代数曲面

The cubic algebraic curves (the Bezier curve and B style curve, etc belong to the parameter curve) is apter to realize the curve transition and fitting to dispersed data than the cubic parametric curves.

比如参数曲面与平面的交线就是一条平面代数曲线；平面参数曲线的等距曲线也是一条代数曲线。

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。

The procedure of transform the parametric form into algebraic form is calledimplicitization.

将参数形式的曲线曲面转化为代数形式的曲线曲面的过程称为隐式化。

Content: This course discusses the line, plane and quadratic surface in space through algebraic methods, and aims at developing the students' ability to solve geometric problems through analytic methods and to relate geometric intuitive image with algebraic quantitative relation.

主要内容：本课程运用代数方法研究空间解析几何中的直线、平面与二次曲面，培养学生运用解析方法解决几何问题的能力，以及将几何直观形象与代数数量关系相联系的能力。

In this paper, we have made a systemic theoretic research on modelling and reshaping several types of spline curves/surfaces in CAGD. And some creative productions are given as follows. At first, the uniform C-B-spline basis is proved to be a normalized totally positive basis, and can be extended to a normalized B-basis. C-B-spline and C-Bézier basis, similar to B-spline basis and Bézier basis in algebraic space, are two bases in algebraic trigonometric space.

本文围绕着CAGD中常用的几种曲线曲面造型和形状调整进行了深入的研究，主要获得了以下一些成果：首先，证明了代数三角空间中的n次均匀C-B-spline基是一组标准全正基，并进一步扩充为一组标准B基。C-B-spline基与C-Bézier基是多项式与三角混合空间中的两组基，是适应工程实践中设计特殊曲线曲面的需要而产生的，类似于多项式空间中的B-spline基与Bézier基，是CAGD中重要的造型工具。

We shall first apply the technique ofμ-bases and local cohomology to study the minimal generators for the moving surface ideals of rational cubic space curves and rational quartic space curves,and then study the implicit equations of rational space curves under different definitions of implicit equations.

在第五章中，我们将同调代数理论与μ基理论相结合，详细分析三次及四次空间有理曲线的动曲面理想生成元，为一般空间有理曲线的动曲面理想生成元的(来源：A21B12ceC论文网www.abclunwen.com)研究开辟可能的道路。

Surface blending in terms of geometric continuous is more practical and welcome with view of geometric interpretation which is different to the traditional differential method based on algebra.

几何连续是对参数曲面中参数连续度量进一步深刻的认识和研究。它是可微性的代数概念的几何抽象，克服了参数连续对曲面苛刻，高阶及结果失真等缺点。

The multivariate polynomial interpolation problem is a classical mathematical problem that is widely used in many fields, such as multivariate function list, surface design and finite element method. In recent years, multivariate polynomial interpolation has been focused by many people, of which the geometric topological struction of sets of interpolation nodes is also much concerned by us.

多元多项式插值问题是一个十分具有研究意义和实际应用价值的数学问题，它广泛应用于多元函数列表，以及曲面的外形设计和有限元法等诸多领域，近年来多元多项式插值越来越受到人们的广泛关注，其中有关插值结点组的几何拓扑结构问题也是人们十分关注的内容。1998年，梁学章和吕春梅在文献[2]中借助代数几何中的有关理论，进一步讨论了沿无重复分量平面代数曲线上的Lagrange插值问题，并应用Cayley ？

It introduces partial fractions of meromorphic functions, product developments of entire functions, Hadamard's theorem, Riemann Zeta functions, Poisson-Jensen's formula; elliptic functions, including simply periodic functions and doubly periodic functions; algebraic functions and algebroid functions, Riemann surface, Nevanlinna theory, including characteristic functions, the first and second fundamental theorems, growth orders, etc; complex differential equations and complex functional equations, etc.

具体为：亚纯函数的部分分式、整函数的无穷乘积展开、Hadamard定理、Riemann Zeta函数、Poisson-Jensen公式；椭圆函数，包括单周期函数、双周期函数；代数函数和代数体函数、Riemann曲面简介；Nevanlinna理论简介，包括特征函数、第一和第二基本定理、增长级等；复微分方程和复函数方程，等等。在教学内容上充分体现基础性、新颖性。

By learning this course, students should proficiently use algebraic methods to solve geometric problems and practical problems, and proficiently grasp the basic ideas and methods of vector algebra and geometric properties of spatial line, plane and quadratic surface.

通过本课程的学习使学生熟练掌握运用代数方法解决几何问题并应用于实际问题之中，并熟练掌握向量代数的基本思想和方法，同时使学生熟练掌握空间的直线、平面和二次曲面及其各种几何性质。

They weren't aggressive, but I yelled and threw a rock in their direction to get them off the trail and away from me, just in case.

他们没有侵略性，但我大喊，并在他们的方向扔石头让他们过的线索，远离我，以防万一。

In slot 2 in your bag put wrapping paper, quantity does not matter in this case.

在你的书包里槽2把包装纸、数量无关紧要。