一类新的细分曲线方法

Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements based on initial control polygon or grid.Usually the curve refinements is the basis of the corresponding surface rules. In this paper we analyze previous subdivision scheme according to theories about convergence of N.Dyn and M.F Hassan. In terms of binary and ternary subdivision schemes general construction about curve‘s refinements are studied.Two approximating curve subdivision schemes with neighboring four control points are derived,the generating limit curves can both reach the smoothness of C^1 over the initial polygon using the two schemes and the tolerances of them are given according to the method of [7].     

CONVERGENCE RATE OF VECTOR SUBDIVISION SCHEME

In this paper,some characterizations on the convergence rate of both the homoge- neous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.     

A unified three point approximating subdivision scheme

In this paper,we propose a three point approximating subdivision scheme,with three shape parameters,that unifies three different existing three point approximating schemes.Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed.The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.     

Digital Cofibration and Digital Lusternik-Schnirelmann Category in the Sense of Subdivision

In this paper, using the notion of subdivision, the authors generalize the definition of cofibration in digital topology and show that this kind of cofibration is injective in the sense of subdivision. Meanwhile, they give the necessary condition under which a digital map is a cofibration. Furthermore, they consider the Lusternik-Schnirelmann category of digital maps in the sense of subdivision and give several fundamental homotopy properties about it.     

Non-stationary subdivision for exponential polynomials reproduction

In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.     

Mean size formula of wavelet subdivision tree on Heisenberg group

The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.     

The classification of bi-quintic parametric polynomial minimal surfaces

Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently,Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic,we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases,we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover,we study some geometric properties of the bi-quintic harmonic surfaces based on the B′ezier representation. Finally, some numerical examples are demonstrated to verify our results.     

A Unified Ternary Curve Subdivision Scheme

A unified $m\,(m>2)$-point ternary scheme with some parameter is proposed. The continuity of subdivision scheme is analyzed based on the relationship between the subdivision scheme and difference scheme. Moreover, the proposed subdivision is extended to asymmetric multi-parameter subdivision and the asymmetric schemes in four cases are presented in detail. Some examples are given to show that the presented scheme has better approximating effect.     

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc′?a-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface.Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.     

The Newton filtration and d-determination of bifurcation problems related to C~0 contact equivalence

In this paper, from the Newton filtration's point of view, we construct the singular Riemannian metric and use the method in singular theory to study the bifurcation problems, and give the sufficient condition of d-determination of bifurcation problems with respect to C0 contact equivalence. The special cases of the main result in this paper are the results of Sun Weizhi and Zou Jiancheng.     

三进制Loop细分曲面的C~1连续性分析

本文利用计算机代数系统,通过对三进制Loop细分算法的细分矩阵和特征映射的构造与分析,证明Loop给出的掩模设计能够保证细分曲面在奇异点是C1连续的,还给出细分矩阵的次优势特征值的一个取值范围,在此范围内利用三进制Loop细分算法生成的细分曲面都是C1连续的.最后给出一种三进制Loop细分算法的新的边点掩模设计方法,在保证细分曲面是C1连续的前提下,比Loop给出的计算公式更简单,细分算法在奇异点附近收敛更快.     

The template‐based procedure of biorthogonal ternary wavelets for curve multiresolution processing

Curve multiresolution processing techniques have been widely discussed in the study of subdivision schemes and many applications, such as surface progressive transmission and compression. The ternary subdivision scheme is the more appealing one because it can possess the symmetry, smaller topological support, and certain smoothness, simultaneously. So biorthogonal ternary wavelets are discussed in this paper, in which refinable functions are designed for cure and surface multiresolution processing of ternary subdivision schemes. Moreover, by the help of lifting techniques, the template‐based procedure is established for constructing ternary refinable systems with certain symmetry, and it also gives a clear geometric templates of corresponding multiresolution algorithms by several iterative steps. Some examples with certain smoothness are constructed.     

ESTIMATING ERROR BOUNDS FOR TERNARY SUBDIVISION CURVES/SURFACES

We estimate error bounds between ternary subdivision curves/surfaces and their control polygons after k-fold subdivision in terms of the maximal differences of the initial control point sequences and constants that depend on the subdivision mask. The bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Our technique is independent of parametrization therefore it can be easily and efficiently implemented. This is useful and important for pre-computing the error bounds of subdivision curves/surfaces in advance in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and so on.     

Approximation order of interpolatory nonlinear subdivision schemes

Linear interpolatory subdivision schemes of C^r smoothness have approximation order at least r+1. The present paper extends this result to nonlinear univariate schemes which are in proximity with linear schemes in a certain specific sense. The results apply to nonlinear subdivision schemes in Lie groups and in surfaces which are obtained from linear subdivision schemes. We indicate how to extend the results to the multivariate case.     

Storing the subdivision of a polyhedral surface

A common structure arising in computational geometry is the subdivision of a plane defined by the faces of a straight-line planar graph. We consider a natural generalization of this structure on a polyhedral surface. The regions of the subdivision are bounded by geodesics on the surface of the polyhedron. A method is given for representing such a subdivision that is efficient both with respect to space and the time required to answer a number of different queries involving the subdivision. For example, given a pointx on the surface of the polyhedron, the region of the subdivision containingx can be determined in logarithmic time. Ifn denotes the number of edges in the polyhedron,m denotes the number of geodesics in the subdivision, andK denotes the number of intersections between edges and geodesics, then the space required by the data structure isO((n +m) log(n +m)), and the structure can be built inO(K + (n +m) log(n +m)) time. Combined with existing algorithms for computing Voronoi diagrams on the surface of polyhedra, this structure provides an efficient solution to the nearest-neighbor query problem on polyhedral surfaces.A preliminary version of this paper appeared at the Second Annual Symposium on Computational Geometry. The support of the Air Force Office of Scientific Research under Contract F-49620-85-K-0009 is gratefully acknowledged.     

Matrix-valued symmetric templates for interpolatory surface subdivisions: I. Regular vertices

The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with “symmetric subdivision templates” (SSTs) for regular vertices. While the precise definition of “symmetry” will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C² interpolatory subdivisions, only one “ring” is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C² interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull–Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C² interpolatory and subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C² splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147–162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C²-splines for quadrilateral subdivisions, Preprint, 2004], C² symmetric interpolatory subdivision schemes associated with refinement equations of C² cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper.     

Progressive transmission of subdivision surfaces

Triangle meshes are a standard representation for surface geometry in computer graphics and virtual reality applications. To achieve high realism of the modeled objects, the meshes typically consist of a very large number of faces. For broadcasting virtual environments over low-bandwidth data connections like the Internet it is highly important to develop efficient algorithms which enable the progressive transmission of such large meshes. In this paper we introduce a special representation for storing and transmitting meshes with subdivision connectivity which allows random access to the detail information. We present algorithms for the decomposition and the reconstruction of subdivision surfaces. With this technique, the receiver can reconstruct smooth approximations of the original surface from a rather small amount of data received.     

Families of univariate and bivariate subdivision schemes originated from quartic B-spline

Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the Lane-Riesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface.     

The distance of a subdivision surface to its control polyhedron

For standard subdivision algorithms and generic input data, near an extraordinary point the distance from the limit surface to the control polyhedron after m subdivision steps is shown to decay dominated by the mth power of the subsubdominant (third largest) eigenvalue. Conversely, for Loop subdivision we exhibit generic input data so that the Hausdorff distance at the mth step is greater than or equal to the mth power of the subsubdominant eigenvalue.In practice, it is important to closely predict the number of subdivision steps necessary so that the control polyhedron approximates the surface to within a fixed distance. Based on the above analysis, two such predictions are evaluated. The first is a popular heuristic that analyzes the distance only for control points and not for the facets of the control polyhedron. For a set of test polyhedra this prediction is remarkably close to the true distance. However, a concrete example shows that the prediction is not safe but can prescribe too few steps. The second approach is to first locally, per vertex neighborhood, subdivide the input net and then apply tabulated bounds on the eigenfunctions of the subdivision algorithm. This yields always safe predictions that are within one step for a set of typical test surfaces.