Minimum Circumscribed Research Articles

Comparison of the Minimum Bounding Rectangle and Minimum Circumscribed Ellipse of Rain Cells from TRMM

Comparison of the Minimum Bounding Rectangle and Minimum Circumscribed Ellipse of Rain Cells from TRMM

A method for measuring geometric information content of area cartographic objects based on discrepancy degree of shape points

In order to improve the comparability between the geometric information content of vector area objects, this article proposes a method for measuring the geometric information content of area objects based on discrepancy degree of shape points. First, the method selects circles with unique geometric feature as the reference shape for extracting geometric features, and the geometric in-formation carried by each shape point of area objects is represented by the discrepancy degree between the area object and the reference circle at the point position. Second, the proposed method measures the geometric information content of area objects from both local and global perspectives. To avoid the subjectivity of assigning feature weights based on empirical experience, the article uses the relationships between the radii of three reference circles (MIC: Maximum Inscribed Circle, EAC: Equal-area circle, and MCC: Minimum Circumscribed Circle) as adaptive weight parameters for local and global structural geometric information. The amount of geometric information at each shape point is obtained by weighted summation, and the total geometric information content of an area object is the sum of the amount of geometric information of all shape points. To verify the effectiveness and rationality of the proposed method, this article designs a noise simulation dataset for simply building area objects and an empirical ranking dataset for evaluating the measurement performance of the proposed method. The experimental results show that the proposed method achieves a Kendall rank correlation coefficient of 0.88 on the empirical ranking dataset, which is higher than that of the nine existing representative methods. The proposed method is more consistent with human cognition and is highly correlated with the amount and intensity of noise information. Moreover, the proposed method achieves the comparability of geometric information content of area objects and the adaptive determination of geometric feature weights. The proposed method is an effective method for measuring the geometric information quantity of area objects.

Analysis and visualization of traceless symmetric tensors. Application to the Hencky strain tensor for large strain tension–torsion

Cyclic multiaxial loadings of soft materials are usually studied throughout experiments involving machines that prescribe a combination of uniaxial tension and torsion. Due to the large strain framework, classical kinematic analyses of strain in uniaxial tension–torsion are usually very complex. Based on this observation, the present paper proposes a method to both analyze and visualize a strain measure during a duty cycle of uniaxial tension–torsion in large strain: based on the mathematical properties of the Hencky strain tensor h, the method consists in projecting h onto a well-chosen tensor basis, whose constituting elements are described in terms of physical meaning. Thanks to this decomposition, the history of h reduces to the time evolution of a 3-component vector α(t). This vector history can then be visualized as a path in 3D space, rendering very visual the complex kinematic phenomenon. As a second result, an original definition of the mean and the amplitude of a strain path, based on the theory of the Minimum Circumscribed Spheres, is proposed. This definition could be useful for fatigue studies, for instance.

Geometric methods for estimating representative sidewalk widths applied to Vienna\u2019s streetscape surfaces database

Space, and in particular public space for movement and leisure, is a valuable and scarce resource, especially in today’s growing urban centres. The distribution and absolute amount of urban space—especially the provision of sufficient pedestrian areas, such as sidewalks—is considered crucial for shaping living and mobility options as well as transport choices. Ubiquitous urban data collection and today’s IT capabilities offer new possibilities for providing a relation-preserving overview and for keeping track of infrastructure changes. This paper presents three novel methods for estimating representative sidewalk widths and applies them to the official Viennese streetscape surface database. The first two methods use individual pedestrian area polygons and their geometrical representations of minimum circumscribing and maximum inscribing circles to derive a representative width of these individual surfaces. The third method utilizes aggregated pedestrian areas within the buffered street axis and results in a representative width for the corresponding road axis segment. Results are displayed as city-wide means in a 500 by 500 m grid and spatial autocorrelation based on Moran’s I is studied. We also compare the results between methods as well as to previous research, existing databases and guideline requirements on sidewalk widths. Finally, we discuss possible applications of these methods for monitoring and regression analysis and suggest future methodological improvements for increased accuracy.

Method for sphericity error evaluation using geometry optimization searching algorithm

According to geometrical characteristics of the sphericity error, a new evaluation algorithm for sphericity error based on geometry optimization searching method has been presented. First, the reference point is established and the initial error is calculated by using the measured points, Second, a regular hexahedron of side length f is collocated by taking the reference point as datum point, and the maximum difference of the radius of all measured points are calculated by regarding each vertex of the hexahedron as the centre of the measured spherical surface. Third, the reference point or side length of the hexahedron is changed by comparing the initial error and the maximum difference of the radius. Step by step, the sphericity error value of corresponding evaluation method (including Minimum Zone Sphere method (MZS), Minimum Circumscribed Sphere method (MCS) and Maximum Inscribed Sphere method (MIS)) are obtained. The principle and the steps of using the algorithm to solve the sphericity error are described in detail and the mathematical formula and program flowchart are given. The experimental results show that the sphericity error can be evaluated effectively and exactly with this algorithm.

Biaxial high-cycle fatigue life assessment of ductile aluminium cruciform specimens

Several results of experimental high-cycle fatigue tests carried out on a new biaxial in plane testing system are presented, covering a wide range of biaxial stress states, including proportional and non-proportional, performed on cruciform specimens machined from ductile aluminium (A1050-H14) sheet plate with 3mm thickness. A specially designed cruciform specimen for crack initiation and low capacity test machines was used to perform fatigue tests. This specimen was optimized for the test machine in order to achieve the maximum uniform stress at the centre gauge area, while keeping the remaining stresses at least 20% lower than the stress at centre. Several of the most common multiaxial fatigue criteria were used to determine an equivalent uniaxial fatigue damage parameter for all the specimens that were tested experimentally and to determine which criteria provide a better correlation to the experimental results. From the results it was shown that most of the criteria provide non-conservative estimations. In overall, better results were found by the Minimum Circumscribed Ellipse criterion and also when using the damage parameter related with the shear stress computed by the Minimum Circumscribed Ellipse in the modified non-linear criterion proposed by Carpinteri and Spagnoli.

Reacher on Ellipticity Error by Use of Minimum Circumscribed Ellipse and Maximum Inscribed Ellipse Algorithm

A data processing method based on a mesh search algorithm for elliptical profile error evaluation is proposed. First ellipse focuses of the least-square are set as reference points, according to a certain rules, a series of grid points are arranged and supposed as hypothetical ideal ellipse focuses respectively , then the distances of supposed ideal ellipse to the measuring points are calculated repeatedly. The minimum circumscribed ellipse and the maximum inscribed ellipse are constructed by comparison and judgment, and the corresponding value of ellipse profile error is obtained. Furthermore, the principle of the grid search algorithm and the process of ellipse profile error evaluation by using the minimum circumscribed ellipse method and the maximum inscribed ellipse method are stated . The results of examples show that the ellipse profile error can be evaluated accurately and effectively by this algorithm.

An accurate and efficient algorithm for determining minimum circumscribed circles and spheres from discrete data points

This paper presents a novel combinatorial search algorithm for determining minimum circumscribed (MC) circles and spheres from discrete data points. The presented algorithm is able to efficiently identify the essential subset of the input data points to construct the MC circle/sphere for the entire data set. The common issue of computational explosion for a large data set due to a greatly increased number of combinatorial searches is thus of no concern. The main feature of this work is the derivation of an innovative geometric property, named the Integrated Property (IP), for a unique three-point subset in 2D or a unique four-point subset in 3D. The significance is that the MC circle/sphere for the identified IP point subset is in fact the MC circle/sphere for the entire data set. As the unique IP point subset can always be found, the presented algorithm is guaranteed to yield exact MC circle/sphere solutions. A related data exchange scheme is formulated to efficiently identify the unique IP point subset from the input point set. The expected computational complexity of the search algorithm is quantified to be O(nlogn) through a large number of computational test cases.

The Relationship between Minimum Zone Sphere and Minimum Circumscribed Sphere and Maximum Inscribed Sphere

Among the four methods (minimum zone sphere, minimum circumscribed sphere, maximum inscribed sphere, and least square sphere), only the minimum zone sphere complies with ANSI and ISO standards and has the minimum sphericity error value. Evaluation of sphericity error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The minimum circumscribed sphere and the maximum inscribed sphere are all easily solved by iterative comparisons, so the relationship between the minimum zone sphere and the minimum circumscribed sphere, the maximum inscribed sphere is proposed to solve efficiently the minimum zone problem. The relationship is implemented and validated with the data available in the literature.

Sphericity Evaluation Based on Minimum Circumscribed Sphere Method

The search of sphericity evaluation is a time-consuming work. The minimum circumscribed sphere (MCS) is suitable for the sphere with the maximum material condition. An algorithm of sphericity evaluation based on the MCS is introduced. The MCS of a measured data point set is determined by a small number of critical data points according to geometric criteria. The vertices of the convex hull are the candidates of these critical data points. Two theorems are developed to solve the sphericity evaluation problems. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the sphericity problems.

Four Methods for Roundness Evaluation

Abstract Whether roundness error can be evaluated accurately and efficiently or not will directly influence the mechanical products performance and life. For this reason, this paper introduces simple and efficient algorithms to evaluate the roundness error from the measured points using four internationally defined methods: Least Squares Circle (LSC), Minimum Circumscribed Circle (MCC), Maximum Inscribed Circle (MIC) and Minimum Zone Circles (MZC). A software has been developed using Matlab to apply these algorithms on the test data.

Mathematical Model for Evaluating Roundness Errors by Minimum Circumscribed Circle Method

An unconstrained optimization model is established for assessing roundness errors by the minimum circumscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the minimun circumscribed circle assessment. One example is given to verify the theoretical results presented.

MAXIMUM- INSCRIBED AND MINIMUM-CIRCUMSCRIBED FITTING FOR CO-ORDINATE MEASURING MACHINE

This paper describes algorithms that fit geometric shapes to data sets according to maximum- inscribed (MI) and minimum- circumscribed (MC) fit. We use these fits to build the CMM’s (Coordinate Measuring Machine) software in cases of circle, sphere and cylinder. For each case, we obtain the fit by two methods: first, by (relative easy) least squares fit method and then refine by MI and MC fit method. Although, the later method is substantially more complicated than the former one, Its results are used to make comparision with the the results of least squares method in order to give more options in the CMM software. In the near future we will continue to develop MI and MC fit with an effective algorithm- Simulated Annealing algorithm.

Establishing circle and circular-cylinder references using computational geometric techniques

Form deviations of cylindrical features present in the manufactured parts are measured using coordinate measuring machine (CMM) and expressed as circularity or cylindricity errors evaluated using appropriate reference features. In the present work, computational geometric techniques are used to establish a circle as reference feature, and a heuristic algorithm is proposed to get a unique convex inner hull. Using the concept of equi-distant lines and diagrams, minimum circumscribed (MC), maximum inscribed (MI), and minimum zone (MZ) circles are established. For the first time, algorithms purely based on computational geometric concepts have been developed in the present work to arrive at MC, MI, and MZ circular cylinders. As the algorithms and the implementation details are explained with simple data sets, the practitioners can easily understand these methods and implement them in CMMs for the evaluation of circularity and cylindricity errors. The algorithms are also tested on larger datasets, and in all cases, accurate results are obtained in less than a second.

Comparative study on biaxial low‐cycle fatigue behaviour of three structural steels

ABSTRACTIn this study the uniaxial/biaxial low‐cycle fatigue behaviour of three structural steels (Ck45 normalized steel, 42CrMo4 quenched and tempered steel and AISI 303 stainless steel) are studied, evaluated and compared. Two parameters are considered for estimating non‐proportional fatigue lives: the coefficient of additional hardening and the factor of non‐proportionality. A series of tests of uniaxial/biaxial low‐cycle fatigue composed of tension/compression with cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Several loading paths were carried out, including proportional and non‐proportional ones, in order to verify the additional hardening caused by different loading paths. The experiments showed that the three materials studied have very different additional hardening behaviour. Generally, the transient process from the initial loading cycle to stabilized loading cycle occurs in a few cycles. The stabilized cyclic stress/strain parameters are controlling parameters for fatigue damage. A factor of non‐proportionality of the loading paths is evaluated based on the Minimum Circumscribed Ellipse approach. It is shown that the microstructure has a great influence on the additional hardening and the hardening effect is dependent on the loading path and also the intensity of the loading.

Evaluation of sphericity error from form data using computational geometric techniques

The measurement data for evaluation of sphericity error can be obtained from inspection devices such as form measuring instruments/set-ups. Due to misalignment and size-suppression inherent in these measurements, sphericity data obtained will be distorted. Hence, the sphericity error is evaluated with reference to an assessment feature, referred to as a limacoid. Appropriate methods based on the computational geometry have been developed to establish Minimum Circumscribed, Maximum Inscribed and Minimum Zone Limacoids. The present methods start with the construction of 3-D hulls. A 3-D convex outer hull is established using computational geometric concepts presently available. A heuristic method is followed in this paper to establish a 3-D inner hull. Based on a new concept of 3-D equi-angular line, 3-D farthest or nearest equi-angular diagrams are constructed for establishing the assessment limacoids. Algorithms proposed in the present work are implemented and validated with the simulated data and the data available in the literature.

A stochastic optimization approach for roundness measurements

In this paper, we develop a vision-based inspection system for roundness measurements. A stochastic optimization approach has been proposed to compute the reference circles of MIC (maximum inscribing circle), MCC (minimum circumscribing circle) and MZC (minimum zone circle) methods. The proposed algorithm is a hybrid optimization method based on simulated annealing and Hooke–Jeeves pattern search. From the experimental results, it is noted that the algorithm can solve the roundness assessment problems effectively and efficiently. The developed vision-based inspection system can be an on-line tool for the measurement of circular components.

Reference software for finding Chebyshev best-fit geometric elements

This paper summarizes work carried out in a project supported by the European Communities' Bureau of Reference (BCR) to develop reference software for finding best-fit geometric elements, under specified criteria, to coordinate data. The geometric elements considered are the line, plane, circle, sphere, cylinder, and cone. The criteria for determining the elements are, generally, minimum zone (MZ) and, where appropriate, minimum circumscribed (MC) and maximum inscribed (Ml). The software developed implements methods founded on optimization theory. Two approaches are described. the first implements mathematical programming methods that exploit the particular structure of the problems considered and provide a unified approach to their solution; the second, applicable to the MZ and, where appropriate, MC problems for the line (in two-dimensions (2-D)], plane, circle (in 2-D) and sphere, implements a combinatorial method that returns all global solutions.

シミュレーションデータによる最小外接中心法および最大内接中心法真球度の算出 ダウンヒルシンプレックス法の適用

This article deals with calculation of three-dimensional sphericity values from minimum circumscribing and maximum inscribing centres, although ISO and JIS define two-dimensional sphericity by means of two or three roundness measurements from minimum circumscribing centre. One of the reasons is the difficulties of measuring technique for the whole spherical surface. In this article, the data to be referred is simulated by applying surface harmonics (Laplace's spherical function). Downhill simplex method, one of nonlinear optimization techniques, is applied for search of minimum circumscribing and maximum inscribing centres. Then, their application conditions are investigated. Furthermore, two- and three-dimensional values of the spherical form errors are compared. In the future, further evaluation methods should be studied for actual three-dimensionally measured spherical form errors.

An investigation of reference criteria used in roundness measurement

The minimum zone, minimum circumscribing and maximum inscribing references (here grouped as 'bounding references') used in the measurement of roundness are investigated. Their behaviour is studied by formulating their definitions mathematically as optimisation problems. The advantages of limacon references, both for accuracy under practical measurement conditions and for computational convenience, are investigated and the relationships between these and circular references are discussed. The formulation leads to the development of efficient algorithms (which can be given simple geometric interpretations) for determining these references. The approach adopted is also believed to give a new insight into the interpretation of data derived from roundness measuring instruments with radius suppression.