Spline Surfaces Research Articles

3D Reconstruction Using Cubic Bezier Spline Curves and Active Contours (Case Study)

Introduction 3D reconstruction of an object from its 2D cross-sections (slices) has many applications in different fields of sciences such as medical physics and biomedical engineering. In order to perform 3D reconstruction, at first, desired boundaries at each slice are detected and then using a correspondence between points of successive slices surface of desired object is reconstructed. Materials and Methods In this study, Gradient Vector Flow (GVF) was used in order to trace the boundaries at each slice. Then, cubic Bezier Spline curves were used to approximate each of obtained contours and to approximate the corresponding points of different contours at successive slices. The reconstructed surface was a bi-cubic Bezier Spline surface which was smooth with G2 continuity. Results Our presented method was tested on SPECT data of JASZCZAK phantom and human's left ventricle. The results confirmed that the presented method was accurate, promising, applicable, and effective. Conclusion Using GVF algorithm to trace boundaries at each slice, and cubic Bezier Spline curves to approximate the obtained rough contours yield to the procedure of reconstruction which was fast and also the final surface was smooth with G2 continuity. So far, some mathematical curves such as spline, cubic spline, and B-spline curves were used to approximate the computed contour during a time consuming procedure. This study presented a 3D reconstruction method based on a combination of GVF algorithm and cubic Bezier Spline curves. There was a good trade-off between speed and accuracy in using cubic Bezier Spline curves which is especially useful for training students.

Spherical DCB-Spline Surfaces with Hierarchical and Adaptive Knot Insertion

This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.

Reliable and Practical City Map generation at 1:1000 or 1:500 scale based on “Five control points" recification of digitized images

Abstract. Using high resolution image for making city map by digital rectification is simple.But five control points can correct image deformation precisely and reliable are not easy particular making 1:500 scale street view map. Nowadays, RTK-GPS becomes the definitive mapping instrument to draw 1:1000 scale maps but get some shortages. That are, 1. Operators can’t have an overall impression of the mapping place. 2 Measuring points are separate so that the lines are not natural and smooth. 3. Data transmission may have 0.1 % mistakes which waste time to check and are troublesome. Given method comes as a substitute, which keeps all advantages and use RTK to definite the key points for drawing. Here are three examples for reference. The map size are extremely large and beyond the custom. (area; 60cm×70cm,80cm×180cm, 60cm×200cm) Which without any splices. The interpolation for rectifying is surface splines which were originally developed for interpolation wing deflections of aircraft; Surface splines are a mathematical tool to interpolate a fuction of two variables. It is Base upon the small deflection equation of an infinite plate. The main adtvantages of the surface splines are that the coordinates of the known points needs not be located in a rectangular array and the function may be differentiated to find slopes. (Contributed by Harderand Desmarais 1972)

Numerical method of static aeroelastic correction and jig-shape design for large airliners

In this paper, a coupled CFD-CSD method based on N-S equations is described for static aeroelastic correction and jig-shape design of large airliners. The wing structural flexibility matrix is analyzed by a finite element method with a double-beam model. The viscous multi-block structured grid is used in aerodynamic calculations. Flexibility matrix interpolation is fulfilled by use of a surface spline method. The load distributions on wing surface are evaluated by solving N-S equations with a parallel algorithm. A flexibility approach is employed to calculate the structural deformations. By successive iterations between steady aerodynamic forces and structural deformations, a coupled CFD-CSD method is achieved for the static aeroelastic correction and jig-shape design of a large airliner. The present method is applied to the static aeroelastic analysis and jig-shape design for a typical large airliner with engine nacelle and winglet. The numerical results indicate that calculations of static aeroelastic correction should employ tightly coupled CFD-CSD iterations, and that on a given cruise shape only one round of iterative design is needed to obtain the jig-shape meeting design requirements.

Aeroelastic trim and flight loads analysis of flexible aircraft with large deformations

A method for static aeroelastic trim analysis and flight loads computation of a flexible aircraft with large deformations has been presented in this paper, which considers the geometric nonlinearity of the structure and the nonplanar effects of aerodynamics. A nonplanar vortex lattice method is used to compute the nonplanar aerodynamics. The nonlinear finite element method is introduced to consider the structural geometric nonlinearity. Moreover, the surface spline method is used for structure/aerodynamics coupling. Finally, by combining the equilibrium equations of rigid motions of the deformed aircraft, the nonlinear trim problem of the flexible aircraft is solved by iterative method. For instance, the longitudinal trim analysis of a flexible aircraft with large-aspect-ratio wings is carried out by both the nonlinear method presented and the linear method of MSC Flightloads. Results obtained by these two methods are compared, and it is indicated that the results agree with each other when the deformation is small. However, because the linear method of static aeroelastic analysis does not consider the nonplanar aerodynamic effects or structural geometric nonlinearity, it is not applicable as the deformations increase. Whereas the nonlinear method presented could solve the trim problem accurately, even the deformations are large, which makes the nonlinear method suitable for rapid and efficient analysis in engineering practice. It could be used not only in the preliminary stage but also in the detail stage of aircraft design.

Free-form splines combining NURBS and basic shapes

We show how to automatically join, into one unified spline surface, C2 tensor-product bi-cubic NURBS and G2 bi-cubic rational splines. The G2 splines are capable of exactly representing basic shapes such as (pieces of) quadrics and surfaces of revolution, including tori and cyclides. The main challenge is to transition between the differing forms of continuity. We transform the G2 splines to splines that are C2 in homogeneous space. This yields Hermite data for a transitional strip of tensor-product splines of degree (6,5) that guarantees overall curvature continuity. We also explain the simpler G1 to C1 transition. Key to the constructions is the C2 parameterization of circles in homogeneous space.

Efficient ALE mesh management for 3D quasi‐Eulerian problems

SUMMARYIn computational solid mechanics, the ALE formalism can be very useful to reduce the size of finite element models of continuous forming operations such as roll forming. The mesh of these ALE models is said to be quasi‐Eulerian because the nodes remain almost fixed—or almost Eulerian—in the main process direction, although they are required to move in the orthogonal plane in order to follow the lateral displacements of the solid. This paper extensively presents a complete node relocation procedure dedicated to such ALE models. The discussion focusses on quadrangular and hexahedral meshes with local refinements. The main concern of this work is the preservation of the geometrical features and the shape of the free boundaries of the mesh. With this aim in view, each type of nodes (corner, edge, surface and volume) is treated sequentially with dedicated algorithms. A special care is given to highly curved 3D surfaces for which a CPU‐efficient smoothing technique is proposed. This new method relies on a spline surface reconstruction, on a very fast weighted Laplacian smoother with original weights and on a robust reprojection algorithm. The overall consistency of this mesh management procedure is finally demonstrated in two numerical applications. The first one is a 2D ALE simulation of a drawbead, which provides similar results to an equivalent Lagrangian model yet is much faster. The second application is a 3D industrial ALE model of a 16‐stand roll forming line. In this case, all attempts to perform the same simulation by using the Lagrangian formalism have been unsuccessful. Copyright © 2012 John Wiley & Sons, Ltd.

Formula omitted]-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.

Geometrical Springback Compensation Algorithm Based on UV Curves for Automobile Structure Parts Die-Face

Springback is the most common defect in sheet metal forming. The compensation method is an effective way to overcome this defect by modifying the die-face. A new geometrical springback compensation algorithm based on UV curves is proposed for automobile structure die-face. The UV discrete technology and space lattice search approach are adopted to get the regular springback lattices from irregular and unordered springback points. The compensation direction and the correction of the pressing direction are considered. The die-face is reconstructed with a high order BEZIER spline surface on the CATIA platform. The high quality geometry compensation die-face obtained from this algorithm can be applied directly in actual machining. The geometry springback compensation of a high strength steel structure part is used to verify the effectiveness of the algorithm.

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for $L_2$ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -- one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

Blending Implicit Algebraic Surfaces by Combining Division with Filling with S-Patches

Based on the method of approximating boundary curves with splines,we use a spline surface of degree 3 to blend two arbitrary implicit algebraic surfaces.Smooth transition is achieved between the spline surface and two base surfaces.Furthermore,we use GB-splines to blend two algebraic surfaces which can exactly represent Lissajous curves,conics,trigonometric function curves,hyperbolic function curves,catenary curves and helixes etc.For several implicit algebraic surfaces,we present the blending method that combines splitting and filling holes with S-patches.G1-continiuty is persisted between splitting blending patches and S-patches.All blending patches are control point patches and the shape of each patch can be adjusted by shape parameters intuitively.

805 スプライン曲面を用いて表現される曲線状繊維複合材の振動最適化

805 スプライン曲面を用いて表現される曲線状繊維複合材の振動最適化

Nonnegativity Preserving Interpolation by C1 Bivariate Rational Spline Surface

This paper is concerned with the nonnegativity preserving interpolation of data on rectangular grids. The function is a kind of bivariate rational interpolation spline with parameters, which is C1 in the whole interpolation region. Sufficient conditions are derived on coefficients in the rational spline to ensure that the surfaces are always nonnegative if the original data are nonnegative. The gradients at the data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled. Some numerical examples are illustrated in the end of this paper.

One-shot spectrometer for several elements using an integrated conical crystal analyzer

Time-resolved x-ray spectrometry using an ultrastrong x-ray source such as an x-ray free electron laser is one of the new trends in the field of x-ray physics. To achieve such time-resolved measurement, the development of an one-shot spectrometer with a wide wavelength range, high efficiency, and good energy resolution is an essential prerequisite. Here we developed an integrated conical Ge crystal analyzer consisting of several conical rings, which were connected using spline surfaces to form a single body using our previously developed hot deformation technique, which can form a Si or Ge wafer into an arbitrary and accurate shape. We simultaneously focused several characteristic lines from an alloy sample onto different positions on a small x-ray charge-coupled device with very high image brightness (gain relative to planar analyzer: 100) and a good spatial resolution of 9-13 eV. The small radius of curvature of the crystal (28-50 mm) enabled us to realize a very short sample-detector distance of 214.4 mm. The present result shows the possibility of realizing a new focusing x-ray crystal spectrograph that can control the focal position as desired.

Extending Dirichlet Approach to Design B Spline Surface of Minimal Area

PDF HTML阅读 XML下载 导出引用 引用提醒 推广Dirichlet 方法用于B 样条极小曲面设计 DOI: 10.3724/SP.J.1001.2011.04009 作者: 作者单位: 作者简介: 通讯作者: 中图分类号: 基金项目: 国家自然科学基金(61070065, 60933007) Extending Dirichlet Approach to Design B Spline Surface of Minimal Area Author: Affiliation: Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:为弥补当前NURBS 系统无法有效设计工程所急需的B 样条极小曲面的缺陷,将构造Bézier 极小曲面的 Dirichlet 方法成功地推广到了B 样条极小曲面设计.提出了插值控制网格边界的B 样条曲面模型,运用B 样条基函数的求导公式及求值割角算法,将计算极小曲面内部控制顶点的问题转化为一个线性方程组的求解,从而避免了强非线性问题所导致的困惑,极大地提高了运算效率.最后,用大量实例对理论和算法进行了验证. Abstract:The current NURBS system is unable to design a B-spline minimal surface effectively which is required for engineering. This paper extends the Dirchlet approach, constructing Bézier minimal surface to the design of B-spline minimal surfaces successfully. The study also proposes a model of B-spline surface which interpolates its control net at the boundary, applying the derivative formulae and cutting-angle evaluation algorithms of B-spline basis. This approach transforms the problem of computing internal control points of the minimal surface to solving a system of linear equations, avoiding the bewilderment brought by a strong nonlinear problem and advancing operational efficiency greatly. Finally, with a large number of examples, the theory and algorithms are verified. 参考文献 相似文献 引证文献

The avoidance of cutter gouging in five-axis machining with a fillet-end milling cutter

This paper proposes a new method of automatic detection and elimination of cutter gouging when using the fillet-end milling cutter to produce a complex surface on the five-axis computer numerical control machine tool. To avoid local cutter gouging at the point where the cutter and part surface make contact with each other, the method of exact curvature matching between the cutter and part surface is presented. The size of cutter radii is more easily determined by this method. To detect if a rear cutter gouging occurs near the contact point, a square grid with horizontal and vertical points is used for illustrating the checking area and checking points. The technique of automatic generation of the square grid points and the method of detection and avoidance of rear cutter gouging are investigated throughout this article. In the end, the studied methodology and algorithms are inspected and verified by using an example of nonuniform rational B spline surface.

NURBS approximation to the flank-milled surface swept by a cylindrical NC tool

In this paper, a method to approximate the flank-milled surface swept by a cylindrical cutter with a non-uniform rational basis spline (NURBS) surface is presented. The swept surface produced by the moving tool can be calculated as a collection of points organized as a series of grazing curves along the surface. The generated NURBS surface closely matches the grazing surface. The deviation between this surface and the grazing surface is calculated and is controlled by increasing the number of control points used to represent the surface.

On local RBF approximation

The purpose of this paper is to investigate RBF approximation with highly nonuniform centers. Recently, DeVore and Ron have developed a notion of the local density of a set of centers—a notion that permits precise pointwise error estimates for surface spline approximation. We give an equivalent, alternative characterization of local density, one that allows effective placement of centers at different resolutions. We compare, also, the pointwise results of DeVore–Ron to previously works of Wu and Schaback and of Duchon.

Triangular bubble spline surfaces

We present a new method for generating a Gn-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patches. The compatible surface strips provide a simple Gn-condition between two neighboring bubble patches, which are used to construct surface patches, connected with Gn-continuity. For n≤2, we describe the obtained Gn-condition in detail. It can be generalized to any n≥3. The construction of a single surface patch is based on Gordon–Coons interpolation for triangles.Our method is a simple local construction scheme, which works uniformly for vertices of arbitrary valency. The resulting surface is a piecewise rational surface, which interpolates the given network of polynomial curves. Several examples of G0, G1 and G2-surfaces are presented, which have been generated by using our method. The obtained surfaces are visualized with reflection lines to demonstrate the order of smoothness.

Constrained design of simple ship hulls with [formula omitted]-spline surfaces

This paper presents a mathematical method to define a ship hull based on numerical constraints directly related to geometric features of the hull surface and on naval architecture parameters that uniquely define a hull form. These geometric parameters have physical, hydrodynamic or stability implications from the design point of view. B -spline curves and surface representations were combined with a constrained approach to produce the final hull display. The presented method follows the traditional design principles of naval architecture, starting with a Sectional Area Curve (SAC) that controls buoyancy and a waterplane curve that ensures stability. A B -spline outline of these two curves is adopted and a nonlinear problem is solved to obtain their constrained definition that agrees with the geometric design parameters. Two boundaries are also introduced into the constrained definition that control the limits of the hull, the centre and deck lines, which are complemented with tangent values at these edges to gain local control of the hull. A net of curves or hull stations is created that matches the previously defined constraints. These curves follow an analytic expression that ensures that a given area, waterline breadth and initial tangent angle are obtained. This is an important advantage of the method because a hull form library or template is not necessary, but limit the type of ship hulls that can be attained. The method continues with a constrained B -spline fitting of points on the analytical curves that ensures the tangent angles and checks the distance from the points to the B -spline. A final lofting surface of the previous B -spline curves produces the hull surface.