Existence Criteria of Blending Surfaces of Three Quadric Surfaces——Wu Wen-tsun's Formulae

Smoothly blending several implicit quadric surfaces with surfaces of the lowest degree is very useful in engineering technology.

USING NUMERICAL METHOD TO GENERATE BLENDING SURFACES

A numerical method is suggested to generate the blends with one of the blended surfaces being analytical.The method is more accurate, less complicated,more efficient and more intuitive.The formulation of equation set is presented in detail and the NURBS representation of the blending surface discussed.The algorithm has been implemented on PC.The method was tested by a couple of samples and got satisfactory results.

Construction of Blending Surfaces

Surface blending is very important in geometric modeling and is widely used in product design and manufacturig. This paper has a comprehensive discussion in surface blending with constant/variable radil, including the generation of rolling-ball blending between a curve and a surface, rolling-ball blending between two surfaces at a given contact curve, sliding-circle blending between two surfaces, etc. All those algorithms developed have been put into the commercial CAD/CAE/CAM software system-CAXA-ME and greatly enhanced its modeling ability

BOUNDARY PENALTY FINITE ELEMENT METHODS FOR BLENDING SURFACES,I BASIC THEORY

When parametric functions are used to blend 3D surfaces, geometric continuity of displacements and derivatives until to the surface boundary must be satisfied. By the traditional blending techniques, however, arbitrariness of the solutions arises to cause a difficulty in choosing a suitable blending surface. Hence to explore new blending techniques is necessary to construct good surfaces so as to satisfy engineering requirements. In this paper, a blending surface is described as a flexibly elastic plate both in partial differential equations and in their variational equations, thus to lead to a unique solution in a sense of the minimal global surface curvature. Boundary penalty finite element methods (BP-FEMs) with and without approximate integration are proposed to handle the complicated constraints along the blending boundary. Not only have the optimal convergence rate O(h(2)) of second order generalized derivatives of the solutions in the solution domain been obtained, but also the high convergence rate O(h(4)) of the tangent boundary condition of the solutions can be achieved, where h is the maximal boundary length of rectangular elements used. Moreover, useful guidance in computation is discovered to deal with interpolation and approximation in the boundary penalty integrals. A numerical example is also provided to verify perfectly the main theoretical analysis made. This paper yields a framework of mathematical modelling, numerical techniques and error analysis to the general and complicated blending problems.

Generating Parametric G^n Blending Surface with Some Constraints

This paper presents a method of generating a parametric G^n blending surfacebased on reparameterizing the partial surface patches in the base surfaces on the basis of ErichHartmann method. This method is expressed as follows Firstly, the partial region near contact curvesin both base surfaces is reparameterized. The contact curves are used as the boundaries of thereparameterized partial region respectively. The reparameterized partial region in two base surfacesis called the reparameterized local base surfaces. Then the parametric G^n blending surface isgenerated by a linear combination of the reparameterized local base surface patches depending on oneof the common parameters. Therefore, generating a Parametric G^n Blending Surface between two basesurfaces is translated into generating a Parametric G^n Blending Surface between the tworeparameterized local base surfaces. This paper illustrates the method to generate the G^n blendingsurface with some constraints by generating a G^2 blending surface between the aerofoil and the bodyof a missile with the constraints of the forward and rear fringe curves. When the G^n blendingsurface with some constraints is generated, the partial region near contact curves in both basesurfaces is reparameterized, and the scale factors, offset, balance factor and thumb weight aredefined by meeting the constraints through using an optimization method. Then the parametric G^nblending surface is generated by the linear combination of the reparameterized local base surfacepatches. The shape of the blending surface can be adjusted by changing the size of thereparameterized local base surface patches.

REVERSE MODELING FOR CONIC BLENDING FEATURE

A novel method to extract conic blending feature in reverse engineering is presented. Different from the methods to recover constant and variable radius blends from unorganized points, it contains not only novel segmentation and feature recognition techniques, but also bias corrected technique to capture more reliable distribution of feature parameters along the spine curve. The segmentation depending on point classification separates the points in the conic blend region from the input point cloud. The available feature parameters of the cross-sectional curves are extracted with the processes of slicing point clouds with planes, conic curve fitting, and parameters estimation and compensation, The extracted parameters and its distribution laws are refined according to statistic theory such as regression analysis and hypothesis test. The proposed method can accurately capture the original design intentions and conveniently guide the reverse modeling process. Application examples are presented to verify the high precision and stability of the proposed method.

A Blend of Subdivision Surfaces and NURBS

In present paper, the contour deletion method is developed both to blend surfaces and to fill N-sided holes, which is used for subdividing the NURBS surface. First, according to the non-uniform Catmull-Clark subdivision principle, surfaces are blended. The non-uniform Catmull-Clark subdivision method is constructed, which build the surface through interpolating comer vertices and boundary curves. Then the contour deletion method is adapted to remove the controlling mesh boundary contour in the process of segmentation iteration. Last, N sided-hole is filled to generate a integral smooth continuous surface. This method not only guarantee that the blending surface and base surface patches have C2 continuity at the boundary, but also greatly improve the smoothness of the N-side hole filling surface. The results show that, this method simplifies the specific computer-implemented process, broads the scope of application of subdivision surfaces, and solves the incompatible problem between the subdivision surface and classical spline. The resulting surface has both advantages of the subdivision surface and classical spline, and also has better filling effect.

METHOD TO EXTRACT BLEND SURFACE FEATURE IN REVERSE ENGINEERING

A new method of extraction of blend surface feature is presented. It contains two steps: segmentation and recovery of parametric representation of the blend. The segmentation separates the points in the blend region from the rest of the input point cloud with the processes of sampling point data, estimation of local surface curvature properties and comparison of maximum curvature values. The recovery of parametric representation generates a set of profile curves by marching throughout the blend and fitting cylinders. Compared with the existing approaches of blend surface feature extraction, the proposed method reduces the requirement of user interaction and is capable of extracting blend surface with either constant radius or variable radius. Application examples are presented to verify the proposed method.

Hyperfinite Interpolation,Wu's Method and Blending of Implicit Algebraic Surfaces

one of the central questions in CAGD[1] is blending surfaces whichprovides the theoretical baJsis for the design technology of space surfaces. We willdiscuss the general theories and algorithms for multivariate hyperfinite interpolationand their aPplication to the blending of implicit algebraic surfaces, and investigate theexistence conditions of hyperfinite interpolation. Based on Wu's theory on blendingimplicit algebraic surfaces, the problem of blending two quadric surfaces is studied.The conditions for the coefficient of gi under which there exists the cubic blendingsurface S(f) (the lowest degree) are obtained and the concrete expressions of f arepresented if they exist. These results can be applied directly to CAGD.

Connolly Surface on an Atomic Structure via Voronoi Diagram of Atoms

One of the most important geometric structures of a protein is the Connolly surface of protein since a Connolly surface plays an important role in protein folding, docking, interactions between proteins, amongst other things. This paper presents an algorithm for precisely and efficiently computing the Connolly surface of a protein using a proposed geometric construct called β-shape based on the Voronoi diagram of atoms in the protein. Given the Voronoi diagram of atoms based on the Euclidean distance from the atom surfaces, the proposed algorithm first computes a β-shape with an appropriate probe. Then, the Connolly surface is computed by employing the blending operation on the atomic complex of the protein by the given probe.

Evaluation of the Impact of Observations on Blended Sea Surface Winds in a Two-Dimensional Variational Scheme Using Degrees of Freedom

This paper presents an evaluation of the observational impacts on blended sea surface winds from a two- dimensional variational data assimilation （2D-Var） scheme. We begin by briefly introducing the analysis sensitivity with respect to observations in variational data assimilation systems and its relationship with the degrees of freedom for signal （DFS）, and then the DFS concept is applied to the 2D-Var sea surface wind blending scheme. Two meth- ods, a priori and a posteriori, are used to estimate the DFS of the zonal （u） and meridional （v） components of winds in the 2D-Var blending scheme. The a posteriori method can obtain almost the same results as the a priori method. Because only by-products of the blending scheme are used for the a posteriori method, the computation time is re- duced significantly. The magnitude of the DFS is critically related to the observational and background error statistics. Changing the observational and background error variances can affect the DFS value. Because the observation error variances are assumed to be uniform, the observational influence at each observational location is related to the background error variance, and the observations located at the place where there are larger background error variances have larger influences. The average observational influence of u and v with respect to the analysis is about 40%, implying that the background influence with respect to the analysis is about 60%.