In this paper, we study the fractal properties of the hyperbolic curve introduced by J. Belair ([Be]). We obtain some conditions of nowhere-differentiability of this kind of curves and its Bouligand dimension, and fin...In this paper, we study the fractal properties of the hyperbolic curve introduced by J. Belair ([Be]). We obtain some conditions of nowhere-differentiability of this kind of curves and its Bouligand dimension, and find a class of curves which are almost everywhere differertiable and have Bouligand dimensions being greater than one simultaneously.展开更多
Cubic algebraic hyperbolic （AH） Bezier curves and AH spline curves are defined with a positive parameter a in the space spanned by {1, t, sinht, cosht}. Modifying the value of a yields a family ofAH Bezier or spline...Cubic algebraic hyperbolic （AH） Bezier curves and AH spline curves are defined with a positive parameter a in the space spanned by {1, t, sinht, cosht}. Modifying the value of a yields a family ofAH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called ＂path of AH curve＂ （AH Bezier and AH spline curves） when a changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.展开更多
The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulat...The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point princi- ple. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.展开更多
In this paper, two new kinds of B-basis functions called algebraic hyperbolic （AH） Bézier basis and AH B-Spline basis are presented in the space Гk=span{ l,t ……f^k-3,sinht,cosht}, in which K is an arbitrary ...In this paper, two new kinds of B-basis functions called algebraic hyperbolic （AH） Bézier basis and AH B-Spline basis are presented in the space Гk=span{ l,t ……f^k-3,sinht,cosht}, in which K is an arbitrary integer larger than or equal to 3. They share most optimal properties as those of the Bézier basis and B-Spline basis respectively and can represent exactly some remarkable curves and surfaces such as the hyperbola, catenary, hyperbolic spiral and the hyperbolic paraboloid. The generation of tensor product surfaces of the AH B-Spline basis have two forms： AH B-Spline surface and AH T-Spline surface.展开更多
We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also...We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also describe how we can draw the picture of an evolute of a hyperbolic plane curvein the Poincaré disk.展开更多
文摘In this paper, we study the fractal properties of the hyperbolic curve introduced by J. Belair ([Be]). We obtain some conditions of nowhere-differentiability of this kind of curves and its Bouligand dimension, and find a class of curves which are almost everywhere differertiable and have Bouligand dimensions being greater than one simultaneously.
基金the National Natural Science Foundation of China (No. 60773179)the National Basic Research Program (973) of China (No. G2004CB318000)the School Scientific Research Foundation of Hangzhou Dianzi University (No. KYS091507070), China
文摘Cubic algebraic hyperbolic （AH） Bezier curves and AH spline curves are defined with a positive parameter a in the space spanned by {1, t, sinht, cosht}. Modifying the value of a yields a family ofAH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called ＂path of AH curve＂ （AH Bezier and AH spline curves） when a changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.
基金Supported by National Natural Science Foundation of China （Grant No. 10971224）
文摘The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point princi- ple. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.
基金Projects supported by the National Natural Science Foundation of China (No. 10371110) and the National Basic Research Program (973) of China (No.G2002CB312101)
文摘In this paper, two new kinds of B-basis functions called algebraic hyperbolic （AH） Bézier basis and AH B-Spline basis are presented in the space Гk=span{ l,t ……f^k-3,sinht,cosht}, in which K is an arbitrary integer larger than or equal to 3. They share most optimal properties as those of the Bézier basis and B-Spline basis respectively and can represent exactly some remarkable curves and surfaces such as the hyperbola, catenary, hyperbolic spiral and the hyperbolic paraboloid. The generation of tensor product surfaces of the AH B-Spline basis have two forms： AH B-Spline surface and AH T-Spline surface.
文摘We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also describe how we can draw the picture of an evolute of a hyperbolic plane curvein the Poincaré disk.
