On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Low degree rational spline interpolation

For a strictly monotone functionf on [a,b] we describe the possibility of finding an interpolating rational splineS of the formS(x)=c ₀ +c ₁ x/(1+d ₁ x) on each subinterval of the grida=x ₀ <x ₁ <...<x _n =b. This leads to a nonlinear system for which we get the local existence and uniqueness of a solution. We prove that ‖S−f‖_∞=O(h ³). Numerical test shows good approximation properties of these splines.     

A degree property of rational matrices

In this paper, the existence of a relationship between the degrees of polynomials in any given rational matrix and the degrees of the polynomials in the inverse of that matrix is explained and proved.     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

The density of rational points on a certain singular cubic surface

We show that the number of nontrivial rational points of height at most B, which lie on the cubic surface x₁x₂x₃=x₄²(x₁+x₂+x₃), has order of magnitude B⁶(logB). This agrees with Manin's conjecture.     

Components of degree two hyperbolic rational maps

Summary We examine the structure of hyperbolic components and some boundary points of these for degree two rational maps. Rather than using quasi-conformal deformation theory, we use a technique from a theorem of Thurston about the existence of rational maps. This involves regarding a rational map as acting on the Teichmüller space of punctured sphere.     

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     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

2-Length and rational characters of odd degree

If G is a finite solvable group of 2-length l, we prove that the number of odd degree rational-valued irreducible characters of G is at least 2 ^l , improving a result of G. Navarro and P. H. Tiep. This bound is best possible, and also provides a new global/local relationship.     

Quasi-determinantal rational surface singularities

In this paper we give explicit equations for quasi-determinantal rational surface singularities, extending previous results for determinantal rational surface singularities.     

Double coverings of rational surface

Let K be a purely trancendental extension of a field k with two variables. We want to classify fields {L}, which are quadratic extension of K. First we prove the existence of a normal form of the defining equation of L. Then we give a rough classification and several results. Lastly we mention of an application to the theory of plane curves.     

空间有理参数多项式曲线的整数级绘制算法

讨论了空间有理曲线中心投影后导数上界的估计，基于曲线各阶差分的递推计算，给出了空间有理参数多项式曲线的快速绘制算法．算法只用到整数的加减法，效率高．