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The study has investigated the implications of three estimation methods, namely L-moments, Maximum Likelihood, and Maximum Product of Spacing (MPS), for fitting the four-parameter Kappa Distribution (KAPD) in extreme value analysis using Monte Carlo simulations. The accuracy of the estimates has been evaluated using root mean square error (RMSE) and bias. The paper also includes an analysis of the effect of the estimation method on the estimated quantiles considering a real-life example of annual maximum peak flows and the Generalized Normal Distribution as the error distribution. Assessment metrics of the empirical analysis include standard error, L-scale, and 90% confidence limits of the estimated quantiles. The results reveal that MPS is a preferred method of estimation of parameters for KAPD, i.e. having the lowest RMSE values, especially in the presence of heavier tail and significant positive skewness for small to very large sample sizes. Secondly, the method of L-moments is recommended due to its low bias while analyzing the distribution of shape parameters having a slightly heavier tail, and slight or moderate positive skewness. The results associated with the quality of estimated quantiles using real-life data are consistent with the findings of simulation outcomes. [ABSTRACT FROM AUTHOR]

Often in reliability studies we observe failure-time data with two dependent failure modes. In this article, by using a bivariate Weibull model with distinct shape parameters, we present a model for reliability data with two dependent failure modes. Inferential methods for the proposed model are discussed. The efficacy of the methods of inference is assessed through a Monte Carlo simulation study, and it is observed that the model and methods perform satisfactorily. An issue of practical interest for reliability engineers is to predict field failures at a future time. Prediction methods are developed in this setting. For illustrative purposes, analysis of a real dataset on failure of a device is presented. In summary, the model and methods presented in this article provide a comprehensive treatment for analyzing reliability data with two dependent failure modes. [ABSTRACT FROM AUTHOR]

Among the many possible metal‐cutting processes, milling is regarded as one of the most widely applied processes to machine different engineering materials productively. During the milling process, there is relative motion between the endmill tool and the workpiece. Energy is required to drive the cutting force used by the tool in separating the chips from the workpiece. Several studies have focused on the effects of milling process parameters on the cutting force, for instance by varying the cutting parameters, machining properties and the extent of cooling/lubrication. Other studies have explored the influence of tool geometry and material properties on the cutting force. Results from these studies have brought about improvements in the design of milling machinery and tools. The current study presents a novel automated approach for tool geometry optimization which is fully coded in Matlab. An automated and parametrized endmill design procedure was created featuring input parameters such as rake angle, relief angle, clearance angle and helix angle, all of which have been shown, through numerous studies, to influence cutting forces. The parameters were here used to generate the shape profile for one cutting edge of a flat endmill. This cutting edge profile is next copied and rotated, as per the design's pitch angle, to form the complete set of cutting flutes for the flat endmill. Next, the surface defined by the completed profile is meshed using quadrilateral elements. The 3D endmill design is then formed by extruding, and twisting as per the helix angle, this quadrilateral mesh to form a solid hexahedral mesh. Finally, an automated iterative optimization process was defined, featuring the above parametrized design process coupled with Abaqus‐based finite element simulation of the milling process. The expected result is that the proposed optimization procedure will be able to identify the endmill design parameters which minimize the cutting force. At the completion of the optimization, a minimized maximum resultant cutting force value is obtained from using the newly optimized endmill. The resultant cutting force was reduced by 17.6%. It is anticipated that the automated design, meshing and simulation‐driven optimization approach is generalizable to other tool design variations. [ABSTRACT FROM AUTHOR]

The shear strength and resistance of granular materials are critical indicators in geotechnical engineering and infrastructure construction. Both sliding and rotation influence the energy evolution of soil granular motion during shear. To examine the effects of particle rotation on shear damage and energy evolution in granular systems, we first describe the transformation of irregularly shaped particles into regular shapes via geometrical parameters, ensuring the invariance of energy density and density. We then analyze the impact of particle rotation on shear-stress variation and energy dissipation through a shear energy evolution equation. Additionally, we establish the relationship between the shear-stress ratio and normal stress, considering particle rotation. Finally, we verify the influence of particle rotation on energy evolution and shear damage through shear tests on irregular calcareous sand and regular silica-bead particles. The results indicate that granular materials do not fully comply with the Coulomb strength criterion. In the initial shear stage, most of the external work is converted into granular rotational-shear energy, whereas in the later stage, it primarily shifts to granular sliding-shear energy. Notably, the sensitivity of the granular rotational energy to a vertical load is significantly greater than that of the granular sliding energy. [ABSTRACT FROM AUTHOR]

This paper introduces a novel method that leverages the capability of the Adam Optimization- Back Propagation model to adaptively select shape parameters in the Multiple-Quadratic Radial Basis Function (MQRBF). This approach has been effectively combined with the Finite Difference (FD) method to generate high-precision solutions for complex Partial Differential Equations (PDEs). The careful selection of shape parameters in MQRBF is crucial to ensure the accuracy of the MQRBF-FD in addressing complex PDE scenarios. We have improved the Random Walk Optimization algorithm and integrated it with Fourier theory and deep learning techniques, establishing a comprehensive framework for adaptive optimization of shape parameters. This significantly enhances the adaptability and accuracy of the MQRBF-FD. A wide range of numerical experiments, including analyses of heat conduction in non-uniform materials and dye transport in fluid channels, highlight the exceptional accuracy, computational efficiency, and versatility of our method. [ABSTRACT FROM AUTHOR]

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors' impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately. [ABSTRACT FROM AUTHOR]

In this paper, a novel surrogate model for shape-parametrized vehicle drag force prediction is proposed. It is assumed that only a limited dataset of high-fidelity CFD results is available, typically less than ten high-fidelity CFD solutions for different shape samples. The idea is to take advantage not only of the drag coefficients but also physical fields such as velocity, pressure, and kinetic energy evaluated on a cutting plane in the wake of the vehicle and perpendicular to the road. This additional "augmented" information provides a more accurate and robust prediction of the drag force compared to a standard surface response methodology. As a first step, an original reparametrization of the shape based on combination coefficients of shape principal components is proposed, leading to a low-dimensional representation of the shape space. The second step consists in determining principal components of the x-direction momentum flux through a cutting plane behind the car. The final step is to find the mapping between the reduced shape description and the momentum flux formula to achieve an accurate drag estimation. The resulting surrogate model is a space-parameter separated representation with shape principal component coefficients and spatial modes dedicated to drag-force evaluation. The algorithm can deal with shapes of variable mesh by using an optimal transport procedure that interpolates the fields on a shared reference mesh. The Machine Learning algorithm is challenged on a car concept with a three-dimensional shape design space. With only two well-chosen samples, the numerical algorithm is able to return a drag surrogate model with reasonable uniform error over the validation dataset. An incremental learning approach involving additional high-fidelity computations is also proposed. The leading algorithm is shown to improve the model accuracy. The study also shows the sensitivity of the results with respect to the initial experimental design. As feedback, we discuss and suggest what appear to be the correct choices of experimental designs for the best results. [ABSTRACT FROM AUTHOR]

The rapid discharge of oxygen on the anode side of a proton exchange membrane electrolytic cell (PEMEC) plays a significant role in the production of hydrogen. This study proposes a serpentine flow field model with circular arc columns on the anode side of the proton exchange membrane electrolysis cell system. In this model, the circular arc columns are incorporated at the bends of the flow field. The Taguchi design is carried out to optimize performance parameters. The advantages of a serpentine flow field model with circular arc columns are analyzed, the results demonstrate that the new model can effectively expel oxygen, thus enhancing hydrogen production performance of the PEMEC. The performance of the improved serpentine flow field model is influenced by the shape parameters, the Taguchi design is carried out to optimize performance parameters. Among them, the upper gap parameter has the greatest impact, followed by the lower gap and angle parameters. The improved performance of the model is attributed to the narrowing of channel, which creates a pressure difference between the two sides of edges, increases fluid velocity, reduces bubble adhesion to the wall surface, and shortens the time for bubbles to pass through the bends, eventually enhances transport of oxygen bubbles. The fitting results show that the oxygen discharge of the optimized model in this study is increased by 22.4% compared to that of the initial serpentine flow field. [Display omitted] • Novel serpentine flow model with circular arc columns enhances PEMEC efficiency. • Improved design boosts oxygen expulsion, enhancing hydrogen production. • Key parameters—upper gap, lower gap, and angle—optimized for oxygen transport. • Enhances fluid velocity, reduces bubble adhesion, and shortens bend passage times. • Achieves 22.4% higher oxygen discharge compared to the initial serpentine flow. [ABSTRACT FROM AUTHOR]

Functional data analysis (FDA) is a method used to analyze data represented in its functional form. The method is particularly useful for exploring both curve and longitudinal data in both exploratory and inferential contexts, with minimal constraints on the parameters. In FDA, the choice of basis function is crucial for the smoothing process. However, traditional basis functions lack flexibility, limiting the ability to modify the shape of curves and accurately represent abnormal details in modern and complex datasets. This study introduced a novel and flexible data smoothing technique for interpreting functional data, employing the beta spline introduced by Barsky in 1981. The beta spline offers flexibility due to the inclusion of two shape parameters. The proposed methodology integrated the roughness penalty approach and generalized cross-validation (GCV) to identify the optimal curve that best fitted the data, ensuring appropriate parameters were considered for transforming data into a functional form. The effectiveness of the approach was assessed by analyzing the GCV color grid chart to determine the optimal curve. In contrast to existing methodologies, the proposed method enhanced flexibility by incorporating the beta spline into the smoothing procedure. This approach was anticipated to effectively handle various forms of time series data, offering improved interpretability and accuracy in data analysis, including forecasting. [ABSTRACT FROM AUTHOR]

Due to heterogenous nature of bimsoils (complex mixture of rock blocks within a soil matrix), studying their behavior through conducting in situ and laboratory tests is really difficult. By developing methods for shear parameters estimation of bimsoils, engineers will be able to have insight about these parameters values without conducting expensive, difficult and sometimes impossible in situ tests. In this study a classification system for estimating friction angle of non-cemented bimsoils has been introduced. In these bimsoils, friction angle is mostly depended on shape and grading parameters of coarse grains. So, in this classification, rates have been allocated to circularity and roundness as shape parameters and uniformity coefficient of coarse-grained part of grading curve (Cuc) and fine gravel percent (FGP) as grading parameters. The rate allocation is based on results of large-scale direct shear tests on samples with different shape and grading properties. The results of verification have shown that this classification can be used for different non-cemented bimsoils in dry condition. Using this classification engineers will be able to have a comparative insight about different bimsoils friction angle simply through sieve analysis and determining circularity and roundness using image processing. The results of tests carried out in this study also have shown the remarkable effect of grains circularity on friction angle and lead to an insight about grains roundness effect on friction angle. Highlights: In this study a classification system for estimating bimsoils friction angle has been introduced. This classification leads engineers to estimate and compare friction angle of different bimsoils just by doing simple tests such as sieve analysis and image processing. Results of this study have shown that circularity, coarse grains grading and roundness are the most effective parameters on friction angle of non-cemented bimsoils, respectively. [ABSTRACT FROM AUTHOR]

In recent research in computer-aided geometric design (CAGD), one of the most popular concerns has been the creation of new basis functions for the Bézier curve. Bézier curves with high degrees often overshoot, which makes it challenging to maintain control over the ideal length of the curved trajectory. To get around this restriction, free-form surfaces and curves can be created using the Caputo Fabrizio basis function. In this study, the Caputo Fabrizio fractional order derivative is used to construct the Caputo Fabrizio basis function, which contains fractional parameter and shape parameters. The Caputo Fabrizio Bézier curve and surface are defined using the Caputo Fabrizio basis function, and their geometric properties are examined. Using fractional and shape parameters in the implementation of the Caputo Fabrizio basis function offers an alternative perspective on how the Caputo Fabrizio basis function can construct complicated curves and surfaces beyond a limited formulation. Curves and surfaces can have additional shape and length control by adjusting a number of fractional and shape parameters without affecting their control points. The Caputo Fabrizio Bézier curve's flexibility and versatility make it more effective in creating complex engineering curves and surfaces. [ABSTRACT FROM AUTHOR]

Functional data analysis (FDA) is a method used to analyze data represented in its functional form. The method is particularly useful for exploring both curve and longitudinal data in both exploratory and inferential contexts, with minimal constraints on the parameters. In FDA, the choice of basis function is crucial for the smoothing process. However, traditional basis functions lack flexibility, limiting the ability to modify the shape of curves and accurately represent abnormal details in modern and complex datasets. This study introduced a novel and flexible data smoothing technique for interpreting functional data, employing the beta spline introduced by Barsky in 1981. The beta spline offers flexibility due to the inclusion of two shape parameters. The proposed methodology integrated the roughness penalty approach and generalized cross-validation (GCV) to identify the optimal curve that best fitted the data, ensuring appropriate parameters were considered for transforming data into a functional form. The effectiveness of the approach was assessed by analyzing the GCV color grid chart to determine the optimal curve. In contrast to existing methodologies, the proposed method enhanced flexibility by incorporating the beta spline into the smoothing procedure. This approach was anticipated to effectively handle various forms of time series data, offering improved interpretability and accuracy in data analysis, including forecasting.

The PM2.5 and PM10 particles were characterized in terms of morphology (size and shape) and surface elemental composition at two different (traffic and industrial) locations in urban region of India and further linked to different morphological defining parameters. The overall PM2.5 and PM10 showed significant daily variability indicating higher PM10 as compared to PM2.5. PM2.5/PM10 ratio was found to be 0.58 ± 0.10 indicating the abundance of PM2.5. Soot aggregates, aluminosilicates, and brochosomes particles were classified based on morphology, aspect ratio (AR), and surface elemental composition of single particles. The linear regression analysis indicates the significant correlation between area equivalent (Daeq) and feret diameter (Dfd) (R2 0.86–0.98). Higher aspect ratio (1.48 ± 0.87–1.43 ± 0.50) was noted at traffic site as compared to industrial site (1.33 ± 0.58–1.29 ± 0.30), while circularity showed the opposite trend. Fractal dimension (Df) of soot aggregates estimated by the soot parameters method (SPM) were found to be 1.70, 1.72, and 1.88, mainly attributed to vehicular emissions, biomass, and industrial emission/coal burning, respectively. This further inferred that freshly emitted soot particles exhibited lacey in nature with spherical shape (Df 1.70) at traffic site, while at industrial location, they were different with compact shapes (Df 1.88) due to particle aging processes. This study inferred the synoptic changes in mass, chemical characteristics, and morphology of aerosol particles which provide the new insights into individual atmospheric particle and their dynamic nature. [ABSTRACT FROM AUTHOR]

The modeling capabilities of trigonometric B-spline curves with form parameters are comparable to those of classical B-spline curves in terms of their significance and use. Through the application of basic functions and shape parameters, this method presents the cubic trigonometric B-spline curves. Constructed by these functions, the concept of designing curves and surfaces possesses all the ideal geometric qualities, such as the partition of unity, convex hull, affine invariance, and variation decreasing. The parameters that are used in the creation of the recommended approach are helpful for several important form characteristics. These vital form aspects include local, global, and biased tension qualities, which provide good control over the curve and allow for shape change as required. In addition, the C 2 strategy is the one that is suggested for use. [ABSTRACT FROM AUTHOR]

Finding a collision-free feasible path for mobile robots is very important because they are essential in many fields such as healthcare, military, and industry. In this paper, a novel Clustering Obstacles (CO)-based path planning algorithm for mobile robots is presented using a quintic trigonometric Bézier curve and its two shape parameters. The CO-based algorithm forms clusters of geometrically shaped obstacles and finds the cluster centers. Moreover, the proposed waypoint algorithm (WP) finds the waypoints of the predefined skeleton path in addition to the start and destination points in an environment. Based on all these points, the predefined quintic trigonometric Bézier path candidates, taking the skeleton path as their convex hull, are then generated using the shape parameters of this curve. Moreover, the performance of the proposed algorithm is compared with K-Means and agglomerative hierarchical algorithms to obtain the quintic trigonometric Bézier paths desired by the user. The experimental results show that the CO-based path planning algorithm achieves better cluster centers and consequently better collision-free predefined paths. [ABSTRACT FROM AUTHOR]

IntroductionSoil pore size distribution curve and using the optimal ranges of the location and shape parameters of this curve can be used to evaluate the soil physical quality. This research was carried out in an area of about 220 hectares of Torogh Agricultural and Natural Resources Research and Education Station, to determine the optimal ranges for soil pore size distribution curve parameters using the soil physical quality index. Different soil textures and the diversity in soil properties are the distinct features of this research station. Materials and MethodsTorogh Agricultural and Natural Resources Research and Education Station of Khorasan-Razavi province, with a semiarid climate, is located in south-east of Mashhad city. For the field measurements and laboratory analysis to determine the soil physical properties and indices, 30 points with different soil textures and structures were selected. Intact soil cores (5 cm diameter by 5.3 cm length) and disturbed soil samples were collected from 0-30 cm depth of each point. After the laboratory analysis and field measurements, 35 soil physical properties were measured and calculated. Soil particle size distribution and five size classes of sand particles, soil bulk, and particle density, dry aggregates mean weight diameter (MWD) and stability index (SI), soil moisture release curve (SMRC) parameters, S-index, soil porosity (POR) and air capacity (AC), soil pore size distribution (SPSD) curves, relative field capacity (RFC), plant available water measured in matric pressure heads of 100 and 330 hPa for the field capacity (PAW100 and PAW330), least limiting water range measured in matric pressure heads of 100 and 330 hPa for the field capacity (LLWR100 and LLWR330), integral water capacity (IWC) and integral energy (EI) of different soil water ranges, were the soil physical properties and indices which were determined in this study. Three parameters of modal, median, and mean pore sizes of the SPSD curves were considered as the location (central tendency), and three parameters of standard deviation, skewness, and kurtosis of the SPSD curves were considered as the shape parameters. Selection of the most important soil physical characteristics using principal component analysis (PCA) method by JMP software (ver. 9.02), weighting and scoring of the selected characteristics using PCA and scoring functions, respectively, and the summation of multiplied characteristics weights by their scores for each soil sample, were the four steps of calculation of the 0-1 value of soil physical quality index (SPQI). Soil samples were classified into four soil physical quality classes by SPQI values. The soils of the first class with the highest SPQIs (> 0.78) were considered to determine the optimal ranges of SPSD curves location and shape parameters. Results and DiscussionThe texture of soil samples were loam (40 %), silt loam (23 %), silty clay loam (17 %), clay loam (13 %), and sandy loam (7 %). Soil organic carbon was between 0.26-1.05 (%), and the average soil bulk density was 1.45 (gr.cm-3). The MWD values of studied soil samples were between 0.94-2.88 (mm), an average of 1.93 (mm). The average modal, median, and mean pore sizes as the location parameters of the SPSD curves were 60.3 (μm), 12.4 (μm), and 6.5 (μm), respectively. The average of standard deviation, skewness, and kurtosis as the shape parameters of the SPSD curves were 71.56 (μm), -0.36 and 1.15, respectively. The average modal pore sizes showed that the pores with a size of 60 (μm) had the highest frequency in soil samples. The range of calculated standard deviation of SPSD curves, along with the difference between the minimum and maximum mean pore sizes (24.6 μm), implied the diversity of pore sizes in the studied soils. The results of PCA showed that the four soil physical properties of PAW330 (0.1-0.2 cm3.cm-3), PORt (0.40-0.51 cm3.cm-3), LLWR100 (0.12-0.22 cm3.cm-3) and SI (0.76-2.61 %) accounted for about 88% of the variance between soil samples and were selected to calculate the SPQIs. The PAW330, PORt, LLWR100, and SI were entered into the calculation of SPQIs with weights of 0.46, 0.31, 0.15, and 0.08, respectively. All the selected physical properties were scored using the scoring function of more is better. The maximum and minimum values of SPQIs for the studied soils were 0.84 and 0.14, respectively. Five soil samples with SPQIs greater than 0.78 were classified as class 1 with the highest physical quality. The ranges between the minimum and maximum values of the SPSD curves, location, and shape parameters of these five soils were proposed as the optimal ranges. In this regard, the ranges of 29-92 (μm), 5-16 (μm), and 2-7 (μm) were suggested for optimal ranges of modal, median, and mean pore sizes, respectively. The optimal ranges of standard deviation, skewness, and kurtosis of the SPSD curves were proposed as 22-81 (μm), (-0.38)-(-0.33), and 1.14-1.15, respectively. ConclusionThe optimal ranges of SPSD curves location and shape parameters suggested in the literature may probably not apply to a wide range of agricultural soils. They must be evaluated in a more extensive range of land uses, soil management, and soil textures. In this research, the soils with the relatively higher physical quality had larger mean pore size and less SPSD curves standard deviation (less diversity of pore size) than the optimal ranges suggested in the literature. The optimal ranges of SPSD curves location and shape parameters proposed in this research are appropriate for medium to coarse-textured soils of regions with the semiarid climate in Iran.

In recent research in computer-aided geometric design (CAGD), one of the most popular concerns has been the creation of new basis functions for the Bézier curve. Bézier curves with high degrees often overshoot, which makes it challenging to maintain control over the ideal length of the curved trajectory. To get around this restriction, free-form surfaces and curves can be created using the Caputo Fabrizio basis function. In this study, the Caputo Fabrizio fractional order derivative is used to construct the Caputo Fabrizio basis function, which contains fractional parameter and shape parameters. The Caputo Fabrizio Bézier curve and surface are defined using the Caputo Fabrizio basis function, and their geometric properties are examined. Using fractional and shape parameters in the implementation of the Caputo Fabrizio basis function offers an alternative perspective on how the Caputo Fabrizio basis function can construct complicated curves and surfaces beyond a limited formulation. Curves and surfaces can have additional shape and length control by adjusting a number of fractional and shape parameters without affecting their control points. The Caputo Fabrizio Bézier curve’s flexibility and versatility make it more effective in creating complex engineering curves and surfaces.

Quantitative characterisation of morphology and shape parameters of pozzolanic materials, as a fundamental problem of characterisation of pozzolanic materials, has received significant consideration in literature. Thus far, previous research works have not paid much attention to the circularity, roundness and solidity of pozzolanic materials including waste brick powder (WBP). This research makes a significant contribution on identification of circularity, roundness and solidity of WBP particles under milling conditions using quantitative image analysis. In particular, the goal was to interrogate the ball milling treatment variables for generating WBP using scanning electron microscopy (SEM) and image analysis. Under the milling conditions of changing sample masses introduced in ball mill, the average circularity values for the specimens were approximately 0.6 whilst the average solidity values for the specimens were approximately 0.71. Moreover, the average roundness values for the specimens were nearly 0.51. It was shown that the trends of shape parameters of WBP under changing fineness levels were not significant. The values of circularity, solidity and roundness in this study therefore collaborate to support the discoveries of hidden shape characteristics of WBP specimens and can tackle the overall behaviour of cement-based composites containing WBP. Quantitative image analysis was therefore observed to be capable of inheriting detailed information from SEM micrographs and remains one of the most outstanding approaches of generating shape parameters.

The quantitative description of morphological features is the basis for the study of three-dimensional morphological reconstruction and mechanical behavior of rock fragments. The characterization of two-dimensional morphological features of fragments was focused. Randomly selected crushed rock fragments of different sizes were studied using digital image processing techniques and the Image-Pro Plus image measurement method to extract basic geometric data of the rock fragments. A morphological feature description system was established based on circularity, sphericity, rectangularity, and fragments size as indicators. Scatter plot fitting and multivariate parameter fitting were performed on the measurement results. The significance of the impact of shape parameters was determined through correlation analysis, eliminating insignificant factors and parameters with low correlation to construct the CSR ( circularity, sphericity, rectangularity) morphological feature description system for rock fragments. The results show that circularity represents the regularity of the two-dimensional morphological features of fragments, while fragments size has lower correlation. Both sphericity and rectangularity are found to be correlated, and regression analysis reveale that sphericity had the most significant impact. Comparative analysis results demonstrated that the CSR index system has a higher accuracy in characterizing the macroscopic two-dimensional morphology of fragments. [ABSTRACT FROM AUTHOR]

The splines with shape parameters have more expressive ability than the traditional splines, and they are widely used in various fields. In this paper, a De Casteljau-type algorithm for a class of αß-Bézier curves with shape parameters was studied. Firstly, the αẞ-Bernstein basis function with shape parameters was extended, the related properties of the basis function were discussed, and the definition of arbitrary degree αß-Bézier curve was given. Then based on the recurrence relationship of basis function, the geometric drawing algorithm of αß-Bézier curve was presented. Similar to the classical De Casteljau algorithm, the algorithm was implemented by a series of linear operations of the control vertices of the curve. Compare to calculating the point value directly based on parameter expression, the algorithm was simpler and more efficient. The calculation examples showed the feasibility and effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The article compares the test results for gimballess electrostatic gyroscopes containing rotors with a light-contrast pattern applied to titanium nitride (TiN) coating or to beryllium. The criteria for comparing the devices with different types of rotors are proposed and justified. The need to modify the rotor manufacturing technology in order to improve the rotor design and mature some technological operations is analyzed. It has been shown that at a certain stage it is rational to mark the raster pattern on the titanium nitride coating, and currently the pattern can be again applied directly to the rotor beryllium surface. [ABSTRACT FROM AUTHOR]

Introduction Soil pore size distribution curve and using the optimal ranges of the location and shape parameters of this curve can be used to evaluate the soil physical quality. This research was carried out in an area of about 220 hectares of Torogh Agricultural and Natural Resources Research and Education Station, to determine the optimal ranges for soil pore size distribution curve parameters using the soil physical quality index. Different soil textures and the diversity in soil properties are the distinct features of this research station. Materials and Methods Torogh Agricultural and Natural Resources Research and Education Station of Khorasan-Razavi province, with a semiarid climate, is located in south-east of Mashhad city. For the field measurements and laboratory analysis to determine the soil physical properties and indices, 30 points with different soil textures and structures were selected. Intact soil cores (5 cm diameter by 5.3 cm length) and disturbed soil samples were collected from 0-30 cm depth of each point. After the laboratory analysis and field measurements, 35 soil physical properties were measured and calculated. Soil particle size distribution and five size classes of sand particles, soil bulk, and particle density, dry aggregates mean weight diameter (MWD) and stability index (SI), soil moisture release curve (SMRC) parameters, S-index, soil porosity (POR) and air capacity (AC), soil pore size distribution (SPSD) curves, relative field capacity (RFC), plant available water measured in matric pressure heads of 100 and 330 hPa for the field capacity (PAW100 and PAW330), least limiting water range measured in matric pressure heads of 100 and 330 hPa for the field capacity (LLWR100 and LLWR330), integral water capacity (IWC) and integral energy (EI) of different soil water ranges, were the soil physical properties and indices which were determined in this study. Three parameters of modal, median, and mean pore sizes of the SPSD curves were considered as the location (central tendency), and three parameters of standard deviation, skewness, and kurtosis of the SPSD curves were considered as the shape parameters. Selection of the most important soil physical characteristics using principal component analysis (PCA) method by JMP software (ver. 9.02), weighting and scoring of the selected characteristics using PCA and scoring functions, respectively, and the summation of multiplied characteristics weights by their scores for each soil sample, were the four steps of calculation of the 0-1 value of soil physical quality index (SPQI). Soil samples were classified into four soil physical quality classes by SPQI values. The soils of the first class with the highest SPQIs (> 0.78) were considered to determine the optimal ranges of SPSD curves location and shape parameters. Results and Discussion The texture of soil samples were loam (40 %), silt loam (23 %), silty clay loam (17 %), clay loam (13 %), and sandy loam (7 %). Soil organic carbon was between 0.26-1.05 (%), and the average soil bulk density was 1.45 (gr.cm-3). The MWD values of studied soil samples were between 0.94-2.88 (mm), an average of 1.93 (mm). The average modal, median, and mean pore sizes as the location parameters of the SPSD curves were 60.3 (سلم), 12.4 سلم)), and سلم) 6.5), respectively. The average of standard deviation, skewness, and kurtosis as the shape parameters of the SPSD curves were 71.56 (سلم), - 0.36 and 1.15, respectively. The average modal pore sizes showed that the pores with a size of سلم) 60) had the highest frequency in soil samples. The range of calculated standard deviation of SPSD curves, along with the difference between the minimum and maximum mean pore sizes (سلم 24.6), implied the diversity of pore sizes in the studied soils. The results of PCA showed that the four soil physical properties of PAW330 (0.1-0.2 cm3.cm-3), PORt (0.40-0.51 cm3.cm-3), LLWR100 (0.12-0.22 cm3.cm-3) and SI (0.76-2.61 %) accounted for about 88% of the variance between soil samples and were selected to calculate the SPQIs. The PAW330, PORt, LLWR100, and SI were entered into the calculation of SPQIs with weights of 0.46, 0.31, 0.15, and 0.08, respectively. All the selected physical properties were scored using the scoring function of more is better. The maximum and minimum values of SPQIs for the studied soils were 0.84 and 0.14, respectively. Five soil samples with SPQIs greater than 0.78 were classified as class 1 with the highest physical quality. The ranges between the minimum and maximum values of the SPSD curves, location, and shape parameters of these five soils were proposed as the optimal ranges. In this regard, the ranges of 29-5,)سلم) 92 - سلم(16), and 2- (سلم) 7 were suggested for optimal ranges of modal, median, and mean pore sizes, respectively. The optimal ranges of standard deviation, skewness, and kurtosis of the SPSD curves were proposed as 22-81 (0.33-)-(سلم), (- 0.38), and 1.14-1.15, respectively. Conclusion The optimal ranges of SPSD curves location and shape parameters suggested in the literature may probably not apply to a wide range of agricultural soils. They must be evaluated in a more extensive range of land uses, soil management, and soil textures. In this research, the soils with the relatively higher physical quality had larger mean pore size and less SPSD curves standard deviation (less diversity of pore size) than the optimal ranges suggested in the literature. The optimal ranges of SPSD curves location and shape parameters proposed in this research are appropriate for medium to coarse-textured soils of regions with the semiarid climate in Iran. [ABSTRACT FROM AUTHOR]

Zircon morphology parameters reflect the physicochemical conditions during crystallization and can be modified by different processes. Zircon populations from Miocene--Pliocene ignimbrites of the Central Anatolian Volcanic Province (CAVP) were studied to reveal relations between the external morphology of zircons and petrogenetic processes. Descriptive grain shape parameters (e.g., minor and major axes, area, perimeter, aspect ratio, roundness, and circularity) were automatically measured from transmitted light images of zircons by a graphical application called AnalyZr. Principal component and cluster analysis were used to determine potential shape descriptors (perimeter, major and minor axis length, and maximum and minimum Feret) for characterizing grains from a particular rock. Accordingly, zircons from ignimbrites display morphological variations, which can be attributed to a specific magmatic process. Zircon isotopic compositions from previous studies indicate that ignimbrites are derived from mantle sources, but due to the distinct contributions of crustal components and periodic mafic recharge, isotopic heterogeneity occurs in the genesis of the ignimbrites. Furthermore, the typological evolution of zircons has already revealed that even a small decrease in saturation and crystallization temperature can significantly alter the external morphology of zircons. The clear imprint of hybridization in the zircon morphology of CAVP ignimbrites might provide information about the development of zircon within silicic melts. This study offers new insight into the integration of data on shape and isotope variations in zircon populations that can be traced back to the magmatic controls on zircon crystal growth. [ABSTRACT FROM AUTHOR]

Phosphogypsum (PG) is an industrial by-product from the wet preparation of phosphoric acid. Phosphorus building gypsum (PBG) can be obtained from PG after high-thermal dehydration. Improving the properties of PBG is of great significance to extending its application range. In this paper, PBG was modified by particle shaping, resultantly its properties have been significantly improved. This study utilized a small-scale ball mill as a power source for simulated shaping to investigate the shape parameters of PBG particles before and after shaping. Further, the effects of different material-to-ball mass ratios, ball milling speeds, and shaping durations on PBG performance were examined. Shaped PBG particles exhibited a reduction in aspect ratio, approaching a roundness of approximately 1. The sharp edges on the particle surfaces decreased, resulting in a particle appearance that tended to approach a spherical form. Moreover, when the material-to-ball mass ratio and ball milling speed were held constant, increasing the shaping duration decreased and then increased the water amount required for normal consistency and setting time of PBG. However, the strength, first increased and then decreased. Under the conditions of a material-to-ball mass ratio of 1:1, ball milling speed of 90 r/min, and a shaping duration of 5 min, PBG exhibited the most significant enhancement in mechanical properties. Finally, the mechanism of how particle shaping affects PBG performance was analyzed using the theory of water film thickness.

When a distributional model is chosen, the analytic relation between its shape parameters and the values taken by some kurtosis indexes, especially if they are unconventional, is rarely known. In addition, different indexes may provide contrasting evidence about the level of global kurtosis, when the parameters of the model are varied. That happens because just few parameters act "plainly" on kurtosis, namely so as to produce consistent modifications of the shape of the graph on both its sides. Many parameters, instead, affect kurtosis along with a change of the skewness of the distribution, that is by "inflating" a single side of the graph (usually a tail) at the expense of the other. Thanks to some relevant examples, this paper tries to provide general indications to recognize the two kinds of parameters above and to interpret their effect on the classical Pearson's standardized fourth moment and on some lesser known kurtosis indexes. Specifically, it is shown that only a decomposed analysis of indexes can help to understand their apparent contradictions, especially when some of them are too sensitive to changes in the tails. Finally, some applications are provided. [ABSTRACT FROM AUTHOR]

With the swift advancement of the geometric modeling industry and computer technology, traditional generalized Ball curves and surfaces are challenging to achieve the geometric modeling of various complex curves and surfaces. Constructing an interpolation curve for the given discrete data points and optimizing its shape have important research value in engineering applications. This article uses an improved golden eagle optimizer to design the shape-adjustable combined generalized cubic Ball interpolation curves with ideal shape. Firstly, the combined generalized cubic Ball interpolation curves are constructed, which have global and local shape parameters. Secondly, an improved golden eagle optimizer is presented by integrating Lévy flight, sine cosine algorithm, and differential evolution into the original golden eagle optimizer; the three mechanisms work together to increase the precision and convergence rate of the original golden eagle optimizer. Finally, in view of the criterion of minimizing curve energy, the shape optimization models of combined generalized cubic Ball interpolation curves that meet the C1 and C2 smooth continuity are instituted. The improved golden eagle optimizer is employed to deal with the shape optimization models, and the combined generalized cubic Ball interpolation curves with minimum energy are attained. The superiority and competitiveness of improved golden eagle optimizer in solving the optimization models are verified through three representative numerical experiments. [ABSTRACT FROM AUTHOR]

The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations. [ABSTRACT FROM AUTHOR]

In multiquadratic radial basis function (MQ-RBF) interpolation, shape parameters have a direct effect on the interpolation accuracy. The paper presents an MQ-RBF interpolation technique with optimized shape parameters for estimating the parameters of sine wave signals. At first, we assessed the impact of basic sinusoidal parameters on the MQ-RBF interpolation outcomes through numerical experiments. The results indicated that the angular frequency of a sine wave is a crucial determinant of the corresponding MQ-RBF interpolation shape parameters. A linear regression method was then used to establish the optimal parameter selection formula for a single-frequency sine wave, based on a large volume of experimental data. For multi-frequency sinusoidal signals, appropriate interpolation shape parameters were selected using the random walk algorithm to create datasets. These datasets were subsequently used to train several regression models, which were then evaluated and compared. Based on its operational cost and prediction accuracy, the random forest algorithm was chosen to establish the shape parameter selection model for multi-frequency sinusoidal signals. The inclusion of the Bayesian optimizer resulted in a highly accurate model. The establishment of this model enabled the adaptive selection of the corresponding shape parameters for any sine wave signal, providing a convenient means of selecting MQ-RBF interpolation shape parameters. Furthermore, the paper proposes an MQ-RBF interpolation subdivision least squares method that significantly improves the estimation accuracy of sine wave parameters. The practicality of the method was validated by successfully applying it in the calibration of the clock delay mismatch of a time-interleaved analog-to-digital converter system. [ABSTRACT FROM AUTHOR]

Bézier surfaces and Q-Bézier surfaces are effective modeling tools for shape design in computer-aided geometric design, computer vision, and computer graphics. The mutual conversion between these two kinds of surfaces is a pivotal and knotty technique in CAD/CAM. In this paper, the conversion between Q-Bézier surfaces and rectangular Bézier surfaces is investigated. However, due to the uncertain parameters of the Q-Bézier surfaces, the approximation conversion from rectangular Bézier surfaces to Q-Bézier surfaces can be regarded as an optimization problem, which is effectively dealt with by swarm intelligence algorithm. In this regard, an enhanced Black Widow Optimization called LDBWO is proposed to find more suitable shape parameters to obtain optimal approximation Q-Bézier surfaces, which are closer to the given Bézier surfaces. The LDBWO algorithm overcomes the shortcomings of standard BWO algorithm such as low accuracy, slow convergence, and is easy to fall into local optimum by introducing golden sine learning strategy and diffusion process. Furthermore, to confirm and validate the performance of the LDBWO, eight well-known intelligent algorithms are compared with the LDBWO on various benchmark functions and engineering examples. Finally, by minimizing the conversion error defined by the L2-norm, the optimization model of the approximation conversion from rectangular Bézier surfaces to Q-Bézier surfaces is established. Several representative numerical examples are provided to illustrate the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The effect of topology on the collapse transition and instantaneous shape of an energy polydisperse polymer (EPP), a model heteropolymer is studied by means of computer simulations. In particular, three different chain topologies, namely, linear (L), ring (R), and trefoil knot (T), are considered. The heteropolymer is modeled by assigning each monomer an interaction parameter, εi, drawn randomly from a Gaussian distribution. Through chain size scaling, the transition temperature, θ, is located and compared among the chains of different topologies. The influence of topology is reflected in the value of θ and observed that θ(L) > θ(R) > θ(T) in a similar fashion to that of the homopolymer counterpart. Also studied chain size distributions, and the shape changes across the transition temperature characterized through shape parameters based on the eigenvalues of the gyration tensor. It is observed that, for the model heteropolymer, in addition to chain topology, the θ‐temperature also depends on energy polydispersity. [ABSTRACT FROM AUTHOR]

Watershed resources analysis and development intervention at the catchment level requires primarily an understanding of the physical morphometry, but it has long been emphasized for hydrological analysis of soil and water conservation initiatives. This research was carried out to assess the morphometric characteristics of the sub-watersheds of Gelana using a geographic information system and remote sensing. The parameters considered were bifurcation ratio (Rb), stream frequency (Fs), drainage density (Dd), drainage texture (Dt), Length of overland flow (LoF), Constant of channel maintenance (CCM), infiltration number (If), elongation ratio (Re), circulatory ratio (Rc), form factor (Ff), compactness coefficient (Cc), basin relief (H), relief ratio (Rh), ruggedness number (Rn), stream power index (SPI), and sediment transport index (STI). Shuttle Radar Topographic Mission Digital Elevation Model 30 m resolution was used as input data to generate the value of these variables. The results showed that linear parameters ranged from Rb = 1.4-5.0; Fs = 0.33-0.55; Dd = 0.54-0.851 while shape parameters revealed that the sub-watersheds (SWs) as being more elongated (Re = 0.46-0.71; Rc = 0.183-0.37; Cc = 1.62-2.33); LoF= 0.27-0.425; IF=0.19-0.46; CCM= 1.175-1.85. This reflects the dominance of dendric drainage patterns and high flooding susceptibility. In addition, the relief parameter also revealed that 75% of 262 km2 of the seven sub-watersheds are relatively gentle relief (Rh<1.146). It can be concluded that sloppy terrain and high surface relief with relatively elongated shapes are observed in the northern and northeastern parts of the study sub-watersheds, implying high soil and water conservation priority, while the central and southern parts are characterized by a flat topography. [ABSTRACT FROM AUTHOR]

The poor adsorption stability of the current air-suction sunflower seed-metering devices under high-speed operation leads to unstable seed adsorption posture, as well as increases in the multiple and miss indices. In order to improve the stability of seed adsorption, an air-suction sunflower seed-metering device based on structure-assisted stable suction posture was designed, the movement mechanism of the seed absorption process and the seed-feeding process is analyzed, following which mathematical models of the processes are established, and the key factors affecting the seeding performance were determined. Furthermore, the shape parameter equation of sunflower seeds is fitted, depending on the dimensions and shape parameters, which provides the design basis for the key components of the seed-metering device. Moreover, in order to optimize the seed-metering devices structure, taking the depth, width, and length of the seed suction grooves as experimental factors, as well as the qualified, multiple, and miss indices as evaluation indices, an orthogonal regression experiment considering three factors and three levels is carried out using the Box–Behnken experimental design. Finally, multiple regression models are established and extreme value theory is used to optimize the parameters of the influencing factors and obtain optimal parameter combinations. The optimal parameter combination is as follows: width of the suction groove, 10.5 mm; depth, 3 mm; and length, 23 mm. Results from verification experiments revealed the corresponding experimental indicators, the qualified index is 91.63%, the multiple index is 5.63%, and the miss index is 2.74%, thus meeting the requirements of effective sunflower seed sowing. [ABSTRACT FROM AUTHOR]

In this paper, we study a GC¹ rational cubic trigonometric spline and develop positivity, monotonicity and constrained curve interpolation schemes by using a GC1 piecewise rational cubic trigonometric spline with five shape parameters. The approximation properties are discussed and confirmed that the expected approximation order is h². [ABSTRACT FROM AUTHOR]

This experimental study is an attempt to apply the constriction size concept for determination of hydraulic conductivity of filter material. Five shapes of granular filter material of different gradations were used for the experimental investigations. The controlling constriction size for selected shapes and gradations were worked out and related to the experimental values of hydraulic conductivity. The empirical model for determination of hydraulic conductivity based on controlling constriction size for each shape of granular filter material was developed. Finally, a combined empirical model for the determination of hydraulic conductivity of granular filters was developed based on shape parameters and the controlling constriction size. The model was validated with the experimental data of a past research work; the predicted values of hydraulic conductivity were found in close approximation to the values obtained from the literature. Results of the present work will help in improving the existing design criteria of protective filters, which form an integral part of the safety of hydraulic structures. HIGHLIGHTS Inclusion of constriction size and shape parameters in the hydraulic conductivity determination of granular filters.; Improvement of the effectiveness of the filter design based on permeability.; Enhancement of water resources management in the essential hydraulic structure design where filters form a critical component.;

Abstract This study is based on some C 1 $C^{1}$ , C 2 $C^{2}$ , and C 3 $C^{3}$ continuous computer-based surfaces that are modeled by using generalized blended trigonometric Bézier (shortly, GBT-Bézier) curves with shape parameters. Initially, generalized blended trigonometric Bernstein-like (shortly, GBTB) basis functions with two shape parameters are derived in explicit expression which satisfied the basic geometric features of the traditional Bernstein basis functions. Moreover, the GBT-Bézier curves and tensor product GBT-Bézier surfaces with two shape parameters are also presented. All geometric features of the proposed GBT-Bézier curves and surfaces are similar to the traditional Bézier curves and surfaces, but the shape-adjustment is the additional feature that the traditional Bézier curves and surfaces do not hold. Finally, a class of some complex computer-based engineering surfaces via GBT-Bézier curves with shape parameters is presented. In addition, two adjacent GBT-Bézier surfaces segments are connected by higher C 2 $C^{2}$ and C 3 $C^{3}$ continuity constraints than the existing only C 1 $C^{1}$ shape adjustable Bézier surfaces. Some practical examples are provided to show the efficiency of the proposed scheme and to prove it as another powerful way for the construction and modeling of various complex composite computer-based engineering surfaces using higher-order continuities.

Recent improvements in robotic arms have increased their interest in many areas such as the industry and biomedical sectors. Path planning is an essential part of the robotic arm, since most automated factories seek to move things from one place to another with obstacles providing the shortest route. This paper presents a novel optimal path planning algorithm based on the 3D cubic Bézier curve with three shape parameters and its geometric properties and hierarchical clustering. The proposed method utilizes a feature vector which is obtained from curvature, torsion, and path length of candidate curves. A hierarchical clustering is applied to determine curve pairs. Then, a multi-objective function is used to determine the best curve pair, which gives the best curve for the robotic arm. Besides forming the optimal 3D cubic Bézier path, the optimal ruled and developable path surfaces are obtained. In addition to presenting theoretical results, this work also demonstrates the proposed method on several Kinova Gen3 robotic arm cases. [ABSTRACT FROM AUTHOR]

In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L, bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.

The B-spline curves have been grasped tremendous achievements inside the widely identified field of Computer Aided Geometric Design (CAGD). In CAGD, spline functions have been used for the designing of various objects. In this paper, new Quadratic Trigonometric B-spline (QTBS) functions with two shape parameters are introduced. The proposed QTBS functions inherit the basic properties of classical B-spline and have been proved in this paper. The proposed scheme associated with two shape parameters where the classical B-spline functions do not have. The QTBS has been used for designing of different parts of airplane like winglet, airfoil, turbo-machinery blades and vertical stabilizer. The designed part can be controlled or changed using free parameters. The effect of shape parameters is also expressed.

With great enthusiasm and admiration we would like to pay tribute to Paul de Faget de Casteljau for his essential contribution to CAGD. Motivated by the development of automated human-computer collaboration for car industry, not only was he the very first pioneer in this field, but his initial geometric approach to creating shapes from poles was even undeniably the simplest and most remarkably effective. Two crucial points in this approach are to keep in mind: firstly, the idea of splitting one variable into several variables to facilitate the algorithmic construction of curves; secondly, the possibility of controlling shapes by means of osculating flats and corner-cutting algorithms. The present article is a partial survey on Chebyshevian blossoms intended to show that his ideas are still alive. • Review Chebyshevian blossoms in keeping with de Casteljau's work. • Review dimension elevation and associated convergence properties with some (new) examples. • Illustrate design and convergence in rational spaces defined by poles. [ABSTRACT FROM AUTHOR]

The aim of this study is to determine the usability of electron microscopy to assess the inhalation exposure and extract morphological information otherwise unobtainable. Generated graphite particles in the size range of 20–150 nm were used as test aerosol and were analyzed by scanning mobility particle spectrometer and collected on membrane filters for scanning electron microscopy analysis. Observations revealed large discrepancies between electromobility diameter and diameter obtained from SEM analysis, as well as changes in particles' morphology. Particles have been quantitatively analyzed and grouped based on their morphological parameters such as solidity and circularity. This proves the feasibility of electron microscopy a complementary technique to real-time spectroscopy measurements, as it allows the identification of nano-objects with a detailed stereological description. [ABSTRACT FROM AUTHOR]

Path planning is an essential step for mobile robots. This paper presents a novel shape parameter search‐ (SP‐Search) based path planning algorithm for mobile robots using quintic trigonometric Bézier curve and its two shape parameters. The proposed search algorithm recursively divides the shape parameter space into a grid and uses a multi‐objective function to determine the shape parameter values. The multi‐objective function is expressed by the sum of four terms multiplied by coefficients: the integral of the squared jerk, the integral of the squared curvature, the mean of minimum distances between the predefined skeleton and quintic trigonometric Bézier path, and the arclength. Optimal shape parameters can be determined according to the required cases by the user using four coefficient values. Experimental results illustrate the feasibility of the proposed SP‐Search algorithm. Also, we have developed another intuitive algorithm, the Brute Force Grid Search algorithm, which is the brute force approach for shape parameter search to compare the performance of the proposed SP‐search algorithm for path planning problems as a baseline. The experimental results show that the SP‐Search algorithm finds better shape parameter values with a considerably small number of searched points. [ABSTRACT FROM AUTHOR]

A key technique for modeling 2D objects is built using a Bézier-like rational quadratic trigonometric function with two form parameters. Since they are generated employing weights, the suggested rational quadratic trigonometric spline curve schemes are helpful for shape modeling. The established method yields a curve with the best geometric properties, such as convex hull, partition of unity, affine invariance, and diminishing variation. The parameters in the suggested method's construction are helpful for several critical shape features such as local, global, and biased tension qualities, which give good control over the curve and allow for shape modification as desired. In addition, the recommended approach is C 2 . Furthermore, the comparative study of both splines is discussed which revealed the proposed method as a superior alternative to rational cubic spline. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters α and λ and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem is given by a weighted A-statistical convergence. A Korovkin-type theorem is provided for the univariate and bivariate cases of the blending-type operators. Moreover, the convergence behavior of the univariate and bivariate new blending basis and new blending operators are exhaustively demonstrated by computer graphics. The studied univariate and bivariate blending-type operators reduce to the well-known Bernstein operators in the literature for the special cases of shape parameters α and λ , and they propose better approximation results. [ABSTRACT FROM AUTHOR]

Abstract A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree m. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The C 3 $C^{3}$ and G 2 $G^{2}$ continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.