This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4...This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.展开更多
Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference...Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control points.Control parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also included.AMS(MOS) : 65D05 , 65D15 , 65D17.展开更多
A smooth interpolatory subdivision algorithm for the generation of surfaces over arbi-trary triangulations is introduced and its convergence properties over nonuniform triangulationsstudied. For uniform data, this met...A smooth interpolatory subdivision algorithm for the generation of surfaces over arbi-trary triangulations is introduced and its convergence properties over nonuniform triangulationsstudied. For uniform data, this method is a generalization of the analysis for univariatesubdivision algorithms and for nonuniform data, an extraordinary point analysis is introducedand the local subdivision matrix anaiysis presented. It is proved that the algorithm producessmooth surfaces over arbitrary triangular networks provided the shape parameters are kept with-in an appropriate range. Finally, two graphical examples of surface interpolation overnonuniform data are given to show the smoothing process of the algorithm.AMS (MOS): 65D05, 65D15,65D17.展开更多
This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed wor...This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.展开更多
Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting v...Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.展开更多
A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to it...A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh M such that limit surface of M would interpolate M. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes. The meshes considered here can be open or closed.展开更多
An improved ternary subdivision interpolation scheme was developed for computer graphics ap- plications that can manipulate open control polygons unlike the previous ternary scheme, with the resulting curve proved t...An improved ternary subdivision interpolation scheme was developed for computer graphics ap- plications that can manipulate open control polygons unlike the previous ternary scheme, with the resulting curve proved to be still C2-continuous. Parameterizations of the limit curve near the two endpoints are given with expressions for the boundary derivatives. The split joint problem is handled with the interpolating ter- nary subdivision scheme. The improved scheme can be used for modeling interpolation curves in computer aided geometric design systems, and provides a method for joining two limit curves of interpolating ternary subdivisions.展开更多
Many-knot spline interpolating is a class of curves and surfaces fitting method presentedin 1974. Many-knot spline interpolating curves are suitable to computer aided geometric design anddata points interpolation. In ...Many-knot spline interpolating is a class of curves and surfaces fitting method presentedin 1974. Many-knot spline interpolating curves are suitable to computer aided geometric design anddata points interpolation. In this paped, the properties of many-knot spline interpolating curves arediscussed and their applications in font design are considered. The differences between many-knotspline interpolating curves and the curves genoaed by exceeding-lacking adjuStment algorithm aregiven.展开更多
We present generalized and unified families of (2n)-point and (2n − 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomialfor any integers n ≥ 2 and p ≥ 3. Almost all existing even-poin...We present generalized and unified families of (2n)-point and (2n − 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomialfor any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-pointinterpolating schemes of lower and higher arity belong to this family of schemes. Wealso present tensor product version of generalized and unified families of schemes.Moreover error bounds between limit curves and control polygons of schemes arealso calculated. It has been observed that error bounds decrease when complexityof the scheme decrease and vice versa. Furthermore, error bounds decrease withthe increase of arity of the schemes. We also observe that in general the continuityof interpolating scheme do not increase by increasing complexity and arity of thescheme.展开更多
文摘This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.
文摘Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control points.Control parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also included.AMS(MOS) : 65D05 , 65D15 , 65D17.
文摘A smooth interpolatory subdivision algorithm for the generation of surfaces over arbi-trary triangulations is introduced and its convergence properties over nonuniform triangulationsstudied. For uniform data, this method is a generalization of the analysis for univariatesubdivision algorithms and for nonuniform data, an extraordinary point analysis is introducedand the local subdivision matrix anaiysis presented. It is proved that the algorithm producessmooth surfaces over arbitrary triangular networks provided the shape parameters are kept with-in an appropriate range. Finally, two graphical examples of surface interpolation overnonuniform data are given to show the smoothing process of the algorithm.AMS (MOS): 65D05, 65D15,65D17.
文摘This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.
文摘Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.
基金supported by NSF of USA under Grant No.DMI-0422126The last author is supported by the National Natural Science Foundation of China under Grant Nos.60625202,60533070.
文摘A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh M such that limit surface of M would interpolate M. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes. The meshes considered here can be open or closed.
基金Supported by the National Natural Science Foundation of China(No. 60273013)the Specialized Research Fund for the DoctoralProgram of Higher Education of China (No. 20010003048)andResearch Grants Council of Hong Kong (RGC) (No. CUHK4189/01E)
文摘An improved ternary subdivision interpolation scheme was developed for computer graphics ap- plications that can manipulate open control polygons unlike the previous ternary scheme, with the resulting curve proved to be still C2-continuous. Parameterizations of the limit curve near the two endpoints are given with expressions for the boundary derivatives. The split joint problem is handled with the interpolating ter- nary subdivision scheme. The improved scheme can be used for modeling interpolation curves in computer aided geometric design systems, and provides a method for joining two limit curves of interpolating ternary subdivisions.
文摘Many-knot spline interpolating is a class of curves and surfaces fitting method presentedin 1974. Many-knot spline interpolating curves are suitable to computer aided geometric design anddata points interpolation. In this paped, the properties of many-knot spline interpolating curves arediscussed and their applications in font design are considered. The differences between many-knotspline interpolating curves and the curves genoaed by exceeding-lacking adjuStment algorithm aregiven.
基金The first author was supported by Pakistan Program for Collaborative Research-foreign visit of local faculty member,Higher Education Commission(HEC)PakistanThe second author was supported by Indigenous Ph.D.Scholarship Scheme of HEC PakistanThe third author was supported by NSF of China(No.61073108)
文摘We present generalized and unified families of (2n)-point and (2n − 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomialfor any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-pointinterpolating schemes of lower and higher arity belong to this family of schemes. Wealso present tensor product version of generalized and unified families of schemes.Moreover error bounds between limit curves and control polygons of schemes arealso calculated. It has been observed that error bounds decrease when complexityof the scheme decrease and vice versa. Furthermore, error bounds decrease withthe increase of arity of the schemes. We also observe that in general the continuityof interpolating scheme do not increase by increasing complexity and arity of thescheme.
