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Maritime collisions are a leading cause of accidents at sea, causing severe casualties, economic losses, and environmental damage. Although numerous methods have been developed to mitigate collisions, existing research primarily emphasises decision-making and path planning, often overlooking motion planning methods that incorporate dynamic constraints and ship-specific kinematics. This paper presents a motion planning method that takes into account ship dynamics to minimise jerk, the acceleration change rate, during ship acceleration and deceleration. The paper demonstrates the differential flatness property of the 3-DOF ship motion model, which allows for the direct use of flat outputs to implement motion planning and facilitate the addition of dynamic constraints. The front-end path search is performed using the traditional A* algorithm. To represent the trajectory as a piecewise Bézier curve, we use the Bernstein polynomial basis function. Based on this curve, we propose a motion planning method that employs jerk as the cost function and includes necessary constraints. This transforms the problem into a typical convex quadratic programming problem, ensuring the solvability of the optimisation problem. Finally, simulations are carried out to evaluate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Aiming at the problem that it is difficult for an orbital photovoltaic panel cleaning robot to span a large distance between photovoltaic panels, a method of designing and optimizing a non-coplanar orbit based on Bezier curves is proposed. Firstly, the robot's motion law is analyzed to obtain trajectory data for a single work cycle. Then, Bezier curves are utilized for trajectory design to ensure a smooth transition during the spanning motion phase. Thirdly, with the average value of the minimum distance between the Bezier curve and the point set data of the spanning motion phase as the optimization objective function, the nonlinear planning based on the SQP algorithm was adopted for the optimization of the upper and lower trajectories. Finally, the results of the case calculations indicate that the standard deviation of the optimized upper and lower trajectories was reduced by 35.63% and 40.57%, respectively. Additionally, the ADAMS simulation validation demonstrates that the trajectory errors of the four wheels decreased by a maximum of 8.79 mm, 23.78 mm, 10.11 mm, and 14.97 mm, respectively, thereby confirming the effectiveness of the trajectory optimization. [ABSTRACT FROM AUTHOR]

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks. [ABSTRACT FROM AUTHOR]

Abstract Currently, many utility‐scale wind turbine generator original equipment manufacturers are dependent on imported rare earth permanent magnets, which are susceptible to market risks from cost instability. To lower the production costs of these generators and stay competitive in the market, several small wind manufacturers are pursuing continuous improvements to both generator design and manufacturing. However, traditional design and manufacturing methods have yielded marginal improvements in wind power performance. This work presents novel methods to redesign a baseline 15‐kW wind turbine generator with reduced rare‐earth permanent magnets by leveraging cutting‐edge three‐dimensional (3D) printed polymer‐bonded permanent magnets and steel. Symmetric, asymmetric, and multimaterial‐magnet parametrization methods are introduced for shape optimization. We extend the symmetric and asymmetric methods to the back iron in the stator to further investigate the impact and opportunities for performance improvements with lesser active materials. We employ a design‐of‐experiments approach with parametric computer‐aided design for shape generation and evaluate different designs by magneto‐thermal modeling and finite‐element analysis. We use adaptive sampling technique to identify better performing designs with lesser magnet mass, higher efficiency, and lower cogging torque when compared with the baseline generator. Asymmetric pole designs resulted in a magnet mass in the range of 4.77–5.37 kg, which was 27%–35% lighter than the baseline generator, suggesting that a new design freedom exists that can be enabled by advanced manufacturing, such as 3D printing. Shaping the back iron in the stator resulted in material savings in electrical steel of up to 14.62 kg, which was 20% lighter than the baseline stator. We conducted a structural analysis to evaluate an optimized asymmetric rotor design from the point of view of mechanical integrity and air‐gap stiffness. The magnetically optimal shape profile was shown as having a positive impact on the radial stiffness, and an optimal solution was discovered to reduce the structural mass by nearly 30 kg, which was 29% lighter than the baseline.

This article explores various techniques for generating fractal-like Bézier curves in both 2D and 3D environments. It delves into methods such as subdivision schemes, Iterated Function System (IFS) theory, perturbation of Bézier curves, and perturbation of Bézier basis functions. The article outlines conditions on subdivision matrices necessary for convergence and demonstrates their use in creating an IFS with an attractor aligned to the convergent point of the subdivision scheme based on specified initial data. Additionally, it discusses conditions for obtaining a one-sided approximation of a given Bézier curve through perturbation. The article also addresses considerations for perturbed Bézier basis functions to construct fractal-like Bézier curves that remain within the convex hull polygon/polyhedron defined by control points. These methods find applications in various fields, including computer graphics, art, and design. [ABSTRACT FROM AUTHOR]

Currently, many utility‐scale wind turbine generator original equipment manufacturers are dependent on imported rare earth permanent magnets, which are susceptible to market risks from cost instability. To lower the production costs of these generators and stay competitive in the market, several small wind manufacturers are pursuing continuous improvements to both generator design and manufacturing. However, traditional design and manufacturing methods have yielded marginal improvements in wind power performance. This work presents novel methods to redesign a baseline 15‐kW wind turbine generator with reduced rare‐earth permanent magnets by leveraging cutting‐edge three‐dimensional (3D) printed polymer‐bonded permanent magnets and steel. Symmetric, asymmetric, and multimaterial‐magnet parametrization methods are introduced for shape optimization. We extend the symmetric and asymmetric methods to the back iron in the stator to further investigate the impact and opportunities for performance improvements with lesser active materials. We employ a design‐of‐experiments approach with parametric computer‐aided design for shape generation and evaluate different designs by magneto‐thermal modeling and finite‐element analysis. We use adaptive sampling technique to identify better performing designs with lesser magnet mass, higher efficiency, and lower cogging torque when compared with the baseline generator. Asymmetric pole designs resulted in a magnet mass in the range of 4.77–5.37 kg, which was 27%–35% lighter than the baseline generator, suggesting that a new design freedom exists that can be enabled by advanced manufacturing, such as 3D printing. Shaping the back iron in the stator resulted in material savings in electrical steel of up to 14.62 kg, which was 20% lighter than the baseline stator. We conducted a structural analysis to evaluate an optimized asymmetric rotor design from the point of view of mechanical integrity and air‐gap stiffness. The magnetically optimal shape profile was shown as having a positive impact on the radial stiffness, and an optimal solution was discovered to reduce the structural mass by nearly 30 kg, which was 29% lighter than the baseline. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In robotics, the ability of quadruped robots to perform tasks in industrial, mining, and disaster environments has already been demonstrated. To ensure the safe execution of tasks by the robot, meticulous planning of its foot placements and precise leg control are crucial. Traditional motion planning and control methods for quadruped robots often rely on complex models of both the robot itself and its surrounding environment. Establishing these models can be challenging due to their nonlinear nature, often entailing significant computational resources. However, a more simplified approach exists that focuses on the kinematic model of the robot's floating base for motion planning. This streamlined method is easier to implement but also adaptable to simpler hardware configurations. Moreover, integrating impedance control into the leg movements proves advantageous, particularly when traversing uneven terrain. This article presents a novel approach in which a quadruped robot employs impedance control for each leg. It utilizes sixth-degree Bézier curves to generate reference trajectories derived from leg velocities within a planar kinematic model for body control. This scheme effectively guides the robot along predefined paths. The proposed control strategy is implemented using the Robot Operating System (ROS) and is validated through simulations and physical experiments on the Go1 robot. The results of these tests demonstrate the effectiveness of the control strategy, enabling the robot to track reference trajectories while showing stable walking and trotting gaits. [ABSTRACT FROM AUTHOR]

In the thin airfoil theory, the camber line and the thickness distribution of general airfoils are mainly extracted by a linear combination of the upper and lower surfaces, giving rise to geometric distortions at the leading edge. Furthermore, despite the recent effort to obtain analytic expressions for the zero-lift angle of attack and quarter-chord moment coefficient, analytic generalizations are needed for the camber line component in the trigonometric series coefficients. In this sense, the present paper proposes a straightforward algorithm to extract the camber line and thickness distribution of general-shaped airfoils based on a finite difference method and the Bézier curve fitting. Integrals in the thin airfoil theory involving a Bernstein basis are performed, leading to series coefficients related to Gegenbauer polynomials. The algorithm is validated against analytical expressions of the NACA airfoils without introducing or adapting geometric parameters, and the results demonstrate good accuracy. In addition, the proposed algorithm indicated a significantly different geometric behavior for the SD7003 and E387 airfoils' camber slope at the leading edge in contrast with the classical linear approximation. Moreover, the method can be coupled conveniently in recent unsteady aerodynamic models established on the thin airfoil theory to obtain closed-form expressions for general airfoils. [ABSTRACT FROM AUTHOR]

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm's theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods. [ABSTRACT FROM AUTHOR]

The dispensing performance of needle-type piezoelectric jet dispenser constitutes a crucial factor that ensures the quality of additive manufacturing processes. In this paper, a novel approach is proposed to enhance the dispensing performance of needle-type piezoelectric jetting dispensers by introducing a more adaptable driving waveform based on Bézier curves. Initially, the approach considers the electromechanical coupling effect of the needle-type piezoelectric dispenser and constructs a high-precision fluid–solid coupling model of the dispensing process. Subsequently, a multi-physics field joint simulation platform combining Matlab and Fluent is established to systematically analyze control strategies in real service conditions. Next, a new driving waveform based on Bézier curves is introduced, and the control parameters are optimized using a genetic algorithm to address issues such as air bubbles in the droplets and instability of the dispensing process. The optimized waveform based on the Bézier curve reduces the volume of air suction during the dispensing process by over 20% compared to the traditional waveform and eliminates the uncontrolled vibration state of the needle in the fluid, ensuring the stability of the entire fluid refill process. Finally, the optimized control strategy is verified through experiments and compared with traditional methods. The experiment demonstrates its advantages in addressing issues with no air bubbles in the droplets and consistency of the droplets. This study provides valuable insights into optimizing the dispensing performance of needle-type piezoelectric jetting dispensers regarding control strategy. [ABSTRACT FROM AUTHOR]

Algal blooms are a common problem in inland waters, which raise growing awareness on monitoring lakes’ conditions. The on site monitoring is expensive and requires large human resources efforts. This work proposes remote monitoring techniques using satellite images and machine learning algorithms to predict chlorophyll $\alpha $ concentration in water bodies and identify algal blooms. The training and test dataset used in this study includes diverse range of lakes in Baltic countries. The lake spectral features obtained from Sentinel-2 satellite images are used as predictors for proposed deep neural network models. The prediction of chlorophyll $\alpha $ concentration with MAE 7.97 mg/ $\text{m}^{3}$ and bloom vs. non-bloom classification with 71.6 % accuracy was achieved. The use of Bèzier curves for smoothing the point-wise prediction is proposed for identification of algal bloom characteristics: the bloom start date, end date, and duration. The results showed lake type impact on the blooming time. The experimental data and code are released with paper.

Aiming at the problem that it is difficult for an orbital photovoltaic panel cleaning robot to span a large distance between photovoltaic panels, a method of designing and optimizing a non-coplanar orbit based on Bezier curves is proposed. Firstly, the robot’s motion law is analyzed to obtain trajectory data for a single work cycle. Then, Bezier curves are utilized for trajectory design to ensure a smooth transition during the spanning motion phase. Thirdly, with the average value of the minimum distance between the Bezier curve and the point set data of the spanning motion phase as the optimization objective function, the nonlinear planning based on the SQP algorithm was adopted for the optimization of the upper and lower trajectories. Finally, the results of the case calculations indicate that the standard deviation of the optimized upper and lower trajectories was reduced by 35.63% and 40.57%, respectively. Additionally, the ADAMS simulation validation demonstrates that the trajectory errors of the four wheels decreased by a maximum of 8.79 mm, 23.78 mm, 10.11 mm, and 14.97 mm, respectively, thereby confirming the effectiveness of the trajectory optimization.

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

The aim of this article is to study the relationships and models among the Van den Dool and Kratz equation, the gas chromatography (GC), and the Bézier curves constructed by aid of the Bernstein polynomials. Another aim of this article is to introduce open problems that contribute to real‐world problems involving mathematics, chemistry, and plant biology, including the Van den Dool and Kratz equation, the GC, and Bézier curves. Searching for the solutions of these problems may have qualities that will create the potential that can enter the field of study of many researchers. As a result of these goals, the usability of Bézier curves was investigated while determining the chemical composition of essential oil obtained from Potentilla aladaghensis Leblebici, by applying the retention index from the Van den Dool and Kratz equation and evaluating chemical compositions of the essential oil are characterized by GC–mass spectrometry (GC–MS). The Van den Dool and Kratz equation results have the potential to be used not only in the chemical compositions of the oils but also in applied mathematics and other fields. Moreover, we construct a new special finite sum. A lower bound and inequality are also given for the finite special sum involving the dead time associated with the isocratic step. Some applications and criticisms are given that include this lower bound and inequality for these sums and its effects on the chemical compositions of essential oil and the Van den Dool and Kratz equation. [ABSTRACT FROM AUTHOR]

Mode transition is an important dynamic process of an adaptive cycle engine (ACE). In order to obtain a smooth mode transition process, the closed-loop controller is designed based on the strong robust augmented linear quadric regulator (ALQR) method, and with the objective of minimizing the thrust fluctuation in the process of mode transition, an open-loop geometrical mechanism control schedules optimization method based on Bézier curves is proposed, so that the closed-loop control and the open-loop control can work in coordination. The simulation results at the subsonic cruise operating point and supersonic cruise operating point show that based on the optimized open-loop geometrical mechanism control schedules and the designed closed-loop ALQR control system, the ACE achieves fast and smooth geometrical mechanisms and engine output transition during the mode transition process with a maximum thrust fluctuation of 2.58%, which is much smaller than that of the traditional linear variation geometric mechanisms with a maximum 4.64% thrust fluctuation, which verifies the effectiveness of the proposed control method. [ABSTRACT FROM AUTHOR]

Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications. [ABSTRACT FROM AUTHOR]

This paper describes the development of a methodology for air propeller optimization using Bezier curves to describe blade geometry. The proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span. The goal of optimization is to identify the appropriate geometry of a propeller that reduces the power required to achieve a given thrust. Because the proposed optimization problem is a constrained optimization process, the technique of generating a penalty function was used to convert the process into a nonconstrained optimization. For the optimization process, a variant of the differential evolution algorithm was used, which includes adaptive techniques of the evolutionary operators and a population size reduction method. The aerodynamic characteristics of the propellers were obtained using the similar to blade element momentum theory (BEMT) isolated section method (ISM) and the XFOIL program. Replacing the angle of geometric twist with the angle of attack of the airfoil section as a design variable made it possible to increase the robustness of the optimization algorithm and reduce the calculation time. The optimization technique was implemented in the OpenVINT code and has been used to design helicopter and tractor propellers for unmanned aerial vehicles. The development algorithm was validated experimentally and using CFD numerical method. The experimental tests confirm that the optimized propeller geometry is superior to commercial analogues available on the market. [ABSTRACT FROM AUTHOR]

In this work, we investigate some approximation properties of blending type univariate and bivariate Schurer-Kantorovich operators based on shape parameter λ ∈ [-1, 1]. We evaluate some moment estimates and obtain several direct theorems. Next, we construct the bivariate version of proposed operators and compute rate of approximation with the partial and complete modulus of continuity. Moreover, we present certain graphical and numerical results for univariate and bivariate versions of these operators. [ABSTRACT FROM AUTHOR]

In the realm of forging processes, the challenge of real-time process control amid inherent variabilities is prominent. To tackle this challenge, this article introduces a Proper Orthogonal Decomposition (POD)-based surrogate model for a one-blow cold upsetting process in copper billets. This model effectively addresses the issue by accurately forecasting energy setpoints, billet geometry changes, and deformation fields following a single forging operation. It utilizes Bézier curves to parametrically capture billet geometries and employs POD for concise deformation field representation. With a substantial database of 36,000 entries from 60 predictive numerical simulations using FORGE® software, the surrogate model is trained using a multilayer perceptron artificial neural network (MLP ANN) featuring 300 neurons across 3 hidden layers using the Keras API within the TensorFlow framework in Python. Model validation against experimental and numerical data underscores its precision in predicting energy setpoints, geometry changes, and deformation fields. This advancement holds the potential for enhancing real-time process control and optimization, facilitating the development of a digital twin for the process. [ABSTRACT FROM AUTHOR]

Hyperuniform disordered solid (HUDS) structures can provide large, uniform, complete, and isotropic light confinement at the nanoscale after precise design. Based on the HUDS structures, in‐plane light confinement for developing photonic integrated circuits is also explored. To improve the performance of HUDS devices, researchers have mainly focused on cell size or cell distribution optimization in HUDS, which suffers from time‐consuming computation or moderate photonic bandgap (PBG) modification. Here, a morphology engineering method is demonstrated to tailor HUDS PBGs and improve HUDS waveguide devices for transverse electric mode. The results show that the Bezier‐curve‐decorated HUDS devices can achieve a maximum of about 75% PBG width increase, 1.5 dB transmittance improvement in a 10 µm long HUDS waveguide, improved quality factors of HUDS‐cladding microring resonators, and device fabrication compatibility with foundry processing. This study opens new avenues for the development of unprecedented devices for exploring light field regulation, nonlinear optics, and sensing. [ABSTRACT FROM AUTHOR]

Numerical simulations are conducted to improve the energy acquisition efficiency of H-type vertical axis wind turbines through the optimization of the related blade airfoil aerodynamic performance. The Bézier curve is initially used to fit the curve profile of a NACA2412 airfoil, and the moving asymptote algorithm is then exploited to optimize the design of the considered H-type vertical-axis wind-turbine blade airfoil for a certain attack angle. The results show that the maximum lift coefficient of the optimized airfoil is 8.33% higher than that of the original airfoil. The maximum lift-to-drag ratio of the optimized airfoil exceeds the maximum lift-to-drag ratio of the original airfoil by 11.22%. Moreover, the power coefficient is increased by 12.19% and the torque coefficient of the wind turbine is significantly improved. [ABSTRACT FROM AUTHOR]

In this study, it has been researched the exponential curve as a $3^{rd},$ $5^{th}$ and $7^{th}$ order B\'{e}zier curve in $\mathbf{E}^{2}$. Also, the numerical matrix representations of these curves have been calculated using the Maclaurin series in the plane via the control points.

Background: In recent years, automatic guided vehicles (AGVs) have developed rapidly and been widely applied in intelligent transportation, cargo assembly, military testing, and other fields. One of the key issues in these applications is path planning. Global path planning results based on known environmental information are used as the ideal path for AGVs combined with local path planning to achieve safe and fast arrival at the destination. The global planning method planning results as the ideal path should meet the requirements of as few turns as possible, short planning time, and continuous path curvature. Methods: We propose a global path-planning method based on an improved A * algorithm. And the robustness of the algorithm is verified by simulation experiments in typical multi obstacles and indoor scenarios. To improve the efficiency of pathfinding time, we increase the heuristic information weight of the target location and avoided the invalid cost calculation of the obstacle areas in the dynamic programming process. Then, the optimality of the number of turns in the path is ensured based on the turning node backtracking optimization method. Since the final global path needs to satisfy the AGV kinematic constraints and the curvature continuity condition, we adopt a curve smoothing scheme and select the optimal result that meets the constraints. Conclusions: Simulation results show that the improved algorithm proposed in this paper outperforms the traditional method and can help AGVs improve the efficiency of task execution by efficiently planning a path with low complexity and smoothness. Additionally, this scheme provides a new solution for global path planning of unmanned vehicles.

The paper presents a new procedure for finding the minimum radius of a Bézier Curves. The authors choose to use the theoretical procedure for a practical application. For this purpose, the authors find the minimum radii of a Bézier Curve consisting of 16 Cubic Bézier Curves on a bike trail situated in Mălini, Romania. In the design of the bike route, the authors consider elevations to design a bike trail on level curves. This decision is made to obtain a slope of less than 20%, for a relaxing route for the family. The route is created on a length of about 5 km completely respecting the design norms of a cycling track, according to the Methodological Guide for regulating the design. Authors choose to present a comparison for the procedure between hand-drawn horizontal curves, and approximating the trail using Cubic Bézier Curves and Python

In robotics, the ability of quadruped robots to perform tasks in industrial, mining, and disaster environments has already been demonstrated. To ensure the safe execution of tasks by the robot, meticulous planning of its foot placements and precise leg control are crucial. Traditional motion planning and control methods for quadruped robots often rely on complex models of both the robot itself and its surrounding environment. Establishing these models can be challenging due to their nonlinear nature, often entailing significant computational resources. However, a more simplified approach exists that focuses on the kinematic model of the robot’s floating base for motion planning. This streamlined method is easier to implement but also adaptable to simpler hardware configurations. Moreover, integrating impedance control into the leg movements proves advantageous, particularly when traversing uneven terrain. This article presents a novel approach in which a quadruped robot employs impedance control for each leg. It utilizes sixth-degree Bézier curves to generate reference trajectories derived from leg velocities within a planar kinematic model for body control. This scheme effectively guides the robot along predefined paths. The proposed control strategy is implemented using the Robot Operating System (ROS) and is validated through simulations and physical experiments on the Go1 robot. The results of these tests demonstrate the effectiveness of the control strategy, enabling the robot to track reference trajectories while showing stable walking and trotting gaits.

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods.

Copyright of Journal of Ordnance Equipment Engineering is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, a cellular automation model developed on the basis of Bezier curves with the use of a cellular automation approach for the creation of digital twins for porous nanostructures of different nature is proposed. Some numerical experiments on the creation of digital twins for the synthesized experimental samples of chitosan-based aerogels are carried out. The structural characteristics of the digital copies and experimental samples are compared, allowing us to conclude that the model is correct. The resulting digital twins can be used for predicting the properties of porous fiber materials, in particular, chitosan-based aerogels, to provide the partial replacement of real experiments by computational ones and, consequently, to decrease the expenditures on the development of new materials with specified properties. [ABSTRACT FROM AUTHOR]

Freeze–thaw cycles (FTCs) are one the most serious causes of damage in RC structures in cold regions affecting both concrete and bond behavior. In this paper, after a comprehensive review on the effects on concrete and bond of freeze–thaw degradation for ordinary concrete, a new constitutive bond stress-slip law is formulated in the framework of a coupled mechanical-environmental damage model. This law is developed exploiting the versatility of Bezier curves and using an optimization process based on genetic algorithms to select the best form among some options (composite quadratic, composite cubic, and a single fourth-order curve). Moreover, a comprehensive enhanced freeze–thaw degradation (E-FTD) model is developed including the effect of FTCs on bond behavior, by means of a new environmental damage parameter included in the proposed bond stress-slip law. The whole model is formulated, calibrated, and validated referring to all the available experimental data to the authors' knowledge both for compressive and pull-out tests for ordinary concrete. The comparison with the experimental results showed the proposed E-FTD model is able to predict bond strength degradation fairly well and generally on the safe side, capturing also the overall bond-slip curves, making it particularly appealing for the structural analysis of RC elements subjected to FTCs according to an engineering-oriented approach. [ABSTRACT FROM AUTHOR]

A metaheuristic method is an optimization technique that is generally inspired by natural or physical processes. The use of metaphors has created a tendency to reproduce existing algorithms with slight modifications or variations rather than encouraging the development of novel algorithmic techniques and principles. On the other hand, a complex network is a mathematical structure whose main characteristic is the ability to capture and analyze the intricate patterns and properties that emerge from the interactions between the elements that it connects. In this paper, a new metaphor-free metaheuristic algorithm based on complex networks and Bezier curves is presented. In this approach, candidate solutions are represented as nodes in a graph, whereas the connections between nodes or edges reflect the differences in their objective function values. Therefore, the graph provides a higher-level representation that captures the essential relationships and dependencies among the solutions. Once the graph is generated, the shortest path between each solution and the best solution is obtained. Then, the nodes obtained from this process are used as control points in the Bezier equation to generate the new agent position. Therefore, during the optimization process, the graph is continuously modified based on the evaluation of new candidate solutions and their objective function values, producing trajectories that allow the exploration and exploitation of the search space. The experimental results demonstrated the effectiveness of our approach by achieving competitive results compared to other well-known metaheuristic algorithms on various benchmark functions.

Navigation relayed by multiple stations (NRMS) is a promising technique that can significantly extend the operational range of unmanned aerial vehicles (UAVs) and hence facilitate the execution of long-range tasks. However, NRMS employs multiple external stations in sequence to guide a UAV to its destination, introducing additional variables and constraints for UAV trajectory planning. This paper investigates the trajectory planning problem for a UAV under NRMS from its initial location to a pre-determined destination while maintaining a connection with one of the stations for safety reasons. Instead of line segments used in prior studies, a piecewise Bézier curve is applied to represent a smooth trajectory in three-dimensional (3D) continuous space, which brings both benefits and complexity. This problem is a bi-level optimisation problem consisting of upper-level station routing and lower-level UAV trajectory planning. A station sequence must be obtained first to construct a flight corridor for UAV trajectory planning while the planned trajectory evaluates it. To tackle this challenging bi-level optimisation problem, a novel efficient decoupling framework is proposed. First, the upper-level sub-problem is solved by leveraging techniques from graph theory to obtain an approximate station sequence. Then, an alternative minimisation-based algorithm is presented to address the non-linear and non-convex UAV trajectory planning sub-problem by optimising the spatial and temporal parameters of the piecewise Bézier curve iteratively. Computational experiments demonstrate the efficiency of the proposed decoupling framework and the quality of the obtained approximate station sequence. Additionally, the alternative minimisation-based algorithm is shown to outperform other non-linear optimisation methods in finding a better trajectory for the UAV within the given computational time. [ABSTRACT FROM AUTHOR]

In this study, it has been researched the exponential curve as a 3rd, 5th and 7th order Bézier curve in E². Also, the numerical matrix representations of these curves have been calculated using the Maclaurin series in the plane via the control points. [ABSTRACT FROM AUTHOR]

In this study, firstly, the residual method, which was developed for initial value problems, is improved to find unknown coefficients without requiring for any system solution. Later, the adaptation of improved residual method is given to find approximate solutions of boundary value problems. Finally, the method improved and adapted for boundary value problems is used to find both critical eigenvalue and eigenfunctions of the one-dimensional Bratu problem. The most significant advantage of the method is finding approximate solutions of nonlinear problems without any linearization or solving any system of equations. Error analysis of the adapted method is given and an upper bound on the approximation error is derived for the eigenfunctions. The numerical results obtained are compared with the theoretical findings. Comparisons and theoretical observations show that the improved and adapted method is very convenient and successful in solving boundary value problems and eigenvalue problems approximately with high accuracy. [ABSTRACT FROM AUTHOR]

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In recent years, new steganographic methods have emerged to hide information in various multimedia files, such as images, videos, audio, text, and more. Such cover files, which are also called media or containers, are freely transmitted via the Internet, so they do not arouse any suspicion. An authorized person who has a secret steganographic key can restore the hidden message. This article discusses techniques for hiding data in vector images. Individual elements of vector graphics are represented through special mathematical objects (points, lines, curves, nodes, tangents, control points, etc.). These objects are used to hide information. But there is a problem, which lays in the fact, that affine vector image transformations can destroy a hidden message. In addition, hiding algorithms usually increase the size of vector image files, which is a certain unmasking feature. The main purpose of our study is to examine these problems. To do this, it is programmatically implemented data hiding and conducted a series of experiments. In particular, it is investigated various concealment techniques (bit method and pattern-based method). For different initial conditions, it is evaluated the resistance of hidden data to affine transformations, resizing of vector graphics files, performance, and so on. Based on the results of numerical experiments, it is substantiated the advantages and conditions of using the studied algorithms. To conduct the research, we have developed a cross-platform software implementation (JavaScript programming language), i.e. our scripts can be run on different types of platforms and browsers. Thus, the developed implementation can be used as a framework for individual web client applications, as well as for writing a backend of complex and professional tasks. [ABSTRACT FROM AUTHOR]

Purpose : To assess with Bézier curves the outcomes of Müllerotomy with anterior graded Müller muscle disinsertion for the treatment of Graves upper eyelid retraction (UER). Methods: Eighty-six eyelids of 52 inactive GO patients operated from November 2018 to June 2021 were included in this study. All measurements were performed on Bézier curves adjusted to the upper lid contour with a previously validated algorithm. Lid contour was classified regarding grade of superposition (GS) as excellent (GS > 90%), good (GS 85–90%) or poor (< 85%). Surgical success was defined as complete or partial if postoperative grade of asymmetry was < 10% with an excellent or good lid contour, respectively. Results: The mean age was 51 ± 10.4 years with a range from 31 to 78 years and a mean follow-up of 14.4 ± 7.4 months. There was a significant improvement of median GS (p < 0.0001) from preoperative (74.3%; 10.7 IQR) to postoperative values (91.7%; 6.3 IQR). A normalization of postoperative contour peak (− 0.69; 1.27 SD) and MPLD90 (4.2 mm; 0.8 SD) was noticed (p < 0.0001). Postoperative lid contour was excellent in 62 (72%), good in 16 (19%) and poor in 8 eyelids (9%). Surgical success was achieved in 42 patients (81%), from which 34 (81%) were complete. Reintervention was required in 14 eyelids (16%). Conclusions: Measuring surgical outcomes with Bézier curves allows an automated, complete and objective assessment, giving more consistency to our data compared to previous reports. Müllerotomy with graded Müller muscle disinsertion is a safe and effective procedure for Graves UER, offering predictable results. [ABSTRACT FROM AUTHOR]