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
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
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
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
A DP curve is a new kind of parametric curve defined by Delgado and Pefla(2003);it has very good properties when used in both geometry and algebra,i.e.,it is shape preserving and has a linear time complexity for evalu...A DP curve is a new kind of parametric curve defined by Delgado and Pefla(2003);it has very good properties when used in both geometry and algebra,i.e.,it is shape preserving and has a linear time complexity for evaluation.It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape,and the disadvantage of the Bézier curve that is shape preserving but slow for evaluation.It also has potential applications in computer-aided design and manufacturing(CAD/CAM) systems.As conic section is often used in shape design,this paper deduces the necessary and sufficient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves.The main idea is based on the transformation relationship between low degree DP basis and Bernstein basis,and the representation theory of conics in rational low degree Bézier form.The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form,i.e.,give positions of the control points and values of the weights of rational cubic or quartic DP conics.Finally,several numerical examples are presented to validate the effectiveness of the method.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
The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equ...The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.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 mappings of the mappings represented by the expressions of ...Using algebraic and geometric methods, functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic 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 parametric angles of the selected point and two endpoints on the conic segment, as展开更多
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.
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.
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 (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.
基金supported by the National Natural Science Foundation of China (Nos.60873111 and 60933007)the Natural Science Foundation of Zhejiang Province,China (No.Y6090211)
文摘A DP curve is a new kind of parametric curve defined by Delgado and Pefla(2003);it has very good properties when used in both geometry and algebra,i.e.,it is shape preserving and has a linear time complexity for evaluation.It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape,and the disadvantage of the Bézier curve that is shape preserving but slow for evaluation.It also has potential applications in computer-aided design and manufacturing(CAD/CAM) systems.As conic section is often used in shape design,this paper deduces the necessary and sufficient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves.The main idea is based on the transformation relationship between low degree DP basis and Bernstein basis,and the representation theory of conics in rational low degree Bézier form.The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form,i.e.,give positions of the control points and values of the weights of rational cubic or quartic DP conics.Finally,several numerical examples are presented to validate the effectiveness of the method.
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 National Natural Science Foundation of China(Grants 61100129)Open Program of Key Laboratory of Intelligent Information Processing,Institute of Computing Technology,Chinese Academy of Sciences(IIP2014-7)
文摘The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.
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 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 parametric angles of the selected point and two endpoints on the conic segment, as
