This paper focuses on the analysis of running conditions and machining processes of conical cam with oscillating follower. We point out the common errors existing in the design and machining of the widely used plane e...This paper focuses on the analysis of running conditions and machining processes of conical cam with oscillating follower. We point out the common errors existing in the design and machining of the widely used plane expansion method of conical cam trough-out line. We show that the motion can be divided into two parts, i.e. the oscillating motion of oscillating bar and the rotary motion of oscillating bar relative to the conical cam. By increasing the rotary motion of oscillating bar, the motion path of tapered roller on oscillating bar (i.e. contour surface of conical cam) can be expanded on the cylinder. Based on these analyses, we present a creative and effective designing and machining method for 3D curve expansion of conical cam with oscillating follower.展开更多
The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation e-quivalent curves:(i) conic curv...The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation e-quivalent curves:(i) conic curves i including parabolas, hyperbolas and ellipses;(ii) generalized monomial curves, including curves of the form x=yr,.r R.r=0,1, in the x-y Cartesian coordinate system;(iii) exponential spiral curves of the form p=Apolar coordinate system.This type of curves has many important properties such as convexity , approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation is or at least in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given.展开更多
In this paper,based on the mean value theorem of differential,a new method of generating conics such as circles and parabolas is given,and the related algorithm for generating conics is designed.
Conical cam mechanism has been widely used in modern machinery and equipment.However,the commonly used planar expansion methods for the design of spatial cam contour produce significant errors,because these methods in...Conical cam mechanism has been widely used in modern machinery and equipment.However,the commonly used planar expansion methods for the design of spatial cam contour produce significant errors,because these methods incorrectly use the distance from the axis of the follower to the main conical cam to replace the corresponding arc length on the conical cam.HSIEH,et al,used analytical methods to achieve higher accuracy,but these analytical methods have their own drawbacks since they are too complicated for practical use.Through the analysis of the errors created during the generation of conical cam contour using the existing expansion methods,this paper proposes to include diverge angle in the calculation of conical cam rotation angle in the equation of conical cam contour expansion.This correction eliminates the error generated by the commonly used methods.Based on the expression of the follower＇s 3D trajectory and the spatial geometry of conical cam,this paper has deduced the planar polar curve equation for determining polar coordinates for the curve of planar expansion outline.Furthermore,this paper provides an example of conical cam contour design based on sinusoidal acceleration variation.According to polar coordinates and the movement of curve equation function expression,this paper applies MATLAB software to solve coordinates for the cam expansion curve and uses AutoCAD software to generate conical cam expansion contour that meets the requirement of the law of motion.The proposed method provides a design process that is simple,intuitive and easy to master and implement.It also avoids the design error in the traditional methods for generating contour of conical cam with oscillating follower that requires high precision.展开更多
In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive ...In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.展开更多
We construct two conical surfaces which take non-coplanar lines as generatrix and rational Bezier curve as ridge-line, and prove that the intersecting line of conical surface has similar properties to Bezier curve. Th...We construct two conical surfaces which take non-coplanar lines as generatrix and rational Bezier curve as ridge-line, and prove that the intersecting line of conical surface has similar properties to Bezier curve. Then, the smoothly blending of two cylinders whose axes are non-coplanar is realized by taking intersecting line of conical surface as axes.展开更多
The classical RSA is vulnerable to low private exponent attacks （LPEA） and has homomorphism. KMOV based on elliptic curve En（a,b） over Zn can resist LPEA but still has homomorphism. QV over En（a,b） not only can ...The classical RSA is vulnerable to low private exponent attacks （LPEA） and has homomorphism. KMOV based on elliptic curve En（a,b） over Zn can resist LPEA but still has homomorphism. QV over En（a,b） not only can resist LPEA but also has no homomorphism. However, QV over En（a,b） requires the existence of points whose order is Mn= 1cm{#Ep（a,b）, #Eq（a,b）}. This requirement is impractical for all general elliptic curves. Besides, the computation over En（a,b） is quite complicated. In this paper, we further study conic curve Cn（a,b） over Zn and its corresponding properties, and advance several key theorems and corollaries for designing digital signature schemes, and point out that Cn（a,b） always has some points whose order is Mn： 1cm{#Ep（a,b）,#Eq（a,b））. Thereby we present an improved QV signature over Cn（a,b）, which inherits the property of non-homomorphism and can resist the Wiener attack. Furthermore, under the same security requirements, the improved QV scheme is easier than that over En（a,b）, with respect plaintext embedding, inverse elements computation, points computation and points＇ order calculation. Especially, it is applicable to general conic curves, which is of great significance to the application of QV schemes.展开更多
The necessary and sufficient conditions are presented for NURBS currves of an arbitrary degree to precisely represent circular arcs. NURBS curves of degree 2 or degree 3 representing circular arcs can be regarded as s...The necessary and sufficient conditions are presented for NURBS currves of an arbitrary degree to precisely represent circular arcs. NURBS curves of degree 2 or degree 3 representing circular arcs can be regarded as special cases of the conditions. It is studied whether two NURBS curves of degree three are equivalent. Classifications of conic section curves represented by cubic or quadratic NURBS curves are proposed.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
Mathematicians are constantly constructing and exploring the properties of abstract objects only because they find them beautiful and interesting. Later, sometimes centuries later, the objects may turn out to be e-nor...Mathematicians are constantly constructing and exploring the properties of abstract objects only because they find them beautiful and interesting. Later, sometimes centuries later, the objects may turn out to be e-normously useful when they are applied to the physical world. There are no more elegant examples of this than the work done in ancient Greece on the four conic-section curve. If a right circular cone is sliced by a plane parallel to its base, the cross section is a circle. Tip the展开更多
基金Project supported by the National Natural Science Foundation of China (No. 50575205)the HiTech Research and Development Program (863) of China (No. 2006AA04Z233)and the Natural Science Foundation of Zhejiang Province (Nos. Y104243 and Y105686), China
文摘This paper focuses on the analysis of running conditions and machining processes of conical cam with oscillating follower. We point out the common errors existing in the design and machining of the widely used plane expansion method of conical cam trough-out line. We show that the motion can be divided into two parts, i.e. the oscillating motion of oscillating bar and the rotary motion of oscillating bar relative to the conical cam. By increasing the rotary motion of oscillating bar, the motion path of tapered roller on oscillating bar (i.e. contour surface of conical cam) can be expanded on the cylinder. Based on these analyses, we present a creative and effective designing and machining method for 3D curve expansion of conical cam with oscillating follower.
文摘The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation e-quivalent curves:(i) conic curves i including parabolas, hyperbolas and ellipses;(ii) generalized monomial curves, including curves of the form x=yr,.r R.r=0,1, in the x-y Cartesian coordinate system;(iii) exponential spiral curves of the form p=Apolar coordinate system.This type of curves has many important properties such as convexity , approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation is or at least in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given.
文摘In this paper,based on the mean value theorem of differential,a new method of generating conics such as circles and parabolas is given,and the related algorithm for generating conics is designed.
基金supported by National Natural Science Foundation of China(Grant No.50645032)Zhejiang Provincial Natural Science Foundation of China(Grant No.Y105686)Ningbo Municipal Natural Science Foundation of China(Grant No.2008A610038)
文摘Conical cam mechanism has been widely used in modern machinery and equipment.However,the commonly used planar expansion methods for the design of spatial cam contour produce significant errors,because these methods incorrectly use the distance from the axis of the follower to the main conical cam to replace the corresponding arc length on the conical cam.HSIEH,et al,used analytical methods to achieve higher accuracy,but these analytical methods have their own drawbacks since they are too complicated for practical use.Through the analysis of the errors created during the generation of conical cam contour using the existing expansion methods,this paper proposes to include diverge angle in the calculation of conical cam rotation angle in the equation of conical cam contour expansion.This correction eliminates the error generated by the commonly used methods.Based on the expression of the follower＇s 3D trajectory and the spatial geometry of conical cam,this paper has deduced the planar polar curve equation for determining polar coordinates for the curve of planar expansion outline.Furthermore,this paper provides an example of conical cam contour design based on sinusoidal acceleration variation.According to polar coordinates and the movement of curve equation function expression,this paper applies MATLAB software to solve coordinates for the cam expansion curve and uses AutoCAD software to generate conical cam expansion contour that meets the requirement of the law of motion.The proposed method provides a design process that is simple,intuitive and easy to master and implement.It also avoids the design error in the traditional methods for generating contour of conical cam with oscillating follower that requires high precision.
基金Supported by the National Natural Science Foundations of China(61070065, 60933007)the Zhejiang Provincial Natural Science Foundation of China(Y6090211)
文摘In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.
文摘We construct two conical surfaces which take non-coplanar lines as generatrix and rational Bezier curve as ridge-line, and prove that the intersecting line of conical surface has similar properties to Bezier curve. Then, the smoothly blending of two cylinders whose axes are non-coplanar is realized by taking intersecting line of conical surface as axes.
基金Supported by the National Natural Science Foundation of China (Grant No. 10128103)
文摘The classical RSA is vulnerable to low private exponent attacks （LPEA） and has homomorphism. KMOV based on elliptic curve En（a,b） over Zn can resist LPEA but still has homomorphism. QV over En（a,b） not only can resist LPEA but also has no homomorphism. However, QV over En（a,b） requires the existence of points whose order is Mn= 1cm{#Ep（a,b）, #Eq（a,b）}. This requirement is impractical for all general elliptic curves. Besides, the computation over En（a,b） is quite complicated. In this paper, we further study conic curve Cn（a,b） over Zn and its corresponding properties, and advance several key theorems and corollaries for designing digital signature schemes, and point out that Cn（a,b） always has some points whose order is Mn： 1cm{#Ep（a,b）,#Eq（a,b））. Thereby we present an improved QV signature over Cn（a,b）, which inherits the property of non-homomorphism and can resist the Wiener attack. Furthermore, under the same security requirements, the improved QV scheme is easier than that over En（a,b）, with respect plaintext embedding, inverse elements computation, points computation and points＇ order calculation. Especially, it is applicable to general conic curves, which is of great significance to the application of QV schemes.
文摘The necessary and sufficient conditions are presented for NURBS currves of an arbitrary degree to precisely represent circular arcs. NURBS curves of degree 2 or degree 3 representing circular arcs can be regarded as special cases of the conditions. It is studied whether two NURBS curves of degree three are equivalent. Classifications of conic section curves represented by cubic or quadratic NURBS curves are proposed.
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Mathematicians are constantly constructing and exploring the properties of abstract objects only because they find them beautiful and interesting. Later, sometimes centuries later, the objects may turn out to be e-normously useful when they are applied to the physical world. There are no more elegant examples of this than the work done in ancient Greece on the four conic-section curve. If a right circular cone is sliced by a plane parallel to its base, the cross section is a circle. Tip the
