conic sections, i.e. parabola, ellipse and hyperbola, but even a straight line segment, are presented and guidelines for rotor profile construction are reported. After setting the size of the compressor, attention is paid to the inter-lobe area, as the sum of the area between two consecutive lobes in the male rotor and of the area of the groove in the female rotor. Actually, this area is strictly related to the volume displacement. Limited to the current case study, the profile including an elliptic segment seems to be the preferable solution for higher inter-lobe area, then for higher displacement, though a number of considerations should be necessary for a broader context.

Fatigue endurance limit and crack front evolution in metallic glass

Fatigue endurance limit is a threshold stress-amplitude under which a solid subjected to a cyclic loading can sustain infinite life. Such a limit has been confirmed in ferrous materials but remains controversial in many advanced new materials, including bulk metallic glasses with superb strength. By using a combination of ultrasonic fatigue (USF) test and conventional fatigue test, we obtain the stress amplitude vs. loading cycle curve of the Vitreloy 1TM metallic glass, with loading cycles up to 109. There is a clear fatigue endurance limit (FEL) which is about 320 MPa, and is 17% of the strength of Vitreloy 1TM. The residual strength of survivals after 109 cycles is nearly the same as those of intact samples. We demonstrate that all fatigue cracks initiate from spherical pores which are inevitable and intrinsic amid casting and fast cooling, and the fatigue crack front resembles an ellipse arc till final rapid rapture of all tested samples. The size and position of a pore can be utilized to extract the fatigue initiation threshold K0th, and the elliptical fatigue crack front is used to obtain the mode I fracture toughness KIC, which are K0th=2.0±0.5MPam and KIC=18-20MPam for Vitreloy 1TM, respectively. The fatigue endurance limit and its intrinsic origin of Vitreloy 1TM pave the way to understand fatigue in other metallic glasses.

Pictorial integration in the Calculus

ABSTRACT Graphical analysis of the relationships between pairs of integrals and their corresponding derivatives resulted in the development of a procedure one may describe as ‘pictorial integration’. The arctangent of the derivative of various functions was analysed to confirm that this accurately reports the magnitude of angles made by integral functions from the horizontal, for polynomial, trigonometric, exponential, and transcendental forms. This was used to generate pictures of certain functions that have no known simple algebraic formula. The pictorial representations for the integral of and for the ellipse arc length integral are described. The pictures were confirmed through the use of Newton (Taylor) series expressions over appropriate domains, the second derivatives of the functions which are the first derivatives of their known derivatives, and electronic integration. Several powerful computer systems are available that quickly and easily graph such complicated integrals and confirm these results, but the analysis here explains for Calculus students the detailed graphical relationships between integrals and their derivatives.

Analytical Determination of Arc Length of an Elliptic Curve

Elliptic curves are commonly used to outline various geophysical objects, natural and artificial bodies, trajectories of the planets, satellites, elementary particles. Utilizing these curves, many theoretical and applied problems in the field of mechanics, geophysics, and mathematical physics can be solved. However, the dependences for finding the length of individual sections (arcs) of the ellipse that are in demand for engineering practice have not been developed up to the present day. The article deals with analytical dependences for determining the length of the arc and entire ellipse that are obtained on the basis of the representation of incomplete and complete elliptic integrals of the 2nd kind via elementary functions. These dependences coincide with the known exact formulas for the boundary counters of an ellipse, namely, a circle and a horizontal segment of a straight line (thin plate). Graphs of the dependence of the ellipse arc length on the central angle and the ratio of the semiaxes are also given. Comparison of the obtained results of solving the test problem by using the proposed method with the reference values found from numerical calculation using the “Mathematic” software has showed a good agreement (≤ 0.5%).

Simultaneous Chain-Forming and Generalization of Road Networks

Streets are essential entities of urban terrain and their automatic extraction from airborne sensor data is cumbersome because of a complex interplay of geometric, topological, and semantic aspects. Given a binary image representing the road class, centerlines of road segments are extracted by means of skeletonization. The focus of this paper lies in a well-reasoned representation of these segments by means of geometric primitives, such as straight line segments as well as circle and ellipse arcs. Thereby, we aim at a fusion of raw segments to longer chains which better match to the intuitive perception of what a street is. We propose a two-step approach for simultaneous chain-forming and generalization. First, we obtain an over-segmentation of the raw polylines. Then, a model selection approach is applied to decide whether two neighboring segments should be fused to a new geometric entity. For this purpose, we propose an iterative greedy optimization procedure in order to find a strong minimum of a cost function based on a Bayesian information criterion. Starting at the given initial raw segments, we thus can obtain a set of chains describing long alleys and important roundabouts. Within the procedure, topological attributes, such as junctions and neighborhood structures, are consistently updated, in a way that for the greedy optimization procedure, accuracy, model complexity, and topology are considered simultaneously. The results on two challenging datasets indicate the benefits of the proposed procedure and provide ideas for future work.

Algorithm for geometrical and metrological characterization of gears with low module and high number of teeth: low-cost gears

In the specific case of low-cost gears, and also in some general cases, the use of expensive instruments such as gear measuring machines may not be appropriated during their verification. In the scientific literature there are many works that allow the development of mathematical models capable of establishing the geometric elements of spur gears, necessary for the determination of deviations in circular pith according to what is included in the ISO 1328-1:2013 standard. In this work, the verification of gears from the adjustment and measurement of their flanks treating them as ellipse arcs is presented as an alternative to classic verification. Most of the software implemented in the coordinate measuring machines and the vision machines, has detection and adjustment algorithms of ellipse arcs. However, it is not so common the availability of gear analysis software, and in any case, it would require the payment of expensive additional licenses. The authors present an alternative way in which, from the elliptical arc adjusted to the flank of the tooth, it is possible to determine the fundamental parameters of the gear (tooth thickness, circular pitch and profile form deviation). Naturally, it is necessary to determine under what circumstances (module, number of teeth, gear quality) the approach of the tooth flank by an ellipse is feasible.

Особенности свойств параболы при ее моделировании

When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task "Contour Construction" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.

On the statistical model of source localization based on Range Difference measurements

In this work we study the statistical model of source localization based on Range Difference measurements, under the assumption of Gaussian noise on the data. Our analysis is based on a previous work of the same authors concerning the localization in a noiseless scenario. We investigate the case of planar localization of a source using a minimal configuration of three non aligned receivers. We have four curved exponential families corresponding to four different, non disjoint, regions of the feasible set. For each family we solve Maximum Likelihood Estimation (MLE). This requires to find the projection of a point on a set of segments and arcs of ellipse. Then, we perform the analytic study of the localization accuracy. In particular, we give formulas for mean square error and bias of MLE, depending on the displacement vectors. We validate the results through Monte Carlo simulations, in a given setup of the receivers. As the set of feasible measurements is a semialgebraic variety, this investigation makes use of techniques from Algebraic Statistics and Information Geometry.

CHAIN-WISE GENERALIZATION OF ROAD NETWORKS USING MODEL SELECTION

Abstract. Streets are essential entities of urban terrain and their automatized extraction from airborne sensor data is cumbersome because of a complex interplay of geometric, topological and semantic aspects. Given a binary image, representing the road class, centerlines of road segments are extracted by means of skeletonization. The focus of this paper lies in a well-reasoned representation of these segments by means of geometric primitives, such as straight line segments as well as circle and ellipse arcs. We propose the fusion of raw segments based on similarity criteria; the output of this process are the so-called chains which better match to the intuitive perception of what a street is. Further, we propose a two-step approach for chain-wise generalization. First, the chain is pre-segmented using circlePeucker and finally, model selection is used to decide whether two neighboring segments should be fused to a new geometric entity. Thereby, we consider both variance-covariance analysis of residuals and model complexity. The results on a complex data-set with many traffic roundabouts indicate the benefits of the proposed procedure.

Multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge

Abstract. Flexure hinges made of superelastic materials is a promising candidate to enhance the movability of compliant mechanisms. In this paper, we focus on the multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge. The objective is to determine a set of optimal geometric parameters that maximizes the motion range and the relative compliance of the flexure hinge and minimizes the relative rotation error during the deformation as well. Firstly, the paper presents a new type of ellipse-parabola shaped flexure hinge which is constructed by an ellipse arc and a parabola curve. Then, the static responses of superelastic flexure hinges are solved via non-prismatic beam elements derived by the co-rotational approach. Finite element analysis (FEA) and experiment tests are performed to verify the modeling method. Finally, a multi-objective optimization is performed and the Pareto frontier is found via the NSGA-II algorithm.

A Practical Method for Generating Trigonometric Polynomial Surfaces over Triangular Domains

A class of trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis functions, a kind of trigonometric polynomial patch over triangular domain, which can be used to construct some surfaces whose boundaries are arcs of ellipse or parabola, is proposed. Without changing the control points, the shape of the trigonometric polynomial patch can be adjusted flexibly in a foreseeable way using the shape parameters. For computing the proposed trigonometric polynomial patch stably and efficiently, a practical de Casteljau-type algorithm is developed. Moveover, the conditions for G1 continuous smooth joining two trigonometric polynomial patches are deduced.