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.

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.

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.

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.

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.

Geometry Modeling of Ship Hull Based on Non-uniform B-spline

In order to generate the three-dimensional (3-D) hull surface accurately and smoothly,a mixed method which is made up of non-uniform B-spline together with an iterative procedure was developed.By using the iterative method the data points on each section curve are calculated and the generalized waterlines and transverse section curves are determined.Then using the non-uniform B-spline expression,the control vertex net of the hull is calculated based on the generalized waterlines and section curves.A ship with tunnel stern was taken as test case.The numerical results prove that the proposed approach for geometry modeling of 3-D ship hull surface is accurate and effective.

Turbulent Variance Characteristics of Temperature and Humidity over a Non-uniform Land Surface for an Agricultural Ecosystem in China

This paper describes the application of the variance method for flux estimation over a mixed agricultural region in China. Eddy covariance and flux variance measurements were conducted in a near-surface layer over a non-uniform land surface in the central plain of China from 7 June to 20 July 2002. During this period, the mean canopy height was about 0.50 m. The study site consisted of grass （10% of area）, beans （15%）, corn （15%） and rice （60%）. Under unstable conditions, the standard deviations of temperature and water vapor density （normalized by appropriate scaling parameters）, observed by a single instrument, followed the Monin-Obukhov similarity theory. The similarity constants for heat （CT） and water vapor （Cq） were 1.09 and 1.49, respectively. In comparison with direct measurements using eddy covariance techniques, the flux variance method, on average, underestimated sensible heat flux by 21% and latent heat flux by 24%, which may be attributed to the fact that the observed slight deviations （20% or 30% at most） of the similarity ＂constants＂ may be within the expected range of variation of a single instrument from the generally-valid relations.

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.

An efficient data structure for calculation of unstructured T-spline surfaces

To overcome the topological constraints of non-uniform rational B-splines,T-splines have been proposed to define the freeform surfaces.The introduction of T-junctions and extraordinary points makes it possible to represent arbitrarily shaped models by a single T-spline surface.Whereas,the complexity and flexibility of topology structure bring difficulty in programming,which have caused a great obstacle for the development and application of T-spline technologies.So far,research literatures concerning T-spline data structures compatible with extraordinary points are very scarce.In this paper,an efficient data structure for calculation of unstructured T-spline surfaces is developed,by which any complex T-spline surface models can be easily and efficiently computed.Several unstructured T-spline surface models are calculated and visualized in our prototype system to verify the validity of the proposed 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.

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

An Approach to Representing Heterogeneous Non-Uniform Rational B-Spline Objects

The representation method of heterogeneous material information is one of the key technologies of heterogeneous object modeling, but almost all the existing methods cannot represent non-uniform rational B-spline （NURBS） entity. According to the characteristics of NURBS, a novel data structure, named NURBS material data structure, is proposed, in which the geometrical coordinates, weights and material coordinates of NURBS heterogene- ous objects can be represented simultaneously. Based on this data structure, both direct representation method and inverse construction method of heterogeneous NURBS objects are introduced. In the direct representation method, three forms of NURBS heterogeneous objects are introduced by giving the geometry and material information of con- trol points, among which the homogeneous coordinates form is employed for its brevity and easy programming. In the inverse construction method, continuous heterogeneous curves and surfaces can he obtained by interpolating discrete points and curves with specified material information. Some examples are given to show the effectiveness of the pro- posed methods.

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.

Extraction of Feature Points for Non-Uniform Rational B-Splines(NURBS)-Based Modeling of Human Legs

Methods of digital human modeling have been developed and utilized to reflect human shape features.However,most of published works focused on dynamic visualization or fashion design,instead of high-accuracy modeling,which was strongly demanded by medical or rehabilitation scenarios.Prior to a high-accuracy modeling of human legs based on non-uniform rational B-splines(NURBS),the method of extracting the required quasi-grid network of feature points for human legs is presented in this work.Given the 3 D scanned human body,the leg is firstly segmented and put in standardized position.Then re-sampling of the leg is conducted via a set of equidistant cross sections.Through analysis of leg circumferences and circumferential curvature,the characteristic sections of the leg as well as the characteristic points on the sections are then identified according to the human anatomy and shape features.The obtained collection can be arranged to form a grid of data points for knots calculation and high-accuracy shape reconstruction in future work.

Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.

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.

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.

Threshold Voltage Model for a Fully Depleted SOI-MOSFET with a Non-Uniform Profile

A novel approximation of the two-dimensional （2D） potential function perpendicular to the channel is proposed,and then an analytical threshold voltage model for a fully depleted SOI-MOSFET with a non-uniform Gaussian distribution doping profile is given based on this approximation. The model agrees well with numerical simulation by MEDICI. The result represents a new way and some reference points in analyzing and controlling the threshold voltage of non-uniform fully depleted （FD） SOI devices in practice.