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The COVID-19 pandemic has severely impacted global healthcare and financial systems, Highlighting the need for an automatic computer-aided diagnosis system using image recognition for chest X-rays(CXR). This study aims to classify COVID-19, normal, and pneumonia patients from CXR images via a modified ResNet-50 pre-trained CNN model. Our experiments are based on Dataset-1, which contains CXR images of COVID-19 and normal, while Dataset-2 also includes pneumonia. Dataset-1 and Dataset-2 were collected from King Abdul-Aziz Medical City in National Guard, Jeddah, Saudi Arabia. Moreover, a sample from the Kaggle repository was added to Dataset-1 and 2 to make two more Datasets. Our results for diagnosis of raspatory diseases have shown reliability and high accuracy of 95.28% (97.66% Sensitivity and 93.12% Specificity), which will be beneficial in aiding physicians and healthcare centres in the global fight against harmful spreading viruses through employing AI and ML techniques in X-ray medical diagnosis. [ABSTRACT FROM AUTHOR]

Based on one 5-point and two 2-point stencils, a nonlinear weighted scheme adaptive to the weighting of multi-order interpolation candidates for shock capturing is proposed. The JS weight function is adopted. The smoothness indicator of the 5-point stencil is specially designed with a higher-order leading term similar to the reference global smoothness indicator τ in the Z weighting. The design maintains that the nonlinear weights satisfy the sufficient condition for the scheme to avoid degradation at extreme points. The form of the weighting is also simplified, obviously reducing the weighting procedure's computational costs. Inviscid and viscous compressible flow problems are simulated to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show no obvious oscillations near the discontinuity, and the resolution of both discontinuities and smooth regions is better than JS, Z, and ZQ weighting. [ABSTRACT FROM AUTHOR]

This paper introduces a modified method for construction of a new design-control 6-point approximating refinement scheme. The construction of the new scheme is based on translation of the points of 4-point approximating refinement scheme to the new position according to the linear combination of certain displacement vectors. The initial and terminal points of these vectors are the refinement points of two 4-point approximating binary schemes. The new scheme contains three design-control parameters. These parameters increase the efficiency and flexibility of the new scheme. The mathematical and graphical analysis of the refinement scheme show that this scheme is good choice for curve modeling. [ABSTRACT FROM AUTHOR]

With the wide application of digital media technology and augmented reality, in order to enhance the design experience of mobile visual media design. This paper uses domain processing, point processing, big picture processing and model processing in numerical algorithms, construction of visual media images based on additive and multiplicative models, integrating the SOR model with the EDT model, build PLS model. Depending on the interactivity, patios-temporal fixity and degree of immersion of the virtual object, analyze the results of PLS experience conceptual model construction by combining system characteristics and cognitive elements of mobile visual media design. To explore the mechanisms by which digital media technologies and augmented reality technologies influence the design factors of mobile visual media, and evaluate the measurement model and structural model. The results of the evaluation found that accuracy 0.325, flexibility 0.472 and completeness 0.328 positively and significantly influenced information quality (R=0.697), system quality (R=0.52) and service quality (R=0.741). The system quality, information quality and service quality of the mobile visual media design products are guaranteed.

In order to improve the ability of micromechanical sensors to update data-matching nodes in real-time and speed up the establishment of a neighbor relationship between nodes, this paper proposes a personalized recommendation algorithm for micromechanical sensors based on the cloud model to improve the stability and real-time performance of micromechanical sensors. The algorithm uses the service attribute values of the cloud computing model and cloud clustering method to set feature term labels and establish a cloud service similarity matrix so as to meet the user-matched manufacturing service requirements. The intra-class clustering technique is applied to measure the clustering effect, and the evaluation function of each clustering number is calculated on the basis of considering the time sequence to determine the best clustering result to complete the personalized recommendation path for micromechanical sensors. To verify the application effect of the personalized recommendation algorithm for micromechanical sensors based on the cloud model, experiments are conducted. The results show that the recommendation algorithm in this paper can always control the node energy consumption below 2.5×103, and the average discovery delay is stable between 41-43 seconds. And the sensor response time is 12.1 seconds, and the average absolute deviation value is 0.23, which is nearly 1.3 times smaller than 0.53 and 0.52 of the collaborative recommendation algorithm and hybrid recommendation algorithm. It can be seen that the recommendation algorithm in this paper solves the problem of excessive neighbor discovery delay in the communication process of micromechanical sensors and effectively improves the personalized recommendation performance of micromechanical sensors.

To enrich the content and form of college civic education, this paper constructs a civic education management innovation platform based on big data technology to explore the innovative development path of college civic education. Using the differentiated fusion algorithm, the information processing of college student management data is carried out, and the educational mining information of college students is identified through feature extraction. Bayesian network is introduced in the process of index weight calculation to improve the data mining efficiency to ensure the validity of obtaining the evaluation index of Civic Education. To verify the feasibility of the innovation platform of Civic Education Management based on big data technology, the application effect was tested. The analysis results show that the platform constructed in this paper can make students’ cognitive awareness of course contents reach 0.98, students’ concentration effect reach 0.89, and the overall effect of teaching feedback reach 90% in selecting teaching resources. The average test score of students in the experimental group was 8.5, and the average length range of the independent learning course was 2.78-4.629 hours. Thus, it can be seen that the platform constructed in this paper is conducive to promoting the high-quality development of Civics teaching and enhancing the effectiveness of Civics education in colleges and universities.

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

Physically based modeling of deformable curves and surfaces has been widely used in computer graphics to meet the needs of both geometry and physics. In this paper, the weighted sum of the first and higher order generalized thin-plate spline-surface energies is introduced for the construction of transition surface. We derive the necessary and sufficient condition that the energy functional of a transition surface reaches its minimum, by which the positions of movable control points can be fully determined within the constraints imposed by the controls and continuity constraints. Furthermore, based on the excellent properties of Bernstein basis functions, we present a simple algorithm to compute energy-minimizing transition surfaces if the given two disjoint tensor product surfaces are in Bézier form. The feasibility of the method is validated by several examples. Compared with previous modeling methods, various physical characteristics are imparted on the modeling of transition surfaces in this paper, thus various energy-minimizing transition surfaces with different characteristics can be obtained. [ABSTRACT FROM AUTHOR]

In the areas of engineering and architecture, smooth curves and surfaces are crucial to produce sleek products and to prevent the structures from breaking, especially when dealing with various shapes and lengths. The designing task of smooth curves can be carried out by following certain standards of fairness. Curvatures and internal energy are two of the metrics that are commonly used to define the fairness of shapes. In this study, the generalized fractional Bézier basis functions is used to construct a fair and smooth curve using different types of continuity. The generalized fractional Bézier curve possesses shape parameters that are used to control the flexibility of the curve. It also has a notable parameter called fractional parameter that enables designers to control the length of constructed curves. The variation of the fractional parameter is directly contributed to the stretch energy. A new type of continuity called the fractional continuity that connects curve segments at different lengths of the first curve will be very conducive in satisfying the geometric conditions as well as fairness conditions. Lastly, the influence of various fractional and shape parameters of generalized fractional Bézier curve to the internal energy is demonstrated. Fractional parameter and continuity will expedite designing tasks when dealing with conflicting requirements. [ABSTRACT FROM AUTHOR]

The World Health Organization (WHO) recommends a six-step hand hygiene technique. Although multiple studies have reported that this technique yields inadequate skin coverage outcomes, they have relied on manual labeling that provided low-resolution estimations of skin coverage outcomes. We have developed a computational system to precisely quantify hand hygiene outcomes and provide high-resolution skin coverage visualizations, thereby improving hygiene techniques. We identified frequently untreated areas located at the dorsal side of the hands around the abductor digiti minimi and the first dorsal interosseous. We also estimated that excluding Steps 3, 6R, and 6L from the six-step hand hygiene technique leads to cumulative coverage loss of less than 1%, indicating the potential redundancy of these steps. Our study demonstrates that the six-step hand hygiene technique could be improved to reduce the untreated areas and remove potentially redundant steps. Furthermore, our system can be used to computationally validate new proposed techniques, and help optimise hand hygiene procedures. [ABSTRACT FROM AUTHOR]

In the field of CAGD/CAD, interpolation curve plays an important role in shape representation and shape design. Based on the traditional quintic Hermite interpolation curve, this paper proposes a novel class of quintic generalized Hermite interpolation curve (QG-Hermite interpolation curve, for short) with local shape parameters, which can satisfy C2 continuity automatically. Since each segment of the constructed curve has independent shape parameters, and its shape can be adjusted locally or globally by modifying the shape parameters. First, to obtain an interpolation curve satisfying certain specific geometric requirements such as the approximate shortest arc length, the strain energy minimization or the curvature variation energy minimization, etc., three shape optimization models of the QG-Hermite interpolation curve are established. Second, an improved gray wolf optimization (I-GWO) algorithm is used to solve the shape optimization models so as to determine the optimal shape parameters. Finally, the modeling examples show that the proposed method is effective and applicable in shape designing. [ABSTRACT FROM AUTHOR]

The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

The developable surface is always a hot issue in CAGD, CAD/CAM and used in many manufacturing planning operations, e.g., for ships, aircraft wing, automobiles and garments. In some special fields, the CAD model of developable surface is designed by interpolating a given spatial characteristic curve. In this paper, we present a class of methods to construct local controlled developable H-Bézier surfaces through a given characteristic curve. First, we introduce a class of generalized cubic H-Bézier basis functions, and utilize them to design the generalized cubic H-Bézier curves with shape parameters. Then we construct generalized cubic developable H-Bézier surfaces through a given space generalized cubic H-Bézier curve which serve as the line of curvature or geodesic. The shapes of the constructed surfaces can be adjusted and altered expediently using the shape parameters. Furthermore, the sufficient and necessary conditions for the interpolating developable H-Bézier surface to be a cylinder or a cone are deduced, respectively. Finally, we give some representative examples to illustrate the convenience and efficiency of the presented methods. [ABSTRACT FROM AUTHOR]

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features. [ABSTRACT FROM AUTHOR]

Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds, we propose the weighted quasi-interpolant spline approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of non-parametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation, and simulation of rainfall precipitations. [ABSTRACT FROM AUTHOR]

The procedure of positioning of pipes, for example for the automobile exhaust construction, by the use of 3D CAD systems is difficult and time consuming because of obstacles in an engine compartment. There are also further strong technical requirements and restrictions. An automatic generation procedure based on the level set method and the numerical solution of the Eikonal equation is proposed. The positions of pipes which satisfy the technical requirements are obtained using spline functions. [ABSTRACT FROM AUTHOR]

Ball curve plays a crucial role in modeling tubular shapes with variable thickness. In this paper, energy functionals for curve design are generalized to Ball curves and the variational design of Ball curves is investigated. Given a Ball curve with some of its control balls movable and having variable radiuses, we propose a method to determine the positions and radiuses of the movable control balls which minimize the energy functional. Based on this, we provide two efficient design tools: (i) to achieve C k continuity across linked Ball curves at their joint, while decreasing their energy as low as possible, (ii) for blending disjoint Ball curves subject to C k continuity constraints and energy minimization. The feasibility of the method is verified by several examples. By adjusting the weighted coefficients, different energy functionals are defined and thus Ball curves with different shapes and thickness can be obtained. [ABSTRACT FROM AUTHOR]

Recently, multi-FPGA platforms have become a popular choice to prototype complex digital systems. This is because of unique advantages such as high frequency and real world testing experience that are offered when compared to other pre-silicon testing techniques. However, one of several challenges faced by multi-FPGA prototyping is the requirement of an efficient back end flow. Partitioning is a key part of the back end flow of multi-FPGA systems and it directly affects the quality of final prototyped design. In this work, we explore two different partitioning approaches: one is multilevel; while the other is hierarchical partitioning approach. For experimentation, we use a suite of fourteen large benchmarks. Experimental results reveal that the multilevel approach gives 12.5% better frequency results for mono-cluster benchmarks while the hierarchical approach gives 13% better results for multi-cluster benchmarks. Furthermore, the hierarchical approach requires, on average, 60% less execution time when compared to the multilevel partitioning approach. [ABSTRACT FROM AUTHOR]

Trigonometric B-spline curves have gained a remarkable attention in computer-aided geometric design (CAGD). This paper presents the cubic and rational cubic trigonometric B-spline curves using new trigonometric functions and shape parameter η ∈ (1 / 2 , 2 ]. The proposed curves inherit the basic properties of classical B-spline and have been proved. For uniform knots, both curves are C 2 continuous. On non-uniform knots, cubic trigonometric curves are C 3 and C 5 continuous, whereas rational trigonometric curves are C 3 continuous and have been derived. The applicability of proposed curves has been checked by constructing open and closed curves. Different models like glass, kettle, human hand, and vase have been designed by both schemes and compared. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel method for constructing developable surfaces using generalized C-Bézier bases with shape parameters. Based on the duality between points and planes in 3D projective space, the generalized developable C-Bézier surfaces, whose shape can be adjusted by changing multiple shape parameters, are designed using control planes with extensional C-Bézier basis functions. With the shape parameters taking different values, a family of developable surfaces can be constructed, which keeps most of characteristics of classic developable Bézier surfaces. Furthermore, some interesting properties of the new developable surfaces, as well as the geometric continuity conditions between two adjacent generalized developable C-Bézier surfaces, are investigated. Finally, we illustrate the convenience and efficiency of the proposed methods by several convictive and representative numerical examples. [ABSTRACT FROM AUTHOR]

Computer-aided diagnosis of lung segmentation is the fundamental requirement to diagnose lung diseases. In this paper, a two-dimensional (2D) Otsu algorithm by Darwinian particle swarm optimization (DPSO) and fractional-order Darwinian particle swarm optimization (FODPSO) is proposed to segment the pulmonary parenchyma from the lung image obtained through computed tomography (CT) scans. The proposed method extracts pulmonary parenchyma from multi-sliced CT. This is a preprocessing step to identify pulmonary diseases such as emphysema, tumor, and lung cancer. Image segmentation plays a significant role in automated pulmonary disease diagnosis. In traditional 2D Otsu, exhaustive search plays an important role in image segmentation. However, the main disadvantage of the 2D Otsu method is its complex computation and processing time. In this paper, the 2D Otsu method optimized by DPSO and FODPSO is developed to reduce complex computations and time. The efficient segmentation is very important in object classification and detection. The particle swarm optimization (PSO) method is widely used to speed up the computation and maintain the same efficiency. In the proposed algorithm, the limitation of PSO of getting trapped in local optimum solutions is overcome. The segmentation technique is assessed and equated with the traditional 2D Otsu method. The test results demonstrate that the proposed strategy gives better results. The algorithm is tested on the Lung Image Database Consortium image collections.

We systematically compare two approaches with the optimal design of multiproduct batch plants that are widely used, e.g., in the chemical industry. Deterministic algorithms like branch-and-bound achieve global optimality, but often require a prohibitively high computation effort. We propose an alternative, hybrid algorithm by combining two metaheuristics: ant colony optimization (ACO) and simulated annealing (SA), in order to find near-optimal solutions in a reasonable time. We develop a parallel implementation of our hybrid approach on graphics processing units using CUDA. We experimentally compare both approaches, and we show that our hybrid metaheuristic approach (ACO + SA) yields almost-optimal solutions, while computing them significantly faster than when using branch-and-bound. [ABSTRACT FROM AUTHOR]

Convergence and normal continuity analysis of a bivariate nonstationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, nonstationary subdivision scheme near an extraordinary vertex/face. [ABSTRACT FROM AUTHOR]

In isogeometric analysis, it is frequently required to handle the geometric models enclosed by four-sided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second, the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers (ADMM) is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method. [ABSTRACT FROM AUTHOR]

In this paper we revisit the problem of monotonicity preservation of curves and surfaces and we provide some new proofs and open problems. In particular, we prove a new formula for the derivation of rational Bézier curves. We also deal with the rational monotonicity preservation of rational Bézier surfaces and a related conjecture is presented.

The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as Gn Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation. [ABSTRACT FROM AUTHOR]

We present a new set of basis functions for H ( curl ) -conforming, H ( div ) -conforming, and L 2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H 1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H 1 -conforming Bernstein–Bézier basis functions form part of the H ( curl ) -conforming basis, and the curl of the high-order, non-gradients H ( curl ) -conforming basis functions form part of the H ( div ) -conforming basis, and the divergence of the high-order, non-curl H ( div ) -conforming basis functions form part of the L 2 -conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piecewise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems. [ABSTRACT FROM AUTHOR]

An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent f1(ξ)=r′(ξ)/|r′(ξ)| and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal-plane vectors. Employing the Euler-Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean-hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal-plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal-plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of G1 and G2 continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design. [ABSTRACT FROM AUTHOR]

Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. In this article we employed B-spline curves to construct the occipital bone fracture using Digital Imaging and Communications in Medicine (DICOM) format data. The construction of occipital bone fracture starts with the boundary extraction followed by corner detection, construction of fractured part inner outer curve for each DICOM data using B-spline curves and finally the construction of fractured part in DICOM format. Method used in this article is based on DICOM data only and does not require any technique such as mirror imaging, technical help, reference skull, or to take average thickness of skull bone. Using the proposed method, the constructed fractured implant is custom made for every individual patient. At the end of this article we present a real case, in which we have constructed the occipital bone fracture using B-spline. The proposed method has been validated using post-operation DICOM data. For practical application, Graphical User Interface (GUI) has been developed. [ABSTRACT FROM AUTHOR]

Non-Uniform Rational B-Spline (NURBS) is the multidisciplinary topic in Mathematics, Computer Science, and Engineering. The NURBS curves together with their geometric properties are widely used in Computer Aided Design, Computer Aided Geometric Design, and Computational Mathematics. When a single weight approaches infinity, the limit of a NURBS curve tends to the corresponding control point. In this paper, a kind of control structure of a NURBS curve, called regular control curve, is defined. We prove that the limit of the NURBS curve is exactly its regular control curve when all of weights approach infinity, where each weight is multiplied by a certain one-parameter function tending to infinity, different for each control point. Moreover, some representative examples are presented to show this property and indicate its application for shape deformation. [ABSTRACT FROM AUTHOR]

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties. [ABSTRACT FROM AUTHOR]

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves. [ABSTRACT FROM AUTHOR]

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 92d-6c+1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage. [ABSTRACT FROM AUTHOR]

To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bézier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bézier rotation surfaces are constructed using a generalized Bézier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bézier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bézier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bézier rotation surfaces provide a valuable way for the design of rotation surfaces. [ABSTRACT FROM AUTHOR]

In this paper, two methods for approximation of offset curves of H-Bézier curves and H-B spline curves are presented, which are based on the approximation of shifting control points and norm of parametric speed. Firstly, after calculating the perturbed vectors, the offset curve can be obtained by shifting the control points of the base curve. Then, both the Chebyshev approximation and optimal trigonometric polynomials approximation of parametric speed of base curve are presented, and two approximation functions of offset curves are also obtained. Finally, some examples are used to demonstrate their practicality. [ABSTRACT FROM AUTHOR]

In order to tackle the problem of shape design and shape adjustment of complex surfaces in engineering, continuity conditions between generalized Bézier-like surfaces with multiple shape parameters are studied in this paper. Firstly, the geometric model of the generalized Bézier-like surfaces is built by blending a number of Bézier-like curves with independent shape parameters. Secondly, based on the terminal properties and linear independence of Bernstein-like basis functions, the conditions for G continuity between two adjacent generalized Bézier-like surfaces are derived, and then simplified by choosing appropriate shape parameters. Finally, some properties and applications of the smooth continuity between generalized Bézier-like surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which can greatly improve the ability to construct complex surfaces by using the generalized Bézier-like surfaces. [ABSTRACT FROM AUTHOR]

NURBS surface is very useful in geometric modeling, animation, image morphing and deformation. Constructing non-self-intersecting (injective) NURBS surfaces is an important process in surface and solid modeling. In this paper, the injective conditions of tensor product NURBS surface are studied, based on the geometric positions of control points, which are equivalent to the surface to be non-self-intersecting for all positive weights. Finally, some representative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Refine-Smooth algorithm to design a family of a -point b -ary approximating subdivision schemes with bell-shaped mask, where a ≥ 3 and b ≥ 2. We use the combination of corner cutting b -ary subdivision scheme and weighted average of ( b + 1 ) -points to construct the proposed family. We demonstrate that the proposed family has smaller complexity and support width and higher continuity than the existing Refine-Smooth subdivision schemes. We also study the shape preserving properties of the proposed family. In addition, it is observed that the proposed family is suitable for fitting the locally noisy, oscillatory, and irregular data. [ABSTRACT FROM AUTHOR]

Composite basis functions for pyramidal elements on the spaces H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) and L 2 ( Ω ) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart–Thomas–Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, etc.) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions. [ABSTRACT FROM AUTHOR]

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc´ıa-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster. [ABSTRACT FROM AUTHOR]

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of $\mathrm{G}^{1}$ and $\mathrm{G}^{2}$ between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design. [ABSTRACT FROM AUTHOR]

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (e.g. Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user. Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it. This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan. Numerical results show that this hybrid method is efficient and robust. [ABSTRACT FROM AUTHOR]