Computing B-Spline Surfaces on Transputer Networks

This paper presents a parallel implementation of computing uniform bicubic B spline surfaces on Transputer networks. The work is essential for building Transputer based CAD and graphics systems.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Ship hull plate processing surface fairing with constraints based on B-spline

The problem of ship hull plate processing surface fairing with constraints based on B-spline is solved in this paper. The algorithm for B-spline curve fairing with constraints is one of the most common methods in plane curve fairing. The algorithm can be applied to global and local curve fairing. It can constrain the perturbation range of the control points and the shape variation of the curve, and get a better fairing result in plane curves. In this paper, a new fairing algorithm with constraints for curves and surfaces in space is presented. Then this method is applied to the experiments of ship hull plate processing surface. Finally numerical results are obtained to show the efficiency of this method.

Reconstruction of B-spline surface by interpolating boundary curves and approximating inner points

In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.

G^0 and G^1 connection between two adjacent B-spline surfaces

In this paper, the smooth connection between two B-spline surfaces is discussed. First, a brief proof of some simple sufficient conditions of Go and G1 continuity is given. On this basis, a novel method for Go or G1 connection between two adjacent B-spline surfaces is presented. A reparameterization step is firstly taken for one of the surfaces such that they have the same parameterization in v direction, then, adjust their boundary control vertices to make them Go or Gl connected. The GI connection parameter is determined by an optimization problem. Compared with the existed methods, our method is simple and easy to be used in practice.

Symmetric alteration of four knots of B-spline and NURBS surfaces

Modifying the knots of a B-spline curve, the shape of the curve will be changed. In this paper, we present the effect of the symmetric alteration of four knots of the B-spline and the NURBS surfaces, i.e., symmetrical alteration of the knots of surface, the extended paths of points of the surface will converge to a point which should be expressed with several control points. This theory can be used in the constrained shape modification of B-spline and NURBS surfaces.

B-Spline Approximation of Ship Waves on the Free Surface

We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.

G^1 Continuity Conditions of B-spline Surfaces

According to the B-spline theory and Boehm algorithm, this paper presents several necessary and sufficient G1 continuity conditions between two adjacent B-spline surfaces. In order to meet the need of application, a kind of sufficient conditions of G1 continuity are developed, and a kind of sufficient conditions of G1 continuity among N(N>2) patch B-spline surfaces meeting at a common corner are given at the end.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Fast B-spline surface extraction algorithm of seismic volume data on GPU

In the process of seismic data interpretation, the extraction of a horizon or a fault is generally needed. In this paper we present a fast extraction method. First select some feature points interactively, then reconstruct the surface according to the selected feature points by using B-spline interpolation curve or surface. In order to solve the error-drawing problem appeared in the process of interactive rendering, which is caused by the change of sampling interval as the view point changes, we combine shear-warp and splatting algorithm to render the surface. The rendering of seismic data and specific surface in our work are achieved on GPU platform using CUDA programming language, which make it able to meet the requirements of real-time rendering.

Modification of B-Spline Surfaces with Optimization

Algorithms of modifying a surface to approximate some scattered points, or pass through some characteristic points/curves are presented. Similar to variational approach, the algorithms are based on optimization. For the deviation between the modified surface and the original one is adopted as the objective functions, the change of the surface shape is as small as possible with the modified surface satisfying the specified requirements.

Regional magnetic anomaly fields:3D Taylor polynomial and surface spline models

We used data from 1960.0,1970.0,1980.0,1990.0,and 2000.0 to study the geomagnetic anomaly field over the Chinese mainland by using the three-dimensional Taylor polynomial（3DTP） and the surface spline（SS） models.To obtain the pure anomaly field,the main field and the induced field of the ionospheric and magnetospheric fields were removed from measured data.We also compared the SS model anomalies and the data obtained with Kriging interpolation（KI）.The geomagnetic anomaly distribution over the mainland was analyzed based on the SS and 3DTP models by transferring all points from 1960.0-1990.0 to 2000.0.The results suggest that the total intensity F anomalies estimated based on the SS and KI for each year are basically consistent in distribution and intensity.The anomalous distributions in the X-,Y-,and Z-direction and F are mainly negative.The 3DTP model anomalies suggest that the intensity in the X-direction increases from-100 nT to 0 nT with longitude,whereas the intensity in the Y-direction decreases from 400 nT to 20 nT with longitude and over the eastern mainland is almost negative.The intensity in the Z-direction and F are very similar and in most areas it is about-50 nT and higher in western Tibet.The SS model anomalies overall reflect the actual distribution of the magnetic field anomalies;however,because of the uneven distribution of measurements,it yields several big anomalies.Owing to the added altitude term,the 3DTP model offers higher precision and is consistent with KI.

B-SPLINE SURFACE SMOOTH JOINING

Optimization techniques are being applied to solve the problems of surface interpolation, approximation, smooth joining and fairing, aiming at corresponding objective functions. This paper focuses on the construction of fair surface interpolating the given mesh of curved boundaries with G 2 adjustment at comers and G 1, G 2 smoothness between adjacent patches. Many papers on surface blending have been presented, but almost all of them are restricted to the discussion of Bezier patches, there are no good results for B-spline surface. This paper gives a solution to the B-spline surface, allowing the surface to degenerate at comer in and have different parameterization along the common boundary of two patches.

An Algorithm of B-Spline Surface Interpolation

A surface interpolation algorithm is presented. By using a special kind of knot vector. a B-spline surface can be constructed to interpolate an array of m ×n positions, including parameter u and v tangent vectors and twist vector at each positions. Single surface interpolation approach is easier to ensure the smoothness of the interpolating surface than multi-patches method. This algorithm can be used to solve the approximating problem of B-spline approximation of general parametric surface.

Best bounds on the distance between 3-direction quartic box spline surface and its control net

We present the best bounds on the distance between 3-direction quartic box spline surface patch and its control net by means of analysis and computing for the basis functions of 3-direction quartic box spline surface.Both the local bounds and the global bounds are given by the maximum norm of the first differences or second differences or mixed differences of the control points of the surface patch.

Theoretical Modeling and Surface Roughness Prediction of Microtextured Surfaces in Ultrasonic Vibration-Assisted Milling

Textured surfaces with certain micro/nano structures have been proven to possess some advanced functions,such as reducing friction,improving wear and increasing wettability.Accurate prediction of micro/nano surface textures is of great significance for the design,fabrication and application of functional textured surfaces.In this paper,based on the kinematic analysis of cutter teeth,the discretization of ultrasonic machining process,transformation method of coordinate systems and the cubic spline data interpolation,an integrated theoretical model was established to characterize the distribution and geometric features of micro textures on the surfaces machined by different types of ultrasonic vibration-assisted milling(UVAM).Based on the theoretical model,the effect of key process parameters(vibration directions,vibration dimensions,cutting parameters and vibration parameters)on tool trajectories and microtextured surface morphology in UVAM is investigated.Besides,the effect of phase difference on the elliptical shape in 2D/3D ultrasonic elliptical vibration-assisted milling(UEVAM)was analyzed.Compared to conventional numerical models,the method of the cubic spline data interpolation is applied to the simulation of microtextured surface morphology in UVAM,which is more suitable for characterizing the morphological features of microtextured surfaces than traditional methods due to the presence of numerous micro textures.The prediction of surface roughness indicates that the magnitude of ultrasonic amplitude in z-direction should be strictly limited in 1D rotary UVAM,2D and 3D UEVAM due to the unfavorable effect of axial ultrasonic vibration on the surface quality.This study can provide theoretical guidance for the design and fabrication of microtextured surfaces in UVAM.

A fast direct point-by-point generating algorithm for B Spline curves and surfaces

Traditional generating algorithms for B Spline curves and surfaces require approximation methods where how to increment the parameter to get the best approximation is problematic; or they take the pixel-based method needing matrix trans- formation from B Spline representation to Bézier form. Here, a fast, direct point-by-point generating algorithm for B Spline curves and surfaces is presented. The algorithm does not need matrix transformation, can be used for uniform or nonuniform B Spline curves and surfaces of any degree, and has high generating speed and good rendering accuracy.

Compact Car-Body Surface Design with T-spline Surface

Creating proper B-spline surface models is a very challenging task for designers in car-body surface design.Due to the tensor-product structure of B-spline surface,some undesirable issues of the redundant control points addition,incomplete surface definition and the difficulty of trimming boundary alteration frequently occur,when designing the car-body surface with B-spline surfaces in local-feature-lines construction,full-boundary-merging and visual surface trimming.A more efficient approach is proposed to design the car-body surface by replacing B-spline surface with classical T-spline surface.With the local refinability and multilateral definition offered by Tspline surface,those designing issues related with B-spline surface can be overcomed.Finally,modeling examples of the door,hood and rear-window are given to demonstrate the advantage of T-spline surface over B-spline surface in car-body surface design.

Research into Freedom Surface Modeling System of Double Beta Spline in Space

In this paper, we present the double Beta spline curved surface which is controlled by double parameters including the algorithm principles, the treatment of boundary conditions, the alternation of projection, the algorithms of elimination hiddle line, the process to display and the primiples to produce the shaded curved surface. Based on all the above, a freedom surface modeling system (FSMS) is designed and some examples developed on FSMS are verified and analyzed.

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis （Adv. Comput. Math., 17 （2002）, pp. 135-152） based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Juttler＇s method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.