Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

Degree elevation of nurbs curves by weighted blossom

An algorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation. The elevation method is introduced that consists of the following steps: (a) decompose the NURBS curve into piecewise rational Bézier curves, (b) elevate the degree of each rational Bézier piece, and (c) compose the piecewise rational Bézier curves into NURBS curve.     

以节点与权因子修改为基础的4阶NURBS受限形状控制

改变k阶NURBS曲线的节点,会产生一个单参数NURBS曲线族,该曲线族的包络是用相同控制顶点定义的k-a阶NURBS曲线,这里a是所改变的节点的重数.论文运用这项理论结果,提出了几种建立在修改一个节点与两个连续权因子基础上的4阶NURBS形状控制方法,该方法要受一定的位置与切线方向的约束.     

On the number of Mordell–Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. In this paper we prove, for a cubic surface containing a pair of skew rational lines over a field with at least 13 elements, that the rational points are generated by just one point. We also prove a cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves over the rationals.     

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.     

基于函数值的有理三次插值样条曲线的区域控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题.构造了一种基于函数值的分母为三次的C~1连续有理三次插值样条.这种有理三次插值样条中含有二个调节参数,因而给约束控制带来了方便.对该种插值曲线的区域控制问题进行了研究,给出了将其约束于给定的折线、二次曲线之上、之下或之间的充分条件.最后给出了数值例子.     

Some families of Mordell curves associated to cubic fields

We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X³+Y³+Z³=0.     

Approximation of a Continuous Curve by its Bernstein-B��zier Operator

In this paper we consider integer and rational parametric Bézier curves and study the distance between the curve and its control polygon. To measure the distance we use first and second order moduli of smoothness of vector-valued function. We consider also NURBS curves with equidistant knots. Some direct approximation theorems will be presented.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

局部形状可调整的三次有理B样条插值曲线

利用三次非均匀有理B样条,给出了一种构造局部插值曲线的方法,生成的插值曲线是C2连续的.曲线表示式中带有一个局部形状参数,随着一个局部形状参数值的增大,所给曲线将局部地接近插值点构成的控制多边形.基于三次非均匀有理B样条函数的局部单调性和一种保单调性的准则,给出了所给插值曲线的保单调性的条件.     

Shape control of curve design by weighted rational spline

Controlling the convexity and the strain energy of the interpolating curve can be carried out by controlling the second-order derivative of the interpolating function. In [1], the rational cubic spline with linear denominator has been used to constrain the convexity and the strain energy of the interpolating curves, but it does not work in some case. This paper deals with the weighted rational cubic spline with linear denominator for this kind of constraint, the sufficient and necessary condition for controlling the convexity and strain energy of the interpolating curves are derived, and a numerical example is given.     

Cubic Curves,Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a ‘shared secret’ related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.     

利用近似能量极小构造平面C1三次Hermite插值曲线

研究了利用近似能量极小构造平面$C^1$三次Hermite插值曲线的方法.该方法的主要的目是求出$C^1$三次Hermite插值曲线的最佳切矢.通过将应变能、曲率变化能和组合能的近似函数极小化,得到了求解最佳切矢的线性方程组.通过求解发现,近似曲率变化能极小不存在唯一解, 而近似应变能极小和近似组合能极小由于方程系统的系数矩阵为严格对角占优故都存在唯一解.最后, 通过实例表明了本文方法构造平面$C^1$三次Hermite插值曲线的有效性.     

有理B-样条曲线、曲面的包络性质

研究有理Bezier曲线和B-样条曲线、曲面的包络性质,愈来愈广泛,因为它从包络磨光的角度解释了曲线、曲面的一种几何构造特征,形象地说明了模型是由多边形或多面体逐步磨光的结果.     

Verifying the Implicitization Formulae for Degree N Rational Bézier Curves

This is a continuation of short communication$^{[1]}$. In [1] a verification of the implicitization equation for degree two rational Bézier curves is presented which does not require the use of resultants. This paper presents these verifications in the general cases, i.e., for degree $n$ rational Bézier curves. Thus some interesting interplay between the structure of the $n×n$ implicitization matrix and the de Casteljau algorithm is revealed.     

NURBS曲线保凸性质的几何证明

NURBS curves are convexity preserving, i.e. once the control polygon is convex, the associated NURBS curve will also be convex. In this paper this property is proved geometrically.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000