The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve ...The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.展开更多
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 t...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 rational Bézier curve is also improved. The conclusion of the extreme value problem is thus further confirmed.展开更多
By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents so...By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents some explicit formulas of the first and second derivatives of rational triangular Bézier surface with respect to each variable (including the mixed derivative) and derives some estimations of bound both on the direction and magnitude of the corresponding derivatives. All the results above have value not only in surface theory but also in practice.展开更多
A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a so...A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a solid model and having coincident edges but no inner-intersection among them. The proposed method calculates the silhouettes of the surfaces without tessellating them into triangle meshes commonly used in previous methods so that arbitrary precision can be obtained. The computed sil- houettes of visible surfaces are projected onto a plane orthogonal to the parallel light. Then their spatial relationship is applied to calculate the boundaries of mutual-occlusion regions. As the connectivity of the surfaces on the solid model is taken into account, a surface clustering technique is also employed and the mutual-occlusion calculation is accelerated. Experimental results showed that our method is efficient and robust, and can also handle complex shapes with arbitrary precision.展开更多
Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theo...Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theorem connected to Poisson curves was obtained. Some examples in this paper were given by the visual results.展开更多
Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive man...Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.展开更多
We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previ...We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previous GPU acceleration algorithm and tessel- lation shader algorithms.展开更多
This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method...This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present formulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to investtigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behavior of laminated shells of revolution.展开更多
Coincidence detection of two curves or two surfaces has wide application in computer-aided design(CAD)and computer-aided geometric design(CAGD).Proper reparameterization is the most complicated part in the detection.T...Coincidence detection of two curves or two surfaces has wide application in computer-aided design(CAD)and computer-aided geometric design(CAGD).Proper reparameterization is the most complicated part in the detection.This paper presents and proves the efficient and necessary coincidence condition for two rational Bézier curves in a new way.It also proposes an effective and efficient proper reparameterization method,Algorithm 1,for detecting a rational Bézier curve which can be degenerated into a new one of a lower degree.A numerical proper reparameterization method,Algorithm 2,and examples are also presented.Algorithm 1 is up to ten times faster than other prevailing methods,and Algorithm 2 is twice as fast and half as close as other prevailing methods.New CAD systems using Algorithm 1 and Algorithm 2 will hold accuracy and little computation time.展开更多
In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and rational Bézier curves can ...In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and rational Bézier curves can be improved greatly.展开更多
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 co...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.展开更多
Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bé...Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.展开更多
基金Project supported by the National Basic Research Program (973) of China (No. 2002CB312101) and the National Natural Science Foun-dation of China (Nos. 60373033 and 60333010)
文摘The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.
文摘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 rational Bézier curve is also improved. The conclusion of the extreme value problem is thus further confirmed.
基金Project supported by the National Natural Science Foundation of China (Nos. 60373033 & 60333010), the National Natural Science Foundation for Innovative Research Groups (No. 60021201), and the National Basic Research Program (973) of China (No. 2002CB312101)
文摘By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents some explicit formulas of the first and second derivatives of rational triangular Bézier surface with respect to each variable (including the mixed derivative) and derives some estimations of bound both on the direction and magnitude of the corresponding derivatives. All the results above have value not only in surface theory but also in practice.
基金Project supported by the National Basic Research Program (973) of China (No. 2002CB312106) and the National Natural Science Foundation of China (Nos. 60533070, and 60403047). The third author was supported by the project sponsored by a Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 200342) and a Program for New Century Excellent Talents in Uni-versity (No. NCET-04-0088), China
文摘A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a solid model and having coincident edges but no inner-intersection among them. The proposed method calculates the silhouettes of the surfaces without tessellating them into triangle meshes commonly used in previous methods so that arbitrary precision can be obtained. The computed sil- houettes of visible surfaces are projected onto a plane orthogonal to the parallel light. Then their spatial relationship is applied to calculate the boundaries of mutual-occlusion regions. As the connectivity of the surfaces on the solid model is taken into account, a surface clustering technique is also employed and the mutual-occlusion calculation is accelerated. Experimental results showed that our method is efficient and robust, and can also handle complex shapes with arbitrary precision.
文摘Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theorem connected to Poisson curves was obtained. Some examples in this paper were given by the visual results.
基金Supported by Natural Science Foundation of China(No.10871208,No.60970097)
文摘Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.
基金Supported by the National Natural Science Foundation of China(61170138 and 61472349)
文摘We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previous GPU acceleration algorithm and tessel- lation shader algorithms.
文摘This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present formulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to investtigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behavior of laminated shells of revolution.
基金supported by the National Natural Science Foundation of China under Grant Nos.61972120 and 62021002 and the Haihe Lab of ITAI Project(22HHXCJC00006)supported by the National Key Research and Development Program of China under Grant Nos.2020YFB1708900 and TC190A4DA/3.
文摘Coincidence detection of two curves or two surfaces has wide application in computer-aided design(CAD)and computer-aided geometric design(CAGD).Proper reparameterization is the most complicated part in the detection.This paper presents and proves the efficient and necessary coincidence condition for two rational Bézier curves in a new way.It also proposes an effective and efficient proper reparameterization method,Algorithm 1,for detecting a rational Bézier curve which can be degenerated into a new one of a lower degree.A numerical proper reparameterization method,Algorithm 2,and examples are also presented.Algorithm 1 is up to ten times faster than other prevailing methods,and Algorithm 2 is twice as fast and half as close as other prevailing methods.New CAD systems using Algorithm 1 and Algorithm 2 will hold accuracy and little computation time.
基金Supported by the National Natural Science Foundation of China (60303015,60333010).
文摘In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and rational Bézier curves can be improved greatly.
文摘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.
基金supported by the Foundation of State Key Basic Research 973 Item(Grant No.2004CB719400)the National Natural Science Foundation of China(Grant Nos.60373033&60333010)National Natural Science Foundation for Innovative Research Groups(Grant No.60021201).
文摘Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.
