Judging or setting weight steady-state of rational Bezier curves and surfaces

Many works have investigated the problem of reparameterizing rational B~zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.     

Geometric meanings of the parameters on rational conic segments

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.     

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc′?a-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface.Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.     

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bézier curve, usually related to an explicit function,is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized monotonicity, called direction monotonicity, is introduced for rational Bézier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine coordinate systems, and it includes the traditional monotonicity as a subcase. By means of it,proper affine coordinate systems may be chosen to make some rational Bézier curves monotonic.Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.     

在几何和参数上表示同一条曲线的两条有理NURBS曲线的权与控制顶点之间的关系

The paper discusses the relationship between weights and control vertices of two rational NURBS curves of degree two or three with all weights larger than zero when they represent the same curve parametrically and geometrically, and gives sufficient and necessary conditions for coincidence of two rational NURBS curves in non-degeneracy case.     

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000,Wu presented two new types of generalized Ball curves,one of which is called an NB1 curve located between the Wang Ball curve and the Said Ball curve.In this article,the authors aim to discuss properties of NB1 curves and surfaces,including the recursive algorithms,conversion algorithms between NB1 and Bézier curves and surfaces, etc.In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.     

The pointwise convergence of p-adic Mbius maps

The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.     

代数等距曲线、曲面的有理性质

A simple method is developed to identify the rationality of offsets to algebraic curves and surfaces which are given either implicitly or parametrically and to parametrize the offsets if they are rational. In particular, we show that offsets to ellipses and hyperbolas are of nonzero genus except for the circles, and somewhat surprisingly, that offsets to paraboioids and hyperboioids with one sheet can all be rationally parametrtised.     

CONVEXITY THEOREM OF PARAMETRIC TRIANGULAR BZIER SURFACES

The author introduces twist vectors of control nets of parametric triangular Bèziersurfaces.With twist vectors and other geometric magnitudes,a sufficient condition forparametric triangular Bèzier surfaces to be convex is offered.Incidently,a necessary andsufficient condition for triangular Bèzier surfaces of parametric form to degenerate to thatof functional form is given.A less stringent sufficient condition is proposed for quadraticBèzier surfaces to be convex.     

GLOBAL STABILITY IN A VIRAL INFECTION MODEL

This paper investigates the global stability of a viral infection model with lytic immune response. If the basic reproductive ratio of the virus is less than or equal to one, by the LaSalle's invariance principle, the disease-free steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is greater than one but less than or equal to a constant, which is defined by the parameters of the model, then the immune-exhausted steady state is globally asymptotically stable. The endemic steady state is globally asymptotically stable if the inverse is valid.     

Pythagorean Bézier速端曲线及其等距线

The Pythagorean Bézier hodograph curves are Bézier curves {x(t), y(t)}, whose hodograph (first-order parametric derivative) components satisfy the Phythagorean condition x'2(t) +y'2(t) = σ'(t) for some polynomial σ(t). For nth degree PB curve (n is odd), its offset curve is represented by rational Bézier curve with (2n - 1)th degree and arc length by polynomial. Specially, the properties of cubic PB curve are studied, its geometric features are discussed and its quasi-Hermite interpolating curve and GC1 composite cubic PB curve are also constructed.     

Lattices, Diophantine equations and applications to rational surfaces

In this article,we study certain quadratic Diophantine equations in Picard lattices of blow-ups of the complex projective plane,and describe their relations with root systems and Weyl group orbits of quasiminuscule fundamental weights.We apply these to study the geometry of certain rational surfaces.     

关于Bernstein多项式的一类不等式及其应用

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.     

Sewing homeomorphism and conformal invariants

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sufficient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Hlder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.     

Global asymptotical stability for a fishery model with seasonal harvesting

A new fishery model is proposed by using the strategy of seasonal harvesting. Sufficient and necessary conditions are established to ensure the existence of a unique equilibrium or a periodic solution by the approach of Poincar′e maps. It is shown that the equilibrium or the periodic solution is globally asymptotically stable. Numerical examples are provided to demonstrate the model dynamics and some biological implications are given as well.     

Number of embeddings of circular and Mbius ladders on surfaces

There are many results on the flexibility of(general) embeddings of graphs,but few are known about that of strong embeddings.In this paper,we study the flexibility of strong embeddings of circular and Mbius ladders on the projective plane and the Klein bottle by using the joint tree model of embeddings.The numbers of(nonequivalent) general embeddings and strong embeddings of circular and Mbius ladders on these two nonorientable surfaces are obtained,respectively.And the structures of those strong embeddings are described.     

EXPONENTIAL STABILITY AND PERIODIC SOLUTION OF HYBRID BIDIRECTIONAL ASSOCIATIVE MEMORY NEURAL NETWORKS WITH DISCRETE DELAYS

In this paper, we study the existence, uniqueness, and the global exponential stability of the periodic solution and equilibrium of hybrid bidirectional associative memory neural networks with discrete delays. By ingeniously importing real parameters di > 0 (i = 1,2, …, n) which can be adjusted, making use of the Lyapunov functional method and some analysis techniques, some new sufficient conditions are established. Our results generalize and improve the related results in [9]. These conditions can be used both to design globally exponentially stable and periodical oscillatory hybrid bidirectional associative neural networks with discrete delays, and to enlarge the area of designing neural networks. Our work has important significance in related theory and its application.     

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

For two rational quadratic B-spline curves with same control vertexes, the cross ratio of four eollinear points are represented; which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the twocurves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Beeier curves, the value is generally related with the loeation of the ray, and the necessary and sufficient condition o5 the ratio being independent of the ray‘s loeation is showed. Alsn another cross ratio o5 the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initlal vertex intersects respectivdy with the curve segmentt the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only whh the ray‘s location, butnot with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.     

Empirical likelihood estimation of discretely sampled processes of OU type

This paper presents an empirical likelihood estimation procedure for parameters of the discretely sampled process of Ornstein-Uhlenbeck type. The proposed procedure is based on the condi- tional characteristic function, and the maximum empirical likelihood estimator is proved to be consistent and asymptotically normal. Moreover, this estimator is shown to be asymptotically efficient under some mild conditions. When the background driving Lévy process is of type A or B, we show that the intensity parameter c...     

本刊英文版2009年52卷第4期（617-848）摘要

Remarks on lines and minimal rational curves MOK Ngaiming & SUN XiaoTao Abstract We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.Keywords: minimal rational curves, lines, moduli spaces, stable bundles MSC(2000): 14H60, 14D20, 14F05