平面曲线

For a general parametric curve/surface, we usually cannot compute its exact implicit form. Even though its exact implicit form can be computed, the curve/surface implicitization involves relatively complicated computation and the degree is higher. Moreover, it may have unexpected components and self-intersections. All these unsatisfied properties limit the applications of the exact implicitization. So finding curve/surface approximate implicitization has become a practical problem. In this paper, we present an algorithm to solve the approximate implicitization of a given parametric curve by using a quadratic algebraic spline curve.

由于精确隐式化过程不一定可以实现，即使可以实现隐式曲线曲面的阶数高计算复杂，并且具有不希望的自交点和奇异分支，从而限制了隐式化的运用，所以寻求参数曲线曲面的近似隐式化问题成为很实际又重要的问题，提出利用二次代数样条曲线来实现一般平面参数曲线近似隐式化的一种算法。

Carried out generalized Mises yield criterion , yield curve on the n flat face was circle curve, and different elected parameters correspond to external angular point circumcircle、 internal angular point circumcircle and incircle of Mohr-Coulomb unequiangular hexagon. According to unequiangular hexagon, circle yield curve which had the same area as hexagon can be found, and corresponding yield criterionwas equilateral circle yield curve.

3广义米赛斯准则在π平面上的屈服曲线为圆曲线，选择的参数不同对应有莫尔—库仑不等角六边形的外角点外接圆，内角点外接圆和内切圆，对于不等角六边形，还可以找到一个面积和六边形相等的圆屈服曲线，对应的屈服准则为等面圆屈服曲线。

The interpolation algorithm of introduced error compensation for complex parametric curve in space is proposed for the high precision interpolation of parametric curve on the basis of studying the first, second order taylor expression parametric curve interpolation algorithm and the plane parametric curves interpolation algorithm.

以一、二阶泰勒展开式参数曲线插补算法和平面参数曲线插补算法研究为基础，提出引入误差补偿值的复杂空间参数曲线高精度插补算法以实现空间曲线的高精度插补。

Based on the Guldin's first theorem, it was deduced that the general formula to calculate the rotating surface area which is shaped by rotating the curve Г in a plane around the L-axis in the space by using curve integral.

依据古鲁金第一定理推证，得到利用曲线积分计算空间某一平面上的曲线Г绕直线L旋转而成的旋转曲面面积的通用公式；利用曲线积分计算xoy面上的曲线c绕直线L旋转而成的旋转曲面面积的通用公式。

The first section introduces some basic concepts[9].The second section introduces Schmidt"s method of proving isoperimetric inequality [2].The third section introduces Hurwitz"s method [9].The fourth section takes from my tutors lecture notes.The method is due to REILLY [10].Finally,I make use of variational method to prove the circle among the curves of length L encloses the biggest area on the plane.

首先介绍了平面上的一些基本概念[9]，其次介绍了文献[2]中Schmidt证明等周不等式的方法，再次介绍了文献[9]中Hurwitz证明等周不等式的方法，然后介绍了导师吴发恩整理的文献[8]中REILLY的方法证明平面上的等周不等式[10]，最后我利用变分的方法证明了平面上等长的曲线围成的面积最大时为圆。

Rational planar curve; rational space curve; rational ruled surface; moving curve/surface;μ-bases; singularity; self-intersection curve; moving curve/moving surface ideal; ideal-theoretic/set-theoretic generators

基础科学，数学，几何、拓扑平面有理曲线；空间有理曲线；有理直纹面；动曲线/曲面；μ基；奇点；自交线；动曲线/动曲面理想；集论/理想论生成元

Through the new research methods, this thesis has made the following contributions in this field: 1. For the first time the curvature radii and relative angles of a planar continuous curve at discrete points are chosen as local parameters to represent the curve, which reflect clearly the local geometrical characteristics of the curve and are easy to be dealt with.

借助于新的研究方法，本文在这一领域作出如下贡献：（1）首次提出用曲线上离散点处曲率半径和相对转角作为局部参数表示平面连续曲线，直观地反映曲线各个局部的几何特征，且便于曲线的修改和变换，并通过机构运动分析，快速求出机构轨迹的这两个局部参数。

This paper discusses singularities, inflection points and convexity of FB-spline curves in terms of their control polygons, and gives the necessary and sufficient conditions for testing when FB-spline curves have one or two inflection points, or a cusp, or a loop, or none of the above points by the envelope theory and the topological mapping method.

文章利用包络理论和拓扑映射的方法，讨论了FB-样条曲线的奇拐点和曲线性质，并给出了依据控制多边形判断FB-样条曲线出现1个或2个拐点、1个尖点、1个二重点以及处处为凸的充分必要条件；最后给出了FB-样条曲线的奇点、拐点以及二重点在λ-μ平面上的分布图。

For example, we study the properties of the"centroid curve" and the"auxiliary curve", and identify two planes in which these two curves are defined respectively; we also use the idea of homotopy method to prove a fact that the orbit of a differential system has no triple contact point with the line we considered. Another key technique, which we use several times, is deformation argument.

例如研究由Abel积分之比定义的"质心曲线"和"辅助曲线"的几何性质，并把这两条在不同平面上定义的曲线放在同一坐标系中考虑；利用拓扑中同伦的思想来证明微分系统的轨线不可能与某条直线三重相切；利用几何方法与分析计算相结合证明了某一曲线是正则的；以及连续变动参数的技巧。

On the other formed, polynomial expressions of intersected point between solution curves of WuestII, reflected shock backward-facing sonic, reflected shock backward-strong/weak separating, Mechanical equilibrium are derived and computed results are marked on the (M0,q1) theoretical solution map of perfect triatomic gas steady Mach reflection.

最后吾人针对WuestII曲线、后向音速曲线、后向强弱曲线、机械平衡曲线，推导它们之间的交点解公式，并将计算的结果标示於平面上之解域图上。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了