几种有理插值函数的逼近性质

1　引　　言在曲线和曲面设计中,样条插值是有用的和强有力的工具.不少作者已经研究了很多种类型的样条插值[1,2,3,4].近些年来,有理插值样条,特别是三次有理插值样条,以及它们在外型控制中的应用,已有了不少工作[5,6,7].有理插值样条的表达式中有某些参数,正是由于这些参数,有理插值样条在外型控制中充分显示了它的灵活性;但也正是由于这些参数,使它的逼近性质的研究增加了困难.因此,关于有理插值样条的逼近性质的研究很少见诸文献.本文在第二节首先叙述几种典型的有理插值样条,其中包括分母为一次、二次的三次有理插值样条和仅基于函数值…     

Inflections and singularity on parametric rational cubic curves

Summary. This paper considers the distribution of inflection points and singularity on the parametric rational cubic curve, using much algebraic manipulation. Its use allows one to find a shape preserving interpolatory rational cubic curve of a planar data. Some numerical examples are given to illustrate usefulness of the method. Received April 30, 1995 / Revised version received January 15, 1996     

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.     

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.     

DEVELOPABLE SURFACE PATCHES BOUNDED BY NURBS CURVES

In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bézier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.     

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.     

空间有理三次Bezier曲线的射影变换和权系数的几何性质

本文讨论了空间有理三次Ｂｅｚｉｅｒ曲线的射影变换和权系数的一系列几何性质。其权系数组成构成了控制四顶点基下的权心的齐次坐标;权心是六个特殊平面的公共交点。含权心和曲线“肩点”的某四个共线点之比恒为常数３;权心可作为有理曲线所在射影坐标系的单位点;此有理曲线是对应整有理曲线在射影变换下的象,此变换把控制四面体的形心映为权心;权系数是此射影变换的特征值（差－常数因子）;权系数是变换前后两曲线上对应点关     

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.     

P~3中3次曲面与平面4次曲线间的关系

本文通过二重覆盖建立了P3中仅有二重的孤立奇点的3次曲面与平面带孤立奇点的4次曲线的对应关系，这种关系使我们能用平面4次曲线的奇点分类及几何性质研究P3中3次曲面的奇点类型及几何性质     

一种四次有理插值样条及其逼近性质

1引言有理样条函数是多项式样条函数的一种自然推广,但由于有理样条空间的复杂性,所以有关它的研究成果不象多项式样条那样完美,许多问题还值得进一步的研究．近几十年来,有理插值样条,特别是有理三次有理插值样条,由于它们在曲线曲面设计中的应用,已有许多学者进行了深入研究,取得了一系列的成果(见[1]-[7])．但四次有理插值样条由于其构造所花费的计算量太大以及在使用上很不方便而让人们忽视了其重要的应用价值,因此很少有人研究他们．实际上,在某些情况下四次有理插值样条有其独特的应用效果,如文[8]建立的一种具有局部插值性质的分母为二次的四次有理样条,即一个剖分     

A two-dimensional Minkowski ?(x) function

A function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to ?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit interval. It is also shown that this new function satisfies an analog to the fact that ?(x), while increasing and continuous, has derivative zero almost everywhere.     

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.     

C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

A family of cyclic cubic polynomials whose roots are systems of fundamental units

We give a family of cyclic cubic polynomials whose roots are systems of fundamental units of the splitting fields. These polynomials are constructed by a linear fractional transformation from Shanks’ polynomials with rational coefficients.     

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.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch

A local C ¹ positivity preserving scheme is developed using Bernstein-Bézier rational cubic function. The domain is triangulated by Delaunay triangulation method. Simple sufficient conditions are derived on the inner and boundary Bézier ordinates to preserve the shape of positive data. These inner and boundary Bézier ordinates involve weights in their definition. In any triangular patch if the Bézier ordinates do not satisfy the derived conditions of positivity, then these are modified by the weights (free parameters) involved in the construction of Bernstein-Bézier rational cubic function to preserve the shape of positive scattered data.     

Permutation invariant properties of primitive cubic quadruples

Based on a specific quadratic Hopf map between the Euclidean spaces of dimension four and three that is associated with Euler’s complete rational parameterization of the four cubes equation, we study the permutation invariant properties of the primitive integer cubic quadruples that solve this equation. Fixing the coordinate with maximum height and taking it positive, our main result describes the six positive primitive triples that leave it invariant under the inverted cubic map to this Hopf map and permute the remaining integer coordinates. The obtained invariant primitive triples are ordered in the so-called integer triple ordering, so that the minimum triple with respect to this ordering determines each primitive cubic quadruple uniquely. Implications for the counting and enumeration of all primitive cubic quadruples are mentioned. A list of all primitive cubic quadruples with positive maximum height below 100 and their minimum invariant triples is given. The relationship with the famous Taxicab and Cabtaxi numbers is also explained.     

McMullen's root-finding algorithm for cubic polynomials

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.