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This study presents a generalized approach to Bézier curves and surfaces by utilizing the blending (α , λ , s) -Bernstein basis. The (α , λ , s) -Bernstein basis introduces shape parameters α , λ , and s, which allow for more flexibility and control over the curve's shape compared to the classical Bernstein basis. The paper explores the properties of these generalized curves and surfaces, demonstrating their ability to maintain essential geometric characteristics, such as convex hull containment and endpoint interpolation, while providing enhanced control over the shape. This work aims to contribute to the fields of computer-aided geometric design and related applications by offering a robust tool for curve and surface modeling. [ABSTRACT FROM AUTHOR]

The deneutrosophication process is a process transforming from neutrosophic values to crisp output values. It is the final step for the operations within a neutrosophic set and system. Neutrosophic set theories are a generalization of intuitionistic fuzzy and fuzzy set theories, focusing on truth, indeterminacy, and falsity memberships independently. However, it isn't easy to generate a geometrical model such as a Bézier surface by using neutrosophic set theory through the deneutrosophication process. Therefore, this paper used an average of triangular footprint method in the deneutrosophication process to construct the neutrosophic Bézier surface (NBS) models by using approximation methods. Before generating the model, the neutrosophic control net (NCN) must first be introduced using the deneutrosophication process. After that, the NCN will be blended with the Bernstein basis function to generate the NBS approximation model. Next, some numerical examples of NBS will be provided. Finally, the deneutrosophication of NBS approximation models will be visualized, and its algorithm will be shown. [ABSTRACT FROM AUTHOR]

The connections between parameter surfaces enable the development of various geometric designs. However, these surfaces are generally connected along their boundaries in a rectangular domain. This study investigated the methods for connecting surfaces along a curve. To this end, we introduced two-variable degenerate functions and utilized their algebraic properties to characterize the form of the parameter surfaces for practical surface construction. The results were used to deform the surfaces along the curve. For application, we presented the examples of deformations using Bézier surfaces and extended them to general surfaces. [ABSTRACT FROM AUTHOR]

The human–computer interaction attribute of the interactive genetic algorithm (IGA) allows users to participate in the product design process for which the product needs to be evaluated, and requiring a large number of evaluations would lead to user fatigue. To address this issue, this paper utilizes an XGBoost proxy model modified by particle swarm optimization and the graphical interaction mechanism (GIM) to construct an improved interactive genetic algorithm (PXG-IGA), and then the PXG-IGA is applied to 3D vase design. Firstly, the 3D vase shape has been designed by using a bicubic Bézier surface, and the individual genetic code is binary and includes three parts: the vase control points, the vase height, and the texture picture. Secondly, the XGBoost evaluation of the proxy model has been constructed by collecting user online evaluation data, and the particle swarm optimization algorithm has been used to optimize the hyperparameters of XGBoost. Finally, the GIM has been introduced after several generations, allowing users to change product styles independently to better meet users' expectations. Based on the PXG-IGA, an online 3D vase design platform has been developed and compared to the traditional IGA, KD tree, random forest, and standard XGBoost proxy models. Compared with the traditional IGA, the number of evaluations has been reduced by 58.3% and the evaluation time has been reduced by 46.4%. Compared with other proxy models, the accuracy of predictions has been improved up from 1.3% to 20.2%. To a certain extent, the PXG-IGA reduces users' operation fatigue and provides new ideas for improving user experience and product design efficiency. [ABSTRACT FROM AUTHOR]

This paper proposes a combined approach of deep deterministic policy gradient (DDPG) and graph attention network (GAT) to the geometry optimization of latticed shells with surface shapes defined by a Bézier control net. The optimization problem is formulated to minimize the strain energy of the latticed structures with heights of the Bézier control points as design variables. The information of the latticed shells, including nodal configurations, element properties and internal forces, and the Bézier control net, consisting of control points and control net, are represented as graphs using node feature matrices, adjacency matrices, and weighted adjacency matrices. A specifically designed DDPG agent utilizes GAT and matrix manipulations to observe the state of the structure through the graphs, and decides which and how Bézier control points to move. The agent is trained to excel in the task through a reward signal computed from changes in the strain energy in each optimization step. As shown in numerical examples, the trained agent can effectively optimize structures of different sizes, control nets, configurations, and initial geometries from those used during the training. The performance of the trained agent is competitive compared to particle swarm optimization and simulated annealing despite using a lower computational cost. Highlights: - A method using a reinforcement learning agent is proposed to optimize the geometry of latticed structures. - The agent is designed to observe the structure and Bézier control net and modify the Bézier control net. - The method yields good results using fewer computations when compared to other conventional methods. [ABSTRACT FROM AUTHOR]

Shape optimization is of great significance in structural engineering, as an efficient geometry leads to better performance of structures. However, the application of gradient-based shape optimization for form finding is limited, which is partly due to the difficulty and the complexity in gradient evaluation. In this work, an efficient framework based on automatic differentiation (AD), the adjoint method and accelerated linear algebra (XLA) is proposed to promote the implementation of gradient-based shape optimization for form finding. The framework is realized by the implementation of the high-performance computing (HPC) library JAX. We leverage AD and the adjoint method in the sensitivity analysis stage. The XLA feature is exploited by an efficient programming architecture that we proposed, which can boost gradient evaluation via just-in-time compilation of computer programs. The proposed framework also supports hardware acceleration such as GPUs. The framework is applied to the form finding of arches and different free-form gridshells: gridshell inspired by Mannheim Multihalle, four-point supported gridshell, and canopy-like structures. Two geometric descriptive methods are used: non-parametric and parametric description via Bézier surface. Unconstrained and constrained shape optimization problems are considered, where the former is solved by gradient descent and the latter is solved by sequential quadratic programming (SQP). Through these examples, the proposed framework is shown to be able to provide structural engineers with an efficient tool for form finding, enabling better design for the built environment. [ABSTRACT FROM AUTHOR]

In this paper, we give a method for positioning a toroidal tool on a tensor product surface with two points of contact. By firing vertical rays and circular rays from the tensor product surface to the toroidal cutting surface, we can find these two points of contact by solving only quadratic equations, whereas previous methods typically solve transcendental equations using iterative methods. The new method is faster than the methods presented in literature and is independent of the method used to define the part surface. In addition to general applicability, test results on three sample surfaces are presented. [ABSTRACT FROM AUTHOR]

This article presents, tests and validates a method for generating representative unit cells of braided composites based on a geometrical approach. The method focuses on the geometry of the tows and uses surface minimization to obtain the minimal equivalent area of a central surface that is used for the generation of volumes and orthotropic mechanical properties of the tows. The homogenization of the yarns is solved using analytical equations and the homogenization of the cell is solved using a representative volume element technique. The robustness of the method is evaluated using a benchmark comparison with experimental and numerical results to validate the generated ge.ometry. The method is also tested and shown to be robust in modeling the geometry for several values of geometrical parameters of the tows without user intervention, including for configurations yielding a high cover factor. • A generator of braided composite unit cells is developed using a geometrical approach. • The method uses surface minimization to generate compatible tow geometries. • The robust method can generate cells of different parameters with high cover factor. • Homogenization is performed analytically/numerically at the yarn/cell level. • Homogenized properties are compared with literature experimental, numerical results. [ABSTRACT FROM AUTHOR]

• Dense 3D point clouds can be reduced to a small number of reference points. • Deliberate arrangement of reference points enables precise surface reconstruction. • The pig population can be described by an average shape and variations from it. • Shape attributes can be specified as size, form, posture and data noise. The recording of pig surfaces allows for the estimation of growth-related body size dimensions from which the animals' condition can be inferred. Furthermore, for purposes such as animal breeding, it is of particular interest to describe the shape variations within the population. Therefore, we built a Statistical Shape Model (SSM) that enables a quantitative and visual exploration of variation in size, form and posture in an intuitive way. We included 4315 images from 582 animals with a body weight ranging from 50 to 140 kg. After data processing, i.e. the removal of data points corresponding to the head and tail, the 3D point clouds were reduced to an equal number of reference points. These reference points were arranged so that the pig surface could be reconstructed via Bézier surface patches. Then, Procrustes Analysis was performed to align the reference points from all shapes. A mean shape and its main modes of variation were derived, some of which were used to correct the posture of the animals. A linear regression for weight prediction resulted in a mean absolute error of 5.06 kg before and 2.84 kg after data processing. The median error of surface reconstruction was 2.5 mm taking eight surface patches with 45 reference points and 1.3 mm taking 32 surface patches with 153 reference points into account. The first six modes of variation from the SSM indicated differences in size and posture. Interestingly, regions prone to noise were also revealed. In conclusion, we present a novel approach that can be used to receive a compact number of corresponding data points, providing data that is easier and more efficient to handle while exposing shape-specific attributes. Our approach is helpful to visualize and compare shapes, including the collection of body size measurements as well as unsupervised and supervised clustering for more objective data-driven decision-making. [ABSTRACT FROM AUTHOR]

Ball surfaces play a crucial role in modeling 3D objects with varying thickness. In this paper, energy functionals for surface design are extended to Ball surfaces and the variational problem of finding the energy-minimizing Ball surface within the constraints imposed by the controls is investigated. Besides, owing to the excellent properties of Bernstein bases, we propose a computationally inexpensive algorithm for the construction of energy-minimizing Ball Bézier surfaces. Finally, an efficient design tool is provided for the construction of Cr continuous energy-minimizing blend surface from two disjoint Ball surfaces. The feasibility of the method is verified by several examples. By adjusting the weighted coefficients, different energy functionals are defined and thus Ball surfaces with different shapes and thickness can be obtained, achieving the deformable modeling of Ball surfaces. [ABSTRACT FROM AUTHOR]

Diagonal curve is one of the most important shape measurements of tensor-product Bézier surfaces. An approach to construct Bézier surfaces with energy-minimizing from two prescribed diagonal curves is presented in this paper. Firstly, a general second order functional energy is formulated with several parameters. This functional includes many common functionals as special cases, such as the Dirichlet energy, the biharmonic functional and the quasi-harmonic functional etc. Secondly, the necessary and sufficient conditions for tensor-product Bézier surface with minimal energy from prescribed diagonal curves are derived. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves are also prescribed. Finally, the effectiveness of the proposed method is illustrated by several surface modeling examples. [ABSTRACT FROM AUTHOR]

We introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.

This paper presents an extension of multipoint machining technique, called the Drop Spin and Tilt (DST) method, that spins the tool on two axes, allowing for the generation of multiple contact points at varying distances around the first point of contact. The multiple DST second points of contact were used to manually generate a toolpath with uniform spacing between the two points of contact. The original DST method used a symbolic algebra package to position the tool on a bi-quadratic surface; our extension is a numerical solution that allows positioning a toroidal tool on a tensor product Bézier surface, which we tested on bi-degrees 2, 3, 4, and 5. On the three bicubic surfaces we tested, the average number of initial seeds for our numerical method was less than 5 across all three surfaces, and the average number of Newton iterations was less than 20 except when the curvature of the tool closely matches the local surface curvature (in which case more iterations were required). Furthermore, we investigate the spread of possible second points of contact as the tool is spun around these two axes, demonstrating the feasibility of using the method to control the machining strip width. [ABSTRACT FROM AUTHOR]

In this paper, the 3-tuple Bézier surface interpolation model is introduced. The 3-tuple control net relation is defined through intuitionistic fuzzy concept. Later, the control net is blended with Bernstein basis function to obtain surface blending function and to produce 3-tuple Bézier surface. The 3-tuple Bézier surface model is illustrated through the interpolation method by using data point with intuitionistic features. Some numerical example is shown. Lastly, the 3-tuple Bézier surface properties is also discussed. [ABSTRACT FROM AUTHOR]

Bezier surfaces are commonly used in Computer-Aided Geometric Design since it enables in geometric modeling of the objects. In this study, the shape operator of the timelike and spacelike surfaces has been analyzed in Minkowski-3 space. Then, the obtained results were applied to a numeric example. [ABSTRACT FROM AUTHOR]

This paper presents a new procedure to numerically compute multi-point tool positions for five-axis machining for tensor product Bézier surfaces. The developed method first drops the toroidal tool along the vertical axis and then iteratively rotates the surface to determine the second point of contact between the tool and the surface. The surface tilt gives the rotation of the tool axis that positions the tool to have multiple points of contact with the unrotated surface. The mathematical model involves higher order rational equations and Newton's method is used to numerically determine the solution. The developed algorithm is implemented in C++ and is tested for several bi-order Bézier patches from bi-cubic up to bi-order eight with uniformly and non-uniformly distributed control points. A comparative study with the earlier developed multi-point tool positioning methods (Duvedi et al. Int J Adv Manuf Technol 95:219–232, 2018, 97: pp. 279–295, 2018) is presented. The study shows that the developed method is computationally more efficient and reliable than previous methods. The accuracy of developed method is verified using the 3D graphical simulations in Maple and by actual five-axis machining. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Computer Engineering & Applications is the property of Beijing Journal of Computer Engineering & Applications Journal Co Ltd. 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.)

The main goal of this paper is to construct Bézier surface modeling for neutrosophic data problems. We show how to build the surface model over a data sample from agriculture science after the theoretical structure of the modeling is introduced. As a sampler application for agriculture systems, we give a visualization of Bézier surface model of an estimation of a given yield of bean seeds grown in a field over a period.

A wonderful unique research developments in modeling surface roughness and optimization of the predominant parameters to get a surface finish of desired level since only suitable selection of cutting parameters can get a better surface finish, so the objective of this work is to study the milling process parameters which include tool diameter, feed rate, spindle speed, and depth of cut resulting in optimal values of the surface roughness during machining AL-alloy 7024. The machining operation implemented on XK7124 3-axis CNC milling machine. The effects of the selected parameters on the chosen characteristics have been accomplished using Taguchi’s parameter design approach. The parameters considered are – depth of cut with two levels (0.2, 0.5 mm), tool diameter with two levels (6, 8 mm), spindle speed with two levels (1000, 2500 rpm), and finally feed rate with two levels (200, 500 mm/min). Analysis of the results showed that the optimal settings for low values of surface roughness are large tool diameter (8 mm), high spindle speed (2500 r.p.m), low feed rate (200 mm/min) and high depth of cut (0.5 mm). Response Table for mean of surface roughness showed that tool diameter has the most effected factors (rank one) followed by feed rate (rank two) then depth of cut which is the third effected factors and finally spindle speed with the less effected factors of surface roughness (rank four).

The Drop and Spin Method (DSM), a multipoint tool positioning technique for five-axis machining of corners formed by the intersection of two bi-quadratic Bézier surfaces with a toroidal end mill is presented in this paper. In the DSM the tool is dropped onto one of the surfaces to identify the first point of contact. The second point of contact is determined on the second surface. These two points of contact ensure that the toroidal tool can be positioned in the closest possible proximity to the two surfaces along the common edge. The method results in a multitude of solutions; our algorithm chooses the solution that fits the tool snug to the edge, resulting in a radiused corner with a radius matching the tool insert radius. A classification method is used to identify the region of the radiused corner and an algorithm is presented to generate tool path for machining it. The DSM was tested and verified by graphical simulations in a custom OpenGL based graphical CNC simulator, a custom simulator developed in Maple graphics™, and test pieces machined using five-axis DSM toolpaths are presented. • A new 5-axis method for machining a radius join between two surfaces meeting along a common edge is developed. • The new method positions a toroidal tool snug to the edge with the tool touching each of the joined surfaces tangently at a point of contact. • The method places the cutting tool being in tangential contact with both surfaces simultaneously, ensuring that the five axis tool paths are gouge free. [ABSTRACT FROM AUTHOR]

We present a robust and efficient surface-surface intersection (SSI) algorithm. The algorithm is based on intersecting surface triangulations and improves the resulting linear spline curve approximations of the exact intersection curves by replacing intersecion points of the surface triangulations by exact intersecion points. We use a specialized data structure, the k-d tree, for the efficient intersection of surface triangulations. The resulting intersection curves are guaranteed to lie within a user-specified tolerance.

A closed loop simulation is used for minimizing of undemanding vibration caused in the dish system. The finite element method is used to model the closed loop control via ANSYS- APDL. In this paper the Bezier function for surface modeling to get the best modeling points for dish is used. A designed surface is represented by sufficient control points, using these control points, the surface has been represented depending on Bezier technique to generate reliable and near-optimal dish surface. The required equations are generated to apply the surfaces and curves efficiently using MATLAP program, then exported to ANSYS to perform closed loop vibration analysis. It can be concluded that the closed loop control system with gain (Kp=4, Ki=1, Kd=0.1) suppression the vibration of the dish with 98% with different thickness and materials of dish. Also the natural frequencies and the mode shapes of the dish is evaluated. Three materials (pure copper, pure aluminum and steel) each with different thickness is taken for the dish (0.8, 0.9, 1, 1.1 and 1.2) mm and for each thickness the responses and the natural frequencies are determined for six modes. The effect of the thickness’s variation on natural frequencies for each material was studied. It can be observed that natural frequency is direct proportional with the thickness of the dish.

Multipoint machining (MPM) strategies are a class of methods that use the flexibility of a five-axis machine to position a tool in close proximity of the desired surface so that the tool touches the surface on at least two points of contact. However, existing multipoint methods cannot control the spacing between the two points of contact. This paper presents a new technique, called the Drop, Spin, and Tilt (DST) method that can generate multiple contact points at varying distances around the first point of contact. The multiple DST second points of contact were used to manually generate a toolpath with uniform spacing between the two points of contact. While a uniform space could be attained at most tool positions, problem cases were encountered demonstrating that more work is required for an algorithmic version of this process. The proposed method was implemented in a symbolic algebra system and verified on a variety of surfaces. [ABSTRACT FROM AUTHOR]

Abstract A singular vertex cannot always be avoided in constructing networks of boundary curves. When three boundary curves, two tangents of which are collinear, meet at a singular vertex, then the region that is bounded by the curves makes T-shape. We give a necessary condition for G 1 interpolation of a 3-valent singular vertex with Bézier surfaces, suggest a subdivision method with three rectangular sub-patches including T-junctions for use when this condition is not met, and give examples of its use. Graphical abstract Highlights • G 1 Bézier surfaces are obtained from a boundary curve network with a 3-valent singular vertex. • A necessary condition for G1 interpolation of the singular vertex is presented. • A subdivision method with three rectangular sub-patches including T-junctions is suggested for the singular vertex. [ABSTRACT FROM AUTHOR]

Copyright of Transactions of Beijing Institute of Technology is the property of Beijing 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 work, a robust tool-drop algorithm coupled with the bisection method is used to speed up the computations in the drop and tilt method (DTM), a multipoint tool positioning technique for 5-axis machining with a toroidal end mill. In the DTM, the tool is dropped on the surface along a predetermined tool-axis direction until it touches the surface tangentially. The position of the circular tool insert (pseudo-insert circle) through the first point of contact is then used to compute the tool-tilt angle that causes the tool to touch the surface tangentially at a second point. A previous implementation of DTM used Newton’s method to solve the system of equations for tilting the tool, but took many iterations and multiple initial seed solutions to converge to a valid solution for tool-tilt orientation. Using the bisection method coupled with the generalized tool-drop algorithm instead of a direct use of Newton’s method to determine the tool-tilt angle improves performance without compromising the accuracy. The method was implemented in C+ +, and a comparative study with an earlier method for several bi-quadratic and bi-cubic Bézier surfaces demonstrates the reliability of the developed method. [ABSTRACT FROM AUTHOR]

The Drop and Tilt tool position method for fiveaxis machining positions a toroidal tool to have two points of contact with a tensor product B'ezier surface. The method results in high order simultaneous non-linear transcendental equations that are slow to solve. In this paper, an efficient numerical implementation of the Drop and Tilt Method of multi-point tool positioning on tensor product surfaces is presented. The method was implemented in C++, and is a direct method that does not convert the tensor product surface into an intermediate representation such as triangles or point clouds. The method is successfully demonstrated on a number of tensor product B'ezier surfaces. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to construct Bézier surface modeling for neutrosophic data problems. We show how to build the surface model over a data sample from agriculture science after the theoretical structure of the modeling is introduced. As a sampler application for agriculture systems, we give a visualization of Bézier surface model of an estimation of a given yield of bean seeds grown in a field over a period. [ABSTRACT FROM AUTHOR]

The polyhedral method is an effective computational approach to the approximation of the gravitational potential and attraction of an irregularly shaped body. Refinement of the approximation by means of the piecewise continuous interpolation of body's surface via Bézier patches is proposed.

One of the most common operations encountered in computational fluid dynamics (CFD) solvers is the evaluation of the caloric and thermal equations of state (EoS) which are required to compute thermodynamic state variables from the conserved values that are typically being advanced by the simulation. The complexity of these calculations can vary widely depending on the nature of the fluid under consideration. We present a method for generating an interpolated representation of the EoS that is fairly inexpensive to evaluate regardless of the complexity of the actual underlying state equations. This approach has the advantage of being agnostic towards the original representation; whether it be a complex analytic expression, expensive iterative method, or interpolated from empirical data.

In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing.

In aspect of surface precision and smoothness, Bezier curves and surfaces, B-spline curves and surfaces, and NURBS curves and surfaces are three ways which have their own characteristics. Some of them has strong communication skills on precision of surface, but sometimes, in aspect of smoothness, it is hard to meet the requirements; For some others, surface smoothness could meet the design requirements, but the accuracy cannot. Combined the characteristics of different surface types, and integrated the advantages of Bezier surfaces and NURBS surfaces, the new curves can be constructed to meet the curved surface accuracy and surface smoothness requirements. [ABSTRACT FROM AUTHOR]

The complexity of state-of-the-art tools for shell optimization may limit their applicability in common practice. We propose two shape parametrizations, inserted into a robust and simple procedure, based on linear finite elements and gradient-based optimization. We represent the mid-surface by triangular Bézier surface and ad-hoc heuristic functions. The first method allows searching for a general shape, while in the second one the functions are chosen according to structural and aesthetical criteria. Small number of design variables ensures efficiency. The procedure is applied to Kresge auditorium at MIT. Both parametrizations provide satisfactory results, with slightly better performances of Bézier surface representation. [ABSTRACT FROM AUTHOR]