Kinds Of Curves Research Articles

Fast alternating fitting methods for trigonometric curves for large data sets

This paper discusses and develops new methods for fitting trigonometric curves, such as circles, ellipses, and dumbbells, to data points in the plane. Available methods for fitting circles or ellipses are very sensitive to outliers in the data, and are time consuming when the number of data points is large. The present paper focuses on curve fitting methods that are attractive to use when the number of data points is large. We propose a direct method for fitting circles, and two iterative methods for fitting ellipses and dumbbell curves based on trigonometric polynomials. These methods efficiently minimize the sum of the squared geometric distances between the given data points and the fitted curves. In particular, we are interested in detecting the general shape of an object such as a galaxy or a nebula. Certain nebulae, for instance, the one shown in the experiment section, have a dumbbell shape. Methods for fitting dumbbell curves have not been discussed in the literature. The methods developed are not very sensitive to errors in the data points. The use of random subsampling of the data points to speed up the computations also is discussed. The techniques developed in this paper can be applied to fitting other kinds of curves as well.

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.

Capacity estimation for lithium-ion battery via a novel health indicator extracted from partial constant voltage charging curve

This paper develops a capacity estimation method via a novel health indicator (HI) extracted from partial constant voltage (CV) charging curve. Firstly, the CV charging curve is analyzed based on the Thevenin model, whose results indicate that the CV charging curves under different battery aging states can be theoretically described by the same functional form. Secondly, with CV charging curve analysis results, regarding the CV charging curve at the first cycle as original curve, and those at other cycles as fitted curves, the coefficient of determination between those two kinds of curves is computed through the early-stage CV charging data and innovatively taken as HI for further capacity estimation. Meanwhile, considering that the early-stage CV charging data may still be insufficient in real application, a simple subsequent current prediction method is proposed to ease HI extraction in the case of insufficient data based on the first data point of CV charging curve. Thirdly, a moving window-based HI processing method is developed to mitigate the negative influence of the possible current measurement disturbances on capacity estimation accuracy. Finally, the proposed method is verified through Underwriters Laboratories Inc. - Purdue University battery aging dataset, whose results demonstrate that both mean absolute error and root mean squared error of capacity estimation results achieved by the model established by only 12 pairs of data points in the premise of appropriately selected original curve can be roughly controlled less than 1% and 1.5% in the case of sufficient data and insufficient data, respectively.

Numerical computing of isophote curves, general helices, and relatively normal‐slant helices in Minkowski 3‐space

In this paper, we present a method for numerical computing of some characteristic kinds of non‐null curves lying on a non‐degenerate surface in Minkowski space . Namely, we obtain the system of the first‐order ordinary differential equations that correspond to general helix, relatively normal‐slant helix, and isophote curve and integrate it under chosen initial conditions by applying the ode45 function of MATLAB and Runge‐Kutta method. Depending on the kind of curve, we assume that parametric equation of the surface, an axis vector, value of the real cosine or hyperbolic cosine of the corresponding pseudo angle between axis vector and Darboux frame's vector, normal curvature, and geodesic torsion of the curve are given. Finally, we provide the related examples of numerically computed characteristic curves.

Analysis of Flexural Buckling Capacity of Stainless Steel Columns under Axial Compression

In order to provide a basis for the compilation of the bending buckling stability part of axially compressed columns in the technical code for stainless steel structures, the design methods and experimental research data of similar components are collected and compared. The results showed that there are some deficiencies in the current European code for the design of stainless steel structures and the American code for the design of cold‐formed stainless steel structures. According to the collected test data of square tube, rectangular tube, circular tube, elliptical tube, welded H‐shaped steel, and cold‐formed C‐shaped steel for 199 section columns and according to the influence of cold‐formed effect and residual stress on the stability of axial compression members, the members are divided into three categories: A, B, and C according to the section form, in which the columns with cold‐formed square, rectangular hollow section, and lipped C‐section were included in A, columns with circular and oval hollow section were sorted into B, columns with welded H‐section were separated into C, and the three categories of axial compression stability curves are expressed in the form of Perry formula. The comparison with the test data shows that the three kinds of curves can well estimate the stability bearing capacity of members, and the discreteness is small. The reliability of the proposed formula is analyzed, and the results show that its reliability index β meets the requirements of the GB 50068 standard.

On isoptics and isochordal-viewed curves

In this paper, some results involving isoptic curves and constant $$\phi $$ -width curves are given for any closed curve. The non-convex case, as well as non-simple shapes with or without cusps are considered. Relating the construction of isoptics to the construction given in Holditch’s theorem, a kind of curves is defined: the isochordal-viewed curves. The explicit expression of these curves is given together with some examples. Integral formulae on the area of their isoptics are obtained and a Barbier-type theorem is derived. Finally, a characterization for isochordal-viewed hedgehogs and curves of constant $$\phi $$ -width is given in terms of an angle function.

Different activation signatures in the primary sensorimotor and higher-level regions for haptic three-dimensional curved surface exploration

Haptic object perception begins with continuous exploratory contact, and the human brain needs to accumulate sensory information continuously over time. However, it is still unclear how the primary sensorimotor cortex (PSC) interacts with these higher-level regions during haptic exploration over time. This functional magnetic resonance imaging (fMRI) study investigates time-dependent haptic object processing by examining brain activity during haptic 3D curve and roughness estimations. For this experiment, we designed sixteen haptic stimuli (4 kinds of curves × 4 varieties of roughness) for the haptic curve and roughness estimation tasks. Twenty participants were asked to move their right index and middle fingers along the surface twice and to estimate one of the two features—roughness or curvature—depending on the task instruction. We found that the brain activity in several higher-level regions (e.g., the bilateral posterior parietal cortex) linearly increased as the number of curves increased during the haptic exploration phase. Surprisingly, we found that the contralateral PSC was parametrically modulated by the number of curves only during the late exploration phase but not during the early exploration phase. In contrast, we found no similar parametric modulation activity patterns during the haptic roughness estimation task in either the contralateral PSC or in higher-level regions. Thus, our findings suggest that haptic 3D object perception is processed across the cortical hierarchy, whereas the contralateral PSC interacts with other higher-level regions across time in a manner that is dependent upon the features of the object.

High rank elliptic curves induced by rational Diophantine triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

Nonlinear regression analysis of length growth in cultured rainbow trout

ABSTRACT Length growth as a function of time has a non-linear relationship, so nonlinear equations are recommended to represent this kind of curve. We used six nonlinear models to calculate the length gain of rainbow trout (Oncorhynchus mykiss) during the final grow-out phase of 98 days under three different feed types in triplicate groups. We fitted the von Bertalanffy, Gompertz, Logistic, Brody, Power Function, and Exponential equations to individual length-at-age data of 900 fish. Equations were fitted to the data based on the least square method using the Marquardt iterative algorithm. Accuracy of the fitted models was evaluated using a model performance metrics combining mean squared residuals (MSR), mean absolute error (MAE) and Akaike's Information Criterion corrected for small sample sizes (AICc). All models converged in all cases tested. Evaluation criteria for the Logistic model indicated the best overall fit (0.67 of combined metric MSR, MAE and AICc) under all different feeding types, followed by the Exponential model (0.185), and the von Bertalanffy and Brody model (0.074, respectively). Additionally, ∆AICc results identify the Logistic and Gompertz models as being substantially supported by the data in 100% of cases. The logistic model can be suggested for length growth prediction in aquaculture of rainbow trout.

Partial microperforated panel and its acoustic siphon effect

Partial microperforated panel (MPP) structures have been proposed for MPP with low sound absorption properties. Its purpose is to improve the sound absorption performance of the MPP in certain frequency bands. The structure was studied by the transfer matrix method, finite element simulation and experimental test. The results show that the three kinds of curves, that is, theoretical and simulation and experimental curves, are in good agreement. In the parallel structure of partial MPP and the cavity, the MPP structure occupying three-quarters of the total area makes perfect sound absorption at 1076 Hz. The sound absorption peak of the MPP structure which occupies half of the total area is 0.89 at 980 Hz. The theory of calculating the partial MPP, which is accurate for fast prediction of sound absorption peaks, is proposed. The sound absorption performance of the multi-partial MPP structure has also been theoretically verified. The acoustic siphon phenomenon of partial MPP is discussed. On this basis, a multi-Helmholtz resonance structure is designed to achieve high sound absorption effect. The sound absorption coefficient is greater than 0.87 at 488–712 Hz.

A novel methodology for predicting and identifying geometric errors of rotary axis in five-axis machine tools

The geometric errors of rotary axes are crucial error sources of a five-axis machine tool. They directly affect the machining accuracy, and therefore become one of the most important items for accuracy design. In this paper, a prediction and identification method for the geometric errors of rotary axes on a five-axis machine tool is proposed. The prediction is realized by calculating the mapping relationship between tolerances and geometric errors of rotary axes, which is based on exploring rotary axes’ motion regulation and Fourier series fitting. Then in order to figure out the practical geometric errors of rotary axes, the identifying model is established based on homogeneous transform matrix (HTM). Double ball-bar (DBB) is adopted to test error motions of rotary axes. Finally, a demonstration experiment has been conducted for verifying the effectiveness and precision of the proposed prediction model. The experimental results show that the predicting model is able to reflect the motion principle of rotary axes’ kinematic errors. The SSE, which expresses residual sum of squares between estimated value and identified point, of ez(c),δx(c),ey(c),δz(c),ex(c), and δy(c), are 7.7716 × 10−11, 1.2064 × 10−4, 2.7838 × 10−10, 2.9639 × 10−7, 1.8966 × 10−10, and 2.7838 × 10−10, respectively. And R2, who represent fitting equation’s coefficient of determination, of abovementioned geometric errors, are 0.8978, 0.9876, 0.9978, 0.9453, 0.9985, and 0.9978, respectively. The computing results show that two kinds of curves are basically coincide, and the proposed method is proven to be feasibility.

Construction of High Rank Elliptic Curves

We list a number of strategies for construction of elliptic curves having high rank with special emphasis on those curves induced by Diophantine triples, in which we have contributed more. These strategies have been developed by many authors. In particular we present a new example of a curve, induced by a Diophantine triple, with torsion $$\mathbb {Z}/ 2 \mathbb {Z}\times \mathbb {Z}/ 4\mathbb {Z}$$ and with rank 9 over $$\mathbb {Q}$$. This is the present record for this kind of curves.

Fuzzy system applied to a hydraulic turbine efficiency curve fitting

This paper reports the results of different fuzzy-based approaches applied to the fitting of a hydraulic turbine efficiency curve. This kind of curve is granted by the turbine manufacturer as a three-dimensional dataset that needs to be properly fitted in order to provide the turbine efficiency for any values of net head and water discharge in the relevant space and, therefore, guarantee an as realistic as possible representation of the hydroelectric power plant’s machines. The clustering algorithm Fuzzy C-Means and the ANFIS and Extreme Learning ANFIS architectures were widely tested and compared to the conventional polynomial adjustment. Since the studied curve is nonlinear, researches involving any kind of nonlinear curve fitting can benefit from this work’s information.

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

The Effective and Safe Way to Use Crusade Microcatheter-Facilitated Reverse Wire Technique to Solve Bifurcated Lesions with Markedly Angulated Target Vessel.

Objectives We aim to present a new way to introduce reverse wire in crusade microcatheter-facilitated reverse wire technique to solve markedly angulated bifurcated lesions. Background Markedly angulated coronary bifurcation lesions are still one of the considerable challenges of treatment with percutaneous coronary intervention especially with severe proximal stenosis. Microcatheter-facilitated reverse wire technique improved the efficacy of crossing a guide wire to such an extremely angulated complex targeted vessel. However, there has been a debate regarding what kind of curve was the best to introduce reverse guide wire in this technique. Methods We analyzed 7 patients who were admitted to Fuwai Hospital and underwent antegrade wiring which failed. Crusade microcatheter-facilitated reverse wire technique with simple short tip one round curve was used successfully to solve in all 7 bifurcation lesions. We investigated the bifurcation lesion's characteristics and details of the reverse wire technique procedures. Results All 7 bifurcations exhibit both a smaller take-off angle and a larger carina angle and severe proximal significant stenosis. After having suitable size of balloon predilation, reverse wire with simple short distal one curve was delivered to distal segment of targeted vessel successfully. We performed all PCI procedures without any complications and no major adverse cardiac event was observed during hospitalization. Conclusions In solving markedly angulated bifurcated lesions, especially with severe proximal stenosis, crusade microcatheter-facilitated reverse wire technique with simple short tip one curve is an effective and safe way of wiring.

Q-Ferguson curves

Ferguson’s curves are widely used in airfoil design. We present a kind of Ferguson’s curves with a shape parameter by integrating the classical Ferguson’s curves with the q-derivatives, called q-Ferguson curves. This kind of curves not only preserves the interpolation properties of classical Ferguson’s curves but also has a shape parameter which provides a freedom variable to construct the desired curves satisfying the interpolation or length constraint.

Unlocking the underlying information of golden ratio in the Hogarth curves

PurposeCurves with various profiles have been demonstrated to be more attractive and decorative than the straight lines by William Hogarth. Among all kinds of curves, Hogarth proposed seven serpentine lines as the most beautiful curves, i.e., Hogarth curves. Those seven Hogarth curves are subsequently applied in a wide range of fields, e.g., sculpture, painting, architecture and fashion design, indicating their significance to the development of the formal beauty. Recently, the beauty of Hogarth curves has been suspected to be induced by their special-designed curvature, which could have the potential relationship with the Golden Ratio. The purpose of this paper is to explore the relationship between the Hogarth curves and golden ratio by comparing the curvature of curves with the Fibonacci sequence.Design/methodology/approachEach of the Hogarth curves was fully restored and divided into two parts according to the turning point of the curvature; the ratios of span, curvature and angles between these two parts were compared with the Fibonacci sequence.FindingsThe experimental results disclosed that the ratio of the fourth Hogarth curve, which was considered as the most beautiful line by Hogarth, was infinitely approaching the golden ratio. Based on the relationship between the fourth Hogarth curve and the golden ratio, the ratios of each curve were employed to define and normalize these curves, providing a quantitative way to redraw the Hogarth curves.Originality/valueThis research work unlocked the information of the relationship between the Hogarth curves and golden ratio, and proposed an effective and convenient mathematic way to quantify the Hogarth curves. The experimental findings disclosed the underlying mechanisms of the beauty of the forth Hogarth curves. Such a fundamental study will promote the establishment of the normalized methods for evaluating the beauty of arts and provide novel ideas for researchers and industrial technologists to use the Hogarth curves.

The relationship between rail side wear and flange wear

Tokyo Metro owns many sharp curved metro lines. When vehicle passed that kind of curves, both wheels and rails are worn. The wear of wheels and rails increase a risk of the derailment. To solve this problem, oil application devices are installed on the curves. However, it couldn’t be a drastic solution. To conduct an efficient maintenance, it needs to grasp a tendency of the wear of wheels and rails. Therefore, in this research, the relationship between wheels and rails is inspected in terms of the degree of the wear. To grasp that relationship would be helpful when it comes to realize condition based maintenance.

Characteristic analysis of the efficiency hill chart of Francis turbine for different water heads

According to several model test results of Francis turbines, complete model efficiency hill charts were constructed. The formation and inevitability of diversified hydraulic phenomena on model efficiency hill chart for typical head range were analyzed and the difference is compared, as well as characteristics and commonness toward the curves are discussed and summarized. Furthermore, hydraulic performance and geometric features are presented by particularly analyzing the efficiency hill charts. The inherent characteristics of Francis turbine is expressed by all kinds of curves on the model efficiency hill charts, and these curves can be adjusted and moved in a small range but cannot be removed out. Due to wide range of unit speed in terms of medium-low-head hydraulic turbines, incipient cavitation curve on suction side can be observed and it is positioned close to the operation zone; however, it fails to be visualized for the high-head turbine. The blade channel vortex curves are in the vicinity of optimum region for low-head hydraulic turbines, while high-head shows reverse trend. The interaction between zero incidence angle and zero circulation curve has a significant influence on isoefficiency circles. All comparisons and analyses could provide hydraulic design basis and retrofit references.

A TOWER OF RIEMANN SURFACES WHICH CANNOT BE DEFINED OVER THEIR FIELD OF MODULI

AbstractExplicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.