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...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.展开更多
花键曲线和表面在 CAD 和计算机图形起一个重要作用。在这份报纸，我们建议立方的一致 B 花键的几延期。然后，我们在场插入内推的延期 -B-spline 基于新 B 花键和单个相配技术。延期的优点是他们有全球、本地的形状参数。而且，我们也...花键曲线和表面在 CAD 和计算机图形起一个重要作用。在这份报纸，我们建议立方的一致 B 花键的几延期。然后，我们在场插入内推的延期 -B-spline 基于新 B 花键和单个相配技术。延期的优点是他们有全球、本地的形状参数。而且，我们也在数据插值和多角形的形状变丑调查他们的应用。展开更多
Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right heli...Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right helicoid and catenoid as special examples. We firstly construct the Bézier-like basis called AHT Bézier basis in the space spanned by {1, t, sint, cost, sinht, cosht}, t∈[0,α], α∈[0,5π/2]. Then we propose the control mesh representation of the generalized helicoid using the AHT Bézier basis. This kind of representation enables generating the minimal surfaces using the de Casteljau-like algorithm in CAD/CAGD mod- elling systems.展开更多
We decompose the problem of the optimal multi-degree reduction of Bézier curves with corners constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree re...We decompose the problem of the optimal multi-degree reduction of Bézier curves with corners constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree reduction, and doing optimal degree reduction without making high order interpolations at the two endpoints. Further, we convert the second subproblem into multi-degree reduction of Jacobi polynomials. Then, we can easily derive the optimal solution using orthonormality of Jacobi polynomials and the least square method of unequally accurate measurement. This method of 'divide and conquer' has several advantages including maintaining high continuity at the two endpoints of the curve, doing multi-degree reduction only once, using explicit approximation expressions, estimating error in advance, low time cost, and high precision. More importantly, it is not only deduced simply and directly, but also can be easily extended to the degree reduction of surfaces. Finally, we present two examples to demonstrate the effectiveness of our algorithm.展开更多
As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane ca...As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane can, therefore, be partitioned into regions labelled according to the characterization of the curve when the fourth point is in each region. This partitioned plane is called a "characterization diagram". By moving one of the control points but fixing the rest, one can induce different characterization diagrams. In this paper, we investigate the relation among all different characterization diagrams of cubic C-curves based on the singularity conditions proposed by Yang and Wang (2004). We conclude that, no matter what the C-curve type is or which control point varies, the characterization diagrams can be obtained by cutting a common 3D characterization space with a corresponding plane.展开更多
In this paper, we present two new unified mathematics models of conics and polynomial curves, called algebraic hyperbolic trigonometric （ AHT） Bezier curves and non-uniform algebraic hyperbolic trigonometric （ NUAH...In this paper, we present two new unified mathematics models of conics and polynomial curves, called algebraic hyperbolic trigonometric （ AHT） Bezier curves and non-uniform algebraic hyperbolic trigonometric （ NUAHT） B-spline curves of order n, which are generated over the space span{sin t, cos t, sinh t, cosh t, 1, t,..., t^n-5}, n 7〉 5. The two kinds of curves share most of the properties as those of the Bezier curves and B-spline curves in polynomial space. In particular, they can represent exactly some remarkable transcendental curves such as the helix, the cycloid and the catenary. The subdivision formulae of these new kinds of curves are also given. The generations of the tensor product surfaces are straightforward. Using the new mathematics models, we present the control mesh representations of two classes of minimal surfaces.展开更多
This paper investigates how to maintain an efficient dynamic ordered set of bit strings, which is an important problem in the field of information search and information processing. Generally, a dynamic ordered set is...This paper investigates how to maintain an efficient dynamic ordered set of bit strings, which is an important problem in the field of information search and information processing. Generally, a dynamic ordered set is required to support 5 essential operations including search, insertion, deletion, max-value retrieval and next-larger-value retrieval. Based on previous research fruits, we present an advanced data structure named rich binary tree （RBT）, which follows both the binary-search-tree property and the digital-search-tree property. Also, every key K keeps the most significant difference bit （MSDB） between itself and the next larger value among K＇s ancestors, as well as that between itself and the next smaller one among its ancestors. With the new data structure, we can maintain a dynamic ordered set in O（L） time. Since computers represent objects in binary mode, our method has a big potential in application. In fact, RBT can be viewed as a general-purpose data structure for problems concerning order, such as search, sorting and maintaining a priority queue. For example, when RBT is applied in sorting, we get a linear-time algorithm with regard to the key number and its performance is far better than quick-sort. What is more powerful than quick-sort is that RBT supports constant-time dynamic insertion/deletion.展开更多
We constructed a single C-Bezier curve with a shape parameter for G^2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the ...We constructed a single C-Bezier curve with a shape parameter for G^2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bezier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples This method has the following three advantages： （1） the pattern is unified; （2） the parameter able to adjust the shape of the transition curve is available; （3） the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement.展开更多
In this paper, we propose a novel free-form deformation （FFD） technique, RDMS-FFD （Rational DMS-FFD）, based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and...In this paper, we propose a novel free-form deformation （FFD） technique, RDMS-FFD （Rational DMS-FFD）, based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and combines more deformation techniques than previous FFD methods in a consistent framework, such as local deformation, control lattice of arbitrary topology, smooth deformation, multiresolution deformation and direct manipulation of deformation. We first introduce the rational DMS-spline volume by directly generalizing the previous results related to DMS-splines. How to generate a tetrahedral domain that approximates the shape of the object to be deformed is also introduced in this paper. Unlike the traditional FFD techniques, we manipulate the vertices of the tetrahedral domain to achieve deformation results. Our system demonstrates that RDMS-FFD is powerful and intuitive in geometric modeling.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 60473130) and the National Basic Research Program (973) of China (No. G2004CB318000)
文摘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.
基金Project supported by the National Natural Science Foundation of China (No. 60473130) and the National Basic Research Program (973) of China (No. 2004CB318000)
文摘Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right helicoid and catenoid as special examples. We firstly construct the Bézier-like basis called AHT Bézier basis in the space spanned by {1, t, sint, cost, sinht, cosht}, t∈[0,α], α∈[0,5π/2]. Then we propose the control mesh representation of the generalized helicoid using the AHT Bézier basis. This kind of representation enables generating the minimal surfaces using the de Casteljau-like algorithm in CAD/CAGD mod- elling systems.
基金supported by the National Natural Science Foundation of China (No. 60873111)the National Basic Research Program (973) of China (No. 2004CB719400)
文摘We decompose the problem of the optimal multi-degree reduction of Bézier curves with corners constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree reduction, and doing optimal degree reduction without making high order interpolations at the two endpoints. Further, we convert the second subproblem into multi-degree reduction of Jacobi polynomials. Then, we can easily derive the optimal solution using orthonormality of Jacobi polynomials and the least square method of unequally accurate measurement. This method of 'divide and conquer' has several advantages including maintaining high continuity at the two endpoints of the curve, doing multi-degree reduction only once, using explicit approximation expressions, estimating error in advance, low time cost, and high precision. More importantly, it is not only deduced simply and directly, but also can be easily extended to the degree reduction of surfaces. Finally, we present two examples to demonstrate the effectiveness of our algorithm.
基金Project supported by the National Natural Science Foundation of China (No. 60473130)the National Basic Research Program(973) of China (No. 2004CB318000)
文摘As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane can, therefore, be partitioned into regions labelled according to the characterization of the curve when the fourth point is in each region. This partitioned plane is called a "characterization diagram". By moving one of the control points but fixing the rest, one can induce different characterization diagrams. In this paper, we investigate the relation among all different characterization diagrams of cubic C-curves based on the singularity conditions proposed by Yang and Wang (2004). We conclude that, no matter what the C-curve type is or which control point varies, the characterization diagrams can be obtained by cutting a common 3D characterization space with a corresponding plane.
基金This work is supported by the National Natural Science Foundation of China under Grant Nos.60473130,10371110the National Grand Fundamental Research 973 Program of China under Grant No.2004CB318000.
文摘In this paper, we present two new unified mathematics models of conics and polynomial curves, called algebraic hyperbolic trigonometric （ AHT） Bezier curves and non-uniform algebraic hyperbolic trigonometric （ NUAHT） B-spline curves of order n, which are generated over the space span{sin t, cos t, sinh t, cosh t, 1, t,..., t^n-5}, n 7〉 5. The two kinds of curves share most of the properties as those of the Bezier curves and B-spline curves in polynomial space. In particular, they can represent exactly some remarkable transcendental curves such as the helix, the cycloid and the catenary. The subdivision formulae of these new kinds of curves are also given. The generations of the tensor product surfaces are straightforward. Using the new mathematics models, we present the control mesh representations of two classes of minimal surfaces.
基金Supported by the National Natural Science Foundation of China (Grant No. 60873111)the National Basic Research Program of China(Grant No. 2004CB719400)
文摘This paper investigates how to maintain an efficient dynamic ordered set of bit strings, which is an important problem in the field of information search and information processing. Generally, a dynamic ordered set is required to support 5 essential operations including search, insertion, deletion, max-value retrieval and next-larger-value retrieval. Based on previous research fruits, we present an advanced data structure named rich binary tree （RBT）, which follows both the binary-search-tree property and the digital-search-tree property. Also, every key K keeps the most significant difference bit （MSDB） between itself and the next larger value among K＇s ancestors, as well as that between itself and the next smaller one among its ancestors. With the new data structure, we can maintain a dynamic ordered set in O（L） time. Since computers represent objects in binary mode, our method has a big potential in application. In fact, RBT can be viewed as a general-purpose data structure for problems concerning order, such as search, sorting and maintaining a priority queue. For example, when RBT is applied in sorting, we get a linear-time algorithm with regard to the key number and its performance is far better than quick-sort. What is more powerful than quick-sort is that RBT supports constant-time dynamic insertion/deletion.
基金supported by the National Natural Science Foundation ofChina (No. 60673031)the National Basic Research Program(973) of China (No. 2004CB719400)
文摘We constructed a single C-Bezier curve with a shape parameter for G^2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bezier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples This method has the following three advantages： （1） the pattern is unified; （2） the parameter able to adjust the shape of the transition curve is available; （3） the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement.
基金supported by the National Natural Science Foundation of China under Grant Nos. 60773179 and 60473130the National Basic Research 973 Program of China under Grant No. 2004CB318000
文摘In this paper, we propose a novel free-form deformation （FFD） technique, RDMS-FFD （Rational DMS-FFD）, based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and combines more deformation techniques than previous FFD methods in a consistent framework, such as local deformation, control lattice of arbitrary topology, smooth deformation, multiresolution deformation and direct manipulation of deformation. We first introduce the rational DMS-spline volume by directly generalizing the previous results related to DMS-splines. How to generate a tetrahedral domain that approximates the shape of the object to be deformed is also introduced in this paper. Unlike the traditional FFD techniques, we manipulate the vertices of the tetrahedral domain to achieve deformation results. Our system demonstrates that RDMS-FFD is powerful and intuitive in geometric modeling.
