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Algorithmic differentiable ray tracing is a new paradigm that allows one to solve the forward problem of how light propagates through an optical system while obtaining gradients of the simulation results with respect to parameters specifying the optical system. Specifically, the use of algorithmically differentiable non-sequential ray tracing provides an opportunity in the field of illumination engineering to design complex optical system. We demonstrate its potential by designing freeform lenses that project a prescribed irradiance distribution onto a plane. The challenge consists in finding a suitable surface geometry of the lens so that the light emitted by a light source is redistributed into a desired irradiance distribution. We discuss the crucial steps allowing the non-sequential ray tracer to be differentiable. The obtained gradients are used to optimize the geometry of the freeform, and we investigate the effectiveness of adding a multi-layer perceptron neural network to the optimization that outputs parameters defining the freeform lens. Lenses are designed for various sources such as collimated beams or point sources, and finally, a grid of point sources approximating an extended source. The obtained lens designs are finally validated using the commercial non-sequential ray tracer LightTools. [ABSTRACT FROM AUTHOR]

Due to the influence of the complex underwater environment, the sound speed constantly changes, resulting in the acoustic signal propagation trajectory being curved, which greatly affects the positioning accuracy of the underwater long baseline (LBL) system. In this paper, an improved LBL positioning method based on a B-spline surface for fitting the effective sound speed table (ESST) is proposed. Firstly, according to the underwater sound speed profile, the discrete ESST of each measurement station is constructed before the positioning test, and then, the node position of the B-spline surface is optimized by particle swarm optimization (PSO) to accurately fit the discrete ESST. Based on this, the improved LBL positioning method is constructed. In the underwater positioning test, the effective sound speed can be quickly found by measuring the time of arrival (TOA) of the acoustic signal and the target depth, and moreover, the target position parameters can be quickly and accurately estimated. The numerical simulation results show that the improved positioning method proposed in this paper can effectively improve the LBL positioning accuracy and provide the theoretical basis and the technical support for the underwater navigation and positioning. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we propose combining the stochastic blockmodel and the smooth graphon model, two of the most prominent modeling approaches in statistical network analysis. Stochastic blockmodels are generally used for clustering networks into groups of actors with homogeneous connectivity behavior. Smooth graphon models, on the other hand, assume that all nodes can be arranged on a one-dimensional scale such that closeness implies a similar behavior in connectivity. Both frameworks belong to the class of node-specific latent variable models, entailing a natural relationship. While these two modeling concepts have developed independently, this paper proposes their generalization towards stochastic block smooth graphon models. This combined approach enables to exploit the advantages of both worlds. Employing concepts of the EM-type algorithm allows to develop an appropriate estimation routine, where MCMC techniques are used to accomplish the E-step. Simulations and real-world applications support the practicability of our novel method and demonstrate its advantages. The paper is accompanied by supplementary material covering details about computation and implementation. [ABSTRACT FROM AUTHOR]

Due to the influence of the complex underwater environment, the sound speed constantly changes, resulting in the acoustic signal propagation trajectory being curved, which greatly affects the positioning accuracy of the underwater long baseline (LBL) system. In this paper, an improved LBL positioning method based on a B-spline surface for fitting the effective sound speed table (ESST) is proposed. Firstly, according to the underwater sound speed profile, the discrete ESST of each measurement station is constructed before the positioning test, and then, the node position of the B-spline surface is optimized by particle swarm optimization (PSO) to accurately fit the discrete ESST. Based on this, the improved LBL positioning method is constructed. In the underwater positioning test, the effective sound speed can be quickly found by measuring the time of arrival (TOA) of the acoustic signal and the target depth, and moreover, the target position parameters can be quickly and accurately estimated. The numerical simulation results show that the improved positioning method proposed in this paper can effectively improve the LBL positioning accuracy and provide the theoretical basis and the technical support for the underwater navigation and positioning.

The surface/surface intersection technique serves as one of the most fundamental functions in modern Computer Aided Design (CAD) systems. Despite the long research history and successful applications of surface intersection algorithms in various CAD industrial software, challenges still exist in balancing computational efficiency, accuracy, as well as topology correctness. Specifically, most practical intersection algorithms fail to guarantee the correct topology of the intersection curve(s) when two surfaces are in near-critical positions, which brings instability to CAD systems. Even in one of the most successfully used commercial geometry engines ACIS, such complicated intersection topology can still be a tough nut to crack. In this paper, we present a practical topology guaranteed algorithm for computing the intersection loci of two B-spline surfaces. Our algorithm well treats all types of common and complicated intersection topology with practical efficiency, including those intersections with multiple branches or cross singularities, contacts in several isolated singular points or highorder contacts along a curve, as well as intersections along boundary curves. We present representative examples of these hard topology situations that challenge not only the open-source geometry engine OCCT but also the commercial engine ACIS. We compare our algorithm in both efficiency and topology correctness on plenty of common and complicated models with the open-source intersection package in SISL, OCCT, and the commercial engine ACIS. [ABSTRACT FROM AUTHOR]

A fuzzy linguistic data set that is uncertain is difficult to analyze and describe in the form of a smooth and continuous generic figure. Therefore, the study aims to develop a new model of a B-spline surface using a different approach of a crisp and fuzzy linguistic point relation with three types of linguistic function: low L , medium M i and high H. These linguistic functions are defined first to introduce the fuzzy linguistic point relation. Then, a new algorithm of the fuzzy linguistic bicubic B-spline surface model is presented to convert fuzzy linguistic data into fuzzy linguistic control points. In addition, a numerical example of fuzzy linguistic data is considered at the end of this study to visualize the suggested model. Thus, the relation between the fuzzy linguistic data points can be analyzed to present another area of knowledge in which symmetry phenomena occur. The symmetry here plays an important role in solving the uncertain fuzzy linguistic data problem by using the suggested model. [ABSTRACT FROM AUTHOR]

The problem of surface reconstruction is a challenging problem in the fields of data visualization, virtual reality and engineering design. In this study, we investigate the topic of fitting B-spline surface to a set of 3D measured data points. Surface reconstruction in the proposed method consists of two main parts: (1) rewrite the problem as a nonlinear least squares optimization problem and compute the Jacobin matrix and (2) employ the diagonal approximation BFGS method to find the control points and the location parameters simultaneously. The space complexity and the time complexity of proposed method are O (n) . We perform numerical experiments with five test problems, including complex shapes, self intersections, large number of data points and high genus to evaluate the performance of the suggested approach. The experimental results demonstrate that the introduced approach is easy to implement, fast convergence rate, extremely small fitting errors, flexibility, very general and applicable to real time simulations. [ABSTRACT FROM AUTHOR]

In this study, we present a novel method for modeling the canopy surface of an umbrella. Our approach involves representing the area between the ribs on the canopy as a trimmed bilinear patch. Furthermore, we conduct an in-depth exploration of various differential geometric properties of the umbrella surface. We introduce a method for unfolding the canopy surface onto a plane, which serves as a valuable technique for fabricating a cardboard template to accurately cut canopy fabrics. To validate the effectiveness of our geometric modeling method, we apply it to several umbrella models, showcasing its practical application and benefits. • A canopy surface between ribs is represented as a trimmed bilinear patch. • A method to unfold the canopy surface onto a plane is proposed. • Parameters that determine the shape of the canopy surface are defined. • An umbrella was fabricated based on our methods. [ABSTRACT FROM AUTHOR]

The accuracy of fringe projection profilometry is compromised when the system incorporates complex distortions that are challenging to accurately model. To address this issue, this paper presents a novel calibration method for fringe projection profilometry. Firstly, a pixel-wise phase-coordinate mapping model is established. Subsequently, the continuous mapping from pixels to XY coordinates is obtained by fitting a B-spline surface, and the pose estimation of calibration plate is achieved by constructing a virtual linear camera. Notably, this method avoids introducing any parameters representing the physical characteristics of lens distortion throughout the entire calibration process and eliminates the need for precision mobile equipment. Experimental results demonstrate that the proposed method exhibits significantly enhanced calibration accuracy compared to the traditional method in fringe projection profilometry system with complex lens distortions. • Effectively address the complex distortion in fringe projection profilometry. • Operate flexibly without requiring precision mobile equipment. • Use B-spline surface to represent the mapping from pixels to XY coordinates. • Construct a virtual linear camera to estimate the poses of the calibration plate. [ABSTRACT FROM AUTHOR]

A 2-D spatial construction method is proposed for the online modeling of distributed parameter systems (DPSs), such as battery thermal process. The proposed method can combine the advantages of spectral method and Karhunen–Loève decomposition (KLD) method. First, the continuous spatial basis functions are designed by the 2-D spatial construction to keep the information between sensing locations. With the 2-D space-time separation and recursive learning, the derived model can preserve the couplings between spatial dimensions and update over time. The radial basis function network is utilized to identify the low-dimensional temporal dynamics. After the space-time synthesis, the constructed spatiotemporal model can provide continuous modeling of the DPS with satisfactory performance. Convergence analysis has been carried out, which proves that the proposed method can guarantee bounded errors. Finally, simulations and experiments on a pouch-type lithium-ion battery with unknown partial differential equations prove the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as developable as possible. Specifically, the terminal ruling lines are treated as hard constraints. Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control, our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely-designed rulings. We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications. The performance and the efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for the path planning in 5-axis computer numerical control (CNC) flank milling. [ABSTRACT FROM AUTHOR]

• Latent quantities and graphon representation have a mutual relationship. • EM-type algorithm enables to estimate latent quantities and graphon simultaneously. • Gibbs sampling provides information about complex joint conditional distribution of latent quantities. • Penalized B-spline regression allows for a smooth graphon fit with relevant side constraints. • Simulation studies and real-world data examples illustrate the superior applicability. This paper proposes the estimation of a smooth graphon model for network data analysis using principles of the EM algorithm. The approach considers both variability with respect to ordering the nodes of a network and smooth estimation of the graphon by nonparametric regression. To do so, (linear) B-splines are used, which allow for smooth estimation of the graphon, conditional on the node ordering. This provides the M-step. The true ordering of the nodes arising from the graphon model remains unobserved and MCMC techniques are employed to obtain position samples conditional on the network. This yields the E-step. Combining both steps gives an EM-based approach for smooth graphon estimation. Unlike common other methods, this procedure does not require the restriction of a monotonic marginal function. The proposed graphon estimate allows to explore node-ordering strategies and therefore to compare the common degree-based node ranking with the ordering conditional on the network. Variability and uncertainty are taken into account relying on the MCMC sequences. Examples and simulation studies support the applicability of the approach. [ABSTRACT FROM AUTHOR]

A fuzzy linguistic data set that is uncertain is difficult to analyze and describe in the form of a smooth and continuous generic figure. Therefore, the study aims to develop a new model of a B-spline surface using a different approach of a crisp and fuzzy linguistic point relation with three types of linguistic function: low L, medium Mi and high H. These linguistic functions are defined first to introduce the fuzzy linguistic point relation. Then, a new algorithm of the fuzzy linguistic bicubic B-spline surface model is presented to convert fuzzy linguistic data into fuzzy linguistic control points. In addition, a numerical example of fuzzy linguistic data is considered at the end of this study to visualize the suggested model. Thus, the relation between the fuzzy linguistic data points can be analyzed to present another area of knowledge in which symmetry phenomena occur. The symmetry here plays an important role in solving the uncertain fuzzy linguistic data problem by using the suggested model.

The goal of visual surface inspection is to analyze an object’s surface and detect defects by looking at it from different angles. Developments over the past years have made it possible to partially automate this process. Inspection systems use robots to move cameras and obtain pictures that are evaluated by image processing algorithms. Setting up these systems or adapting them to new models is primarily done manually. A key challenge is to define camera viewpoints from which the images are taken. The number of viewpoints should be as low as possible while still guaranteeing an inspection of the desired quality. System engineers define and evaluate configurations that are improved based on a time-consuming trial-and-error process leading to a sufficient, but not necessarily optimal, configuration. With the availability of 3D surface models defined by triangular meshes, this step can be done virtually. This paper presents a new scalable approach to determine a small number of well-placed camera viewpoints for optical surface inspection planning. The initial model is approximated by B-spline surfaces. A set of geometric feature functionals is defined and used for an adaptive, non-uniform surface sampling that is sparse in geometrically low-complexity areas and dense in regions of higher complexity. The presented approach is applicable to solid objects with a given 3D surface model. It makes camera viewpoint generation independent of the resolution of the triangle mesh, and it improves previous results considering number of viewpoints and their relevance. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor product B-spline surfaces and bivariate B-spline surfaces on criss-cross triangulations. Finally, we present graphical results.

• Employ peridynamics to solve inverse scattering problems without remeshing a domain. • Propose a novel bond parameterization scheme using B-spline surface. • Use efficient and accurate adjoint sensitivity for transient peridynamics. • Identify precise scatterer geometry using a gradient-based optimization method. • Applicable to the scatterers of complicated geometry using multi-step optimization. A shape identification, which is an inverse problem finding the configuration of scatterers using discrepancy or an error functional between real wave data with scatterers and estimated wave data, is investigated employing peridynamic theory and gradient-based optimization in plane elastic medium. The particle-based and non-local characteristics of the peridynamic theory enable the direct modeling of interface between scatterers and medium, avoiding remeshing difficulty when employing domain-based methods. The boundary of scatterers is parameterized using B-spline surfaces and determined by bond parameters in peridynamics material. The required design sensitivity of the error functional is obtained by an efficient adjoint variable method. The peridynamic adjoint sensitivity involving history-dependent variables in transient peridynamics is accurately obtained by using an identical path in both adjoint and response analyses. Numerical examples of various geometry are demonstrated to verify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract This work deals with the optimum design of variable angle-tow (VAT) composites in the theoretical framework of B-Spline surfaces. The design problem is formulated as a constrained optimisation problem in the context of the multi-scale two-level (MS2L) optimisation methodology. The proposed approach aims at optimising all parameters characterising the VAT laminate at each pertinent scale (mesoscopic and macroscopic ones). In particular, this study focuses on the first level of the MS2L optimisation strategy that aims at determining, at the VAT laminate macroscopic scale, the optimum distribution of the laminate mechanical properties over the structure. At this stage, the behaviour of the VAT composite is described by means of the polar formalism and the spatial distribution of the polar parameters is represented through a set of suitable B-Spline surfaces. The expression of the gradient of objective and constraint functions is determined in a closed form by exploiting the properties of the B-Spline surfaces. Moreover, the effect of the discrete variables tuning the shape of the B-Spline surfaces (mainly the number of control points) on the quality of the optimum solution is investigated. The effectiveness of the proposed approach is tested through some meaningful benchmarks taken from literature. [ABSTRACT FROM AUTHOR]

Surface skinning is a surface generation method that uses a set of given cross-sectional curves, and it is widely used in free-form surface design. In the B-spline surface skinning, the given B-spline curves should be compatible, that is, the curves should have the same degree and knot sequence. While making the curves compatible, lots of control vertices are generated. Although T-spline surface skinning methods have been introduced to reduce the number of control vertices, the T-spline skinning method that was proposed by Nasri et al. (2012) can generate a wiggled surface when the given B-spline curves are not sufficiently compatible. The intermediate cross sections that were introduced for T-spline surface skinning cannot preserve the shape of the given B-spline curves if the adjacent B-spline curves do not have sufficient common knots, and it can cause wiggles on the surface. In this paper, we analyze this issue and suggest a modified method to remove the wiggles on the skinned T-spline surface. Furthermore, we propose an algorithm for shape preservation of the surface. Our approach is verified by suggesting some examples compared to the Nasri et al. (2012)’s method. [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.

The paper presents a new interactive technique for correcting CAD/CAM data containing errors. An algorithm is described that can be used to correct surface data containing undesirable discontinuities (''gaps''/ ''holes'', and ''overlaps'') and intersections among surface patches. Such surface problems commonly arise in the aircraft, automobile, and ship industry and make later processing of the data difficult, if not impossible. The new method provides a tool to correct wrong data requiring minimal user interaction. The imput for the scheme can be a set of parametrically defined surfaces (e.i., Bezier, B-spline, or NURBS surfaces) or a set of triangles (or quadrilaterals) discretizing the original geometry. The output is a set of G0 or G1 (tangent plane)continuous, bicubic B-splines surfaces approximating the given data. Each of these B-spline surfaces is constructed using just four user-specified boundary curves.

Recent advance of Additive Manufacturing technologies allows us to manufacture various parts used in real-world products. Consequently, product tracking of such 3D printed parts is an important issue. Quick Response (QR) code which is a two-dimensional matrix barcode invented by Denso, a Japanese automotive industry, in 1994, can be used for this purpose. It can store more data than the 1D barcode in a smaller space, and using a smartphone as a scanner, one can directly visit a website where all the information of the parts is stored. However, QR codes require secondary procedures to add them to products and are also vulnerable to wear and tear. Moreover, QR codes cannot be added to freeform surfaces, but only to developable surfaces. In this paper we propose a novel technique to embed QR codes onto CAD models consisting of freeform surfaces represented by B-spline surfaces, which produces 3D QR codes. 3D QR codes work similar to 2D QR codes and can be read by existing QR scanners, but are designed by grooving the surface to obtain light and dark regions caused by ambient occlusion. Unlike conventional QR codes, 3D QR codes do not fall off from the part and can even be painted if necessary. Furthermore, we do not need to prepare dark-colored and light-colored materials for 3D printing as the dark color is provided by the grooving. We demonstrate the effectiveness of our technique with various examples. [ABSTRACT FROM AUTHOR]

• A deflectometric method for discontinuous specular surface is proposed by iteration of B-spline surface. • A direct deflectometric method is conducted to reconstruct specular surface initially and distinguish isolated surface. • With the initial reconstruction, a procedure of iteration is followed to solve control points of B-spline surface. • The validation, accuracy and efficiency of the proposed method is verified by experimental results. There is an over-growing demand for accurate three-dimension model of industrial components. In the measurement of specular surface, deflectometry has been widely used for its convenience and high accuracy. However, it is a challenge to measure discontinuous specular surface by the traditional deflectometry. In order to solve this problem, this paper proposes a deflectometric method based on iteration of control points of B-spline surface. In this method, a screen at two positions displays fringe patterns which will be reflected by specular surface under test and captured by a camera. According to the undistorted and absolute unwrapped phase map, a continuous zone containing all discontinuous area which is distinguished by a direct deflectometric method, is fitted by a B-spline surface with an estimated depth on the focus plane of the camera. Meanwhile, empty points and basis functions bridging isolated surface are removed. The control points of the fitted surface are solved by iterative calculation. With the solved control points, discontinuous specular surface is reconstructed. Simulated and actual experiments are carried out and the results show high accuracy and stability of the proposed method. [ABSTRACT FROM AUTHOR]

Asymptote is widely used in astronomy, mechanics, architecture and relevant subjects. In this paper, by analyzing the Frenet frame and the Darboux frame of a curve on the surface, the necessary and sufficient conditions for a quadrilateral boundary being asymptotic of a surface are derived. This quadrilateral is called asymptotic quadrilateral. Given corner data including positions, tangents and curvatures of a cubic B-spline quadrilateral with six control points in each boundary, a family of asymptotic quadrilaterals are constructed after solving the identification conditions of the control points. An optimized one is obtained by minimizing the strain energy of the boundary curves. Then, the transverse tangent vectors along the boundaries of the B-spline surface can be obtained by the asymptotic conditions and the resulting B-spline surface is of bi-quintic degree. Two arrays of control points of the surface along the quadrilateral are obtained from combinations transverse tangent vectors and the boundaries which are elevated from the cubic B-spline curves. For the given inner control points, B-spline surface of bi-quintic degree interpolating the cubic B-spline asymptotic quadrilateral is constructed. The optimized surface is the one with the minimized thin plate spline energy. The method is verified by some representative examples including the boundary curves with lines and inflections. Such interpolation scheme for the construction of the tensor-product B-spline surfaces is compatible with the CAD systems.

Freeform surfaces like B-splines have proven to be a suitable tool to model laser scanner point clouds and to form the basis for an areal data analysis, for example an areal deformation analysis. A variety of parameters determine the B-spline's appearance, the B-spline's complexity being mostly determined by the number of control points. Usually, this parameter type is chosen by intuitive trial-and-error-procedures. In [] the problem of finding an alternative to these trial-and-error-procedures was addressed for the case of B-spline curves: The task of choosing the optimal number of control points was interpreted as a model selection problem. Two model selection criteria, the Akaike and the Bayesian Information Criterion, were used to identify the B-spline curve with the optimal number of control points from a set of candidate B-spline models. In order to overcome the drawbacks of the information criteria, an alternative approach based on statistical learning theory was developed. The criteria were evaluated by means of simulated data sets. The present paper continues these investigations. If necessary, the methods proposed in [] are extended to areal approaches so that they can be used to determine the optimal number of B-spline surface control points. Furthermore, the methods are evaluated by means of real laser scanner data sets rather than by simulated ones. The application of those methods to B-spline surfaces reveals the datum problem of those surfaces, meaning that location and number of control points of two B-splines surfaces are only comparable if they are based on the same parameterization. First investigations to solve this problem are presented. [ABSTRACT FROM AUTHOR]

How to express the designer's creative intent in a simple and intuitive way is the main problem in 3d modeling, especially for novice designers. This paper presents a free shape 3d modeling system for creative design based on modified Catmull-Clark subdivision. The system contains a series of easy but novel operations which can be used to change the topology of models, such as creating holes and handles. In order to create sharp features, feature marking operations are provided to specify where the sharp feature is. This system also provides surface conversion function to make the modeling results be compatible with the traditional CAD systems. Firstly, a simple but efficient quad domain division scheme is adopted to generate quad sub-meshes. In order to improve the smoothness at the regular vertices, long boundary curves which across multiple sub-meshes are used to be the boundary constraints while fitting. In this way, the smoothness at regular vertex can be C continuous. We perform experiments for both skilled and novice designers. Results show that our system is easy to operate and can be used to construct complex models with less time. [ABSTRACT FROM AUTHOR]

B-spline surfaces, extracted from scanned sensor data, are usually required to represent objects in inspection, surveying technology, metrology and reverse engineering tasks. In order to express a large object with a satisfactory accuracy, multiple scans, which generally lead to overlapping patches, are always needed due to, inter-alia, practical limitations and accuracy of measurements, uncertainties in measurement devices, calibration problems as well as skills of the experimenter. In this paper, we propose an action sequence consisting of division and merging. While the former divides a B-spline surface into many patches with corresponding scanned data, the latter merges the scanned data and its overlapping B-spline surface patch. Firstly, all possible overlapping cases of two B-spline surfaces are enumerated and analyzed from a view of the locations of the projection points of four corners of one surface in the interior of its overlapping surface. Next, the general division and merging methods are developed to deal with all overlapping cases, and a simulated example is used to illustrate aforementioned detailed procedures. In the sequel, two scans obtained from a three-dimensional laser scanner are simulated to express a large house with B-spline surfaces. The simulation results show the efficiency and efficacy of the proposed method. In this whole process, storage space of data points is not increased with new obtained overlapping scans, and none of the overlapping points are discarded which increases the representation accuracy. We believe the proposed method has a number of potential applications in the representation and expression of large objects with three-dimensional laser scanner data. [ABSTRACT FROM AUTHOR]

The objective of this paper is to develop a mathematical methodology that allows the available sight distance calculation of a roadway directly in three-dimensional space and in a fully automatic and accurate manner. The presented methodology overcomes the conventional approach of studying the three-dimensional roadway configuration separately and sequentially in its horizontal and vertical alignment. As far as the road surface is concerned, it is simulated as a 3-d B-spline surface with continuous side barriers whose road centerline has been in turn, simulated as a 3-d B-spline curve. Through this approximation, the road centerline as well as the right and left borderlines which play a crucial role in the sight distance calculation are mathematically defined both geometrically and analytically. The sight distance calculation can be made in each point of the road surface, due to the integrated information existing in the model through its three-dimensional character. These calculations are clearly more reliable, since they are applied on a realistic 3-d road surface. It is worth mentioning that this methodology can be applied on existing roads as well as during the design process by modifying the pseudogeodesic and/or pseudonormal curvature of the roadway in order to obtain the desired available sight distance. Finally, the computational part of this methodology is attained by expressing all the functions and algorithms in code through the programming language offered by the software Mathematica, whereas the whole process is demonstrated as a case study accompanied by Figures and Tables. [ABSTRACT FROM AUTHOR]

This paper is concerned with construction of a B-spline surface that interpolates a B-spline curvilinear quadrilateral as boundary geodesics. The construction consists of two parts. First, from the corner data (i.e., position, unit tangent vector and curvature at the end points of each curve), a four-quartic B-spline curvilinear quadrilateral with minimum strain energy is constructed to satisfy the constraints of the crossing geodesics on a surface. Second, a tensor-product B-spline surface of degree (5, 5) is constructed to interpolate the quadrilateral as boundary geodesics. The control points of the interpolation surface are determined by two steps, and the surface which adheres to the NURBS standard and employs geometric shape handles meeting the requirements of the design scenario. The method is illustrated with several computational examples. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C 1 quadratic splines on criss–cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the construction of a B-spline surface satisfying prescribed angle distribution (with respect to a chosen vector) of tangent planes along its boundary curve. This problem arises e.g. in a creation of a parametric geometric model of a Pelton turbine bucket, where specific angle distributions along a splitter and an outlet curve have to be fulfilled in order to control the flow of water into and out of the bucket. We prove that for a given B-spline curve c ( t ) , t ∈ [ 0 , 1 ] , the exact solution exists only in very special cases (for a special form of an angle function f ( t ) ). Further, we formulate an algorithm for finding an approximate solution. We also derive a bound on its approximation error and give a numerical evidence that the approximation order of the proposed algorithm is four. Finally, the method is demonstrated on several examples. [ABSTRACT FROM AUTHOR]

The need for complex surfaces in CAD motivates researchers for methods which can produce smooth and visually pleasing surfaces. In this research, a new method is presented for creating compatible cross-sectional curves for surface fitting to certain sections or lofting. In this method, the distribution of sections' data points along with basis knot vectors are improved in order to reach a desired smooth surface. In compatibility process, the section curves' degrees and their knot vectors must be set equal before implementing lofting process. Based on proposed algorithm, in this research, the constructed smooth and faired surfaces can be used in many engineering applications such as reverse engineering, biomedical engineering, quality control, etc. The main focus of the method is improvement of data points' distributions and their assigned parameters in a way that by a few iterations, data points' distribution are improved in order to reach a common knot vector for all cross-sectional curves. The method is implemented on some benchmarking examples and its efficiency are confirmed. In addition, the amount of final data points' deviation from the initial section curve is analyzed using the vigorous Hausdorff method. It is worth mentioning that the quality of obtained final surface is visually pleasing. In order to quantitatively confirm that the proposed method will result in smooth and fair surfaces, MVS is used. Finally the application of the method in modeling the rootjoint zone of a wind turbine blade is presented. [ABSTRACT FROM AUTHOR]

The improvement of the simulation process requires an integration of the design and analysis models. There are two essential tasks in the design analysis process: (i) computer-aided design (CAD) which provides the geometric description of the model and (ii) the finite element method (FEM) used for mechanical behaviour simulations. The interoperability between these two tasks reduces costs and improves product quality through the acceleration of design analysis loops. Our activity fits into this research orientation by providing a method to link the FE analysis and the CAD model. This is done by reconstructing the CAD model from the FE analysis results (deformed mesh). This paper proposes a method to update the CAD geometry from the deformed mesh. This approach allows for rebuilding the CAD model after analysis by extracting geometric information from the deformed mesh. An illustration of the developed method is discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

We introduce a new method to interpolate scattered 3D data points with a single B-spline surface patch which is globally smooth. Given a set of scattered 3D data points and their corresponding parametrization, our method first constructs a set α of B-spline bases using a weighted strategy, and inserts knots to the knot vectors based on α. Then, the knot insertion procedure is iterated until a set β of B-spline bases exists, which indicates the existence of the interpolation surface. Finally, by applying the fairing energy minimizing with interpolation constraints, a globally smooth B-spline surface which interpolates the data points can be produced. Experimental results demonstrate that the generated B-spline surfaces often have fewer control points than those of traditional methods, while keeping the scattered data points interpolated accurately. The implementation of our algorithm and the scripts to reproduce all the results are available at https://github.com/wangbolun300/sparse_data. • An iterative knot insertion algorithm is introduced to guarantee the existence of B-spline interpolation surfaces when fitting scattered 3D data points. • The generated B-spline surfaces often use fewer control points than those of tranditional interpolation methods. • A novel parameter δ is introduced to improve the stability of control point solving. • The control points can be solved by a global fairing energy minimization procedure with strict interpolation constraints. Thus, the generated B-spline surface is globally smooth. [ABSTRACT FROM AUTHOR]

The geometric structure of woven fabric affects the appearance and physical properties and can be used to predict the weavability of the fabric. An accurate description of the structure is beneficial to predict the characteristics and the appearance of the woven fabric. The purpose of the study is to build an accurate, realistic, and stable three-dimensional (3D) geometric structure model based on the mechanics for plain woven fabrics. Parameters such as the initial Young's modulus of the yarn materials, thicknesses, warp density, pre-loaded tension on the yarns, and the amount of letting-off and taking-up are considered in the model. The yarns are simplified as a series of spring-sliders that are stretched in the fabric and moved along the direction of the resultant forces step by step according to the changing forces. The yarns stop moving when all the forces reach equilibrium. The tests demonstrate that the algorithm conforms to the weaving principles. The 3D images of the geometric structure of the woven fabric at different steps are displayed by B-spline surface modeling technology. A Keyence VH600 micro-measurement system is used to measure the 3D coordinates of the real fabric accurately without destroying the fabric structure. The similarity of the real geometric structure and the calculated geometric model is evaluated by calculating the discrete Fréchet distances. The result validates that the similarity of the calculated geometric structure and the measured value is more than 90% for both the warp and weft direction. [ABSTRACT FROM PUBLISHER]

A lifting surface optimization method is coupled with a blade alignment procedure for the design of propeller in marine applications. The method is characterized by the use of a vortex lattice discretization of the true blade mean camber surface for the determination of the optimum circulation distribution, which is found by numerically solving a variational problem. The blade mean camber surface is deformed to satisfy the kinematic boundary condition that the normal component of velocity becomes zero on that surface, in an iterative way with the optimization problem. The smooth blade geometry and the robust solution of the alignment are guaranteed by the use of a B-spline surface representation. A validation calculation is carried out for DTNSRDC propeller 4119 operating in uniform flow. A comparative study with literature for some DTNSRDC parametric propeller series is given to demonstrate the proposed algorithm for the design of propellers that have complex geometry. [ABSTRACT FROM AUTHOR]

A group of function containing three functions with parameter m is constructed. This group of function is linearly independent, which is named mB basis. The mB basis has non-negativity, normalization, symmetry and special endpoint property. Based on the mB basis, a new kind of spline curve is defined, which is called mB curve. The mB curve segment can be converted into the form of Bézier curve. By virtue of the de Casteljau algorithm of Bézier curve, the recursive evaluation algorithm of the mB curve is given. The mB curve has the same endpoint behavior with the quadratic uniform B-spline curve, that is to say, the mB curve interpolates the midpoints of the first and the last sides of the control polygon, and tangent to the first and the last sides of the control polygon. Besides, the shape and continuity of the mB curve can be freely adjusted by the parameter m. And the adjustment can be either integral, or local. Using the tensor product method, the mB curve is extended to mB surface. The mB surface has similar properties with the mB curve. [ABSTRACT FROM AUTHOR]