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A subdivision Scheme for Rational Triangular Bézier Surfaces

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Journal of Computer Science & Technology 1996年第1期11卷 9-16页

作者：胡事民 State Key Laboratory of CAD & CG Zhejiang University Hangzhou

An explicit formula is developed to decompose a rational triangular Bezierpatch into three non-degenerate rational rectangular B6zier patches of the samedegree. This formula yields a stable algorithm to compute the control verticesof those three rectallgular subpatches. Some properties of the subdivision arediscussed and the formula is illustrated with an example.

关键词： Rational Bézier surface subdivision reparametrization

A crystal plasticity FE study of macro-and micro-subdivision in aluminium single crystals{001}<110>multi-pass rolled to a high reduction

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Journal of Materials Science & Technology 2021年第17期76卷 231-246页

作者： Hui Wang Cheng Lu Kiet Tieu Yu Liu School of Mechanical Materials and Mechatronic EngineeringUniversity of WollongongNew South Wales 2522Australia Research Center for Advanced Science and Technology The University of TokyoJapan

In this study,the substructure formation in a multi-pass rolled aluminium single crystal{001}was investigated by the crystal plasticity finite element model,and the predations were validated by experimental observations at both macro-and micro-scale.A finite element model for multi-pass rolling was developed to follow the real experimental rolling scheme,by which the through-thickness macroscopic subdivision was successfully predicted up to a 90%*** macro-subdivision was featured by forming matrix bands through the thickness,and the deformation behaviours,in terms of slip activity,shear strain and crystal rotation,alternated between matrix *** development of matrix bands,stability of crystal orientations,and correlation between slip activity,shear strain and crystal rotation have been *** modelling method,Submodel,was used to exceedingly increase the mesh resolution in smaller regions of interest,and the experimentally observed microstructure,i.e.,micro-subdivision was explicitly and spatially *** predictions were obtained in Submodels with different element sizes,which proves the feasibility of this method in predicting microstructure *** was found that the substructure formation by varying slip activity and crystal rotation between domains is energy *** procedure of substructure formation was explained based on the predictions,three types of substructure have been identified,and the substructure formation was discussed.

关键词： Crystal plasticity FE Single crystal Multi-pass rolling Submodel subdivision Substructure

Non-Uniform Doo-Sabin subdivision Surface via Eigen Polygon

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Journal of Systems Science & Complexity 2021年第1期34卷 3-20页

作者： ALAM Md Nur LI Xin School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China

This paper constructs a new non-uniform Doo-Sabin subdivision scheme via eigen *** authors proved that the limit surface is always convergent and is G1 continuous for any valence and any positive knot intervals under a minor assumption, that λ is the second and third eigenvalues of the subdivision matrix. And then, a million of numerical experiments are tested with randomly selecting positive knot intervals, which verify that our new subdivision scheme satisfies the *** this is not true for the other two existing non-uniform Doo-Sabin schemes in Sederberg, et al.(1998), Huang and Wang(2013). In additional, numerical experiments indicate that the quality of the new limit surface can be improved.

关键词： Doo-Sabin non-uniform splines subdivision

Blending Parameric Patches with subdivision Surfaces

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Journal of Computer Science & Technology 2002年第4期17卷 498-506页

作者：李桂清李华 KeyLaboratoryofIntelligentInformationProcessing InstituteofComputingTechnologyTheChineseAcademyofSciencesBeijing100080P.R.China KeyLaboratoryofIntelligentInformationProcessing InstituteofComputingTe

In this paper the problem of blending parametric surfaces using subdivision patches is discussed.A new approach,named removing-boundary,is presented to generate piecewise-smooth subdivision surfaces through discarding the outmost quadrilaterals of the open meshes derived by each subdivision *** the approach is employed both to blend parametric bicubic B-spline surfaces and to fill n-sided *** is easy to produce piecewise-smooth subdivision surfaces with both convex and concave corners on the boundary,and limit surfaces are guaranteed to be C^2 continuous on the boundaries except for a few singular points by the removing-boundary approach Thus the blending method is very efficient and the blending surface generated is of good effect.

关键词： surface blending n-sided hole filling subdivision piecewise-smoothsurface

Blending Basic Shapes By C-Type Splines and subdivision Scheme

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Computer Modeling in Engineering & Sciences 2019年第7期120卷 45-62页

作者： Xiang Li Mei-E Fang Qian Qi School of Computer Science and Cyber Engineering Guangzhou UniversityGuangzhouChina School of Computer Hangzhou Dianzi UniversityHangzhouChina

In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle *** C-type spline belongs to theωB-spline category of splines that are capable of blending polynomial,trigonometric and hyperbolic *** used basic shapes can be exactly represented by these types of *** derive explicit formulas for the convenience of modeling the basic *** entire blending curve is C^*** comparison with the existing best blending method by rational G^2 splines,which are rational splines of degree 3,the proposed method allows simpler representation and blending of the basic curves,and it can represent numerous basic shapes including the hyperbolic *** also design a subdivision method to generate blending curves;this method is precise for the basic curves and approximate for the blending *** subdivision process is efficient for modeling and *** has also proven to be C^1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision *** addition,we extend the proposed methods to cases involving the modeling and blending of basic *** provide many examples that illustrate the merits of our methods.

关键词： Basic shapes blending C-type splines subdivision C^1-continuous

The independence of initial vectors in the subdivision schemes

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Science China Mathematics 2002年第11期45卷 1439-1445页

作者：陈迪荣李落清 1. Department of Applied Mathematics Beijing University of Aeronautics and Astronautics 100083 Beijing China 2. Department of Mathematics Hubei Institute for Nationalities 445000 Enshi China 3. Department of Mathematics Hubei University 430062 Wuhan China

Starting with an initial vector λ=(λ(k)) k∈? ∈ ?p(?), the subdivision scheme generates a sequence (S a n λ) n=1 ∞ of vectors by the subdivision operator $$S_a \lambda (k) = \sum\limits_{j \in \mathbb{Z}} {\lambda (j)a(k - 2j)} , k \in \mathbb{Z}$$ . subdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting to understand under what conditions the sequence (S a n λ) n=1 ∞ converges to an Lp-function in an appropriate sense. This problem has been studied extensively. In this paper we show that the subdivision scheme converges for any initial vector in ?p(?) provided that it does for one nonzero vector in that space. Moreover, if the integer translates of the refinable function are stable, the smoothness of the limit function corresponding to the vector A is also independent of λ.

关键词： subdivision scheme, joint spectral radius, critical exponent.

Continuity of non-uniform recursive subdivision surfaces

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Science China(Technological Sciences) 2000年第5期43卷 461-472页

作者：秦开怀王华维 Department of Computer Science & Technology Tsinghua University Beijing 100084 China

Since Doo-Sabin and Catmull-Clark surfaces were proposed in 1978, eigenstructure, convergence and continuity analyses of stationary subdivision have been performed very well, but it has been very difficult to prove the convergence and continuity of non-uniform recursive subdivision surfaces (NURSSes, for short) of arbitrary topology. In fact, so far a problem whether or not there exists the limit surface as well as G1 continuity of a non-uniform Catmull-Clark subdivision has not been solved yet. Here the concept of equivalent knot spacing is introduced. A new technique for eigenanaly-sis, convergence and continuity analyses of non-uniform Catmull-Clark surfaces is proposed such that the convergence and G1 continuity of NURSSes at extraordinary points are proved. In addition, slightly improved rules for NURSSes are developed. This offers us one more alternative for modeling free-form surfaces of arbitrary topologies with geometric features such as cusps, sharp edges, creases and darts, while elsewhere maintaining the same order of continuity as B-spline surfaces.

关键词： Catmull-Clark non-uniform recursive subdivision surface convergence continuity.

Improvement of the termination criterion for subdivision of the rational Bézier curves

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Journal of Zhejiang University Science 2003年第1期4卷 47-52页

作者：章仁江王国瑾 InstituteofComputerImagesandGraphics ZhejiangUniversityHangzhou310027China／／StateKeyLabofCADandCGZhejiangUniversityHangzhou310027China／／ChinaInstituteofMetrologyHangzhou310034China InstituteofComputerImagesandGraphics ZhejiangUniversityHangzhou310027China／／StateKeyLabofCADandCGZhejiangUniversityHangzhou310027China

By using some elementary inequalities, authors in this paper makes further improvement for estimating the heights of Bézier curve and rational Bézier curve. And the termination criterion for subdivision of the ratio... 详细信息

关键词： Rational Bézier curves subdivision Termination criterion.

Expansion properties for finite subdivision rules I Dedicated to the Memories of William P. Thurston and Lei Tan

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Science China Mathematics 2018年第12期61卷 2237-2266页

作者： William J.Floyd Walter R.Parry Kevin M.Pilgrim Department of Mathematics Virginia Tech Department of Mathematics and Statistics Eastern Michigan University Department of Mathematics Indiana University

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion—combinatorial, dynamical, algebraic,and coarse-geometric.

关键词： finite subdivision rule expanding map postcritically finite Thurston map