曲面

In this paper, we give explicit constructions and formulations for harmonic maps from R1,1 into classical real semisimple Lie groups by using Darboux transformation. We also discuss pluriharmonic maps from complex manifoldsinto symmetric spaces and Willmore surfaces in Sn. By converting geometric conditions satisfied by these maps into integrable systems, and using the the-ory of integrable systems, we give explicit constructions for pluriharmonic maps from complex manifolds into symmetric spaces and the Willmore surfaces in Sn respectively. Finally, we classify hypersurfaces in Sn+1 with three distinct prin-ciple curvatures and zero Mobius form using the theory of Mobius geometry. The paper consists of four chapters.

本文首先利用Darboux变换的方法给出了从Lorentz平面R~(1，1)到经典实半单Lie群的调和映照的具体构造，并给出其显式表示；其次研究了复流形到对称空间的多重调和映照及球空间S~n中Willmore曲面，将这些映照所满足的几何条件转化为可积系统，然后利用可积系统理论分别给出复流形到对称空间的多重调和映照与S~n中Willmore曲面的构造；最后利用Mōbius几何的理论给出S~(n+1)中具有三个不同主曲率且Mōbius形式为零的超曲面的分类。

Specifically, we apply the polar form in the conversion between triangular Bezier surfaces and rectangular Bezier surfaces. This problem helps us to understand the theory of polar form in a unified way. At the last half of this paper, we consider the problem of geometric continuity of parametric curves and parametric surfaces.

我们利用Polar形式实现了三角Bézier曲面和四边Bézier曲面之间的相互转化，同时通过应用，我们对Polar形式也有了一个统一、全面的认识；论文的后半部分考虑了参数曲线曲面间的几何连续性问题。

In the final chapter, we discuss interval polynomial approximation of rational surfaces. Firstly, we briefly state the interval-surface approximation based on the Taylor Expansion, and later, we give out a better interval-surface approximation based on the optimization method, which is also the main work of this paper.

第四章主要介绍了有理曲面的区间多项式的逼近，首先简单介绍了基于泰勒展开来做的区间曲面逼近，后面是本文的主要工作，我们基于优化方法得到了更好的区间曲面逼近，它也是本文的重要部分。

This paper introduces the application fields of reverse engineering , discusses the techniques of coordinate data acquisition and the features and the used scopes of all kinds of measure methods, elaborates the principal techniques of surface reconstruction of reverse engineering,analyses the problems of surface reconstruction., addresses a kind of algorithm of the scattered "points cloud" preprocessing and a method of employing CAD modeling software to finish surface modeling,develops a cloud data optimum processing programme.

本文介绍了反求工程的应用领域，讨论了坐标点的测量技术及各种测量方法的特点与适用范围，阐述了反求工程中曲面重建技术，分析了曲面重建中存在的主要问题，提出了离散"点云"的预处理算法及利用CAD造型软件进行曲面造型的方法，并开发了"点云"数据优化处理程序。

On ray tracing algorithm, line-surface intersection is the basic operation. we have studied the intersection method between line and plane region, quadric surface, NURBS surface.

在光线跟踪算法方面，对精确模型下曲面的直接光线跟踪进行了尝试，提出了对平面区域、简单二次曲面及NURBS曲面直接进行光线跟踪的算法。

Based on data cloud, which is measured from 3-coordinate measuring machine or so, an algorithm to search lopological structure from 3D unorganized points using Envelopment-box technology is proposed in this thesis. Based on this method, we searching neighbor points of sampling point. We also improve the Max-Min angle criteria to realize local triangulation, and then get the normal of sampling point from the triangulation. Using the normal and neighbor relationship, classification of scattered data points is realized according to the quadric surface.

本文以测量得到的曲面数据点点云为基础，提出了一种基于包围盒的自动寻找三维散乱数据点之间拓扑结构的方法，采用该方法寻找采样点的邻域结点，并对三角剖分中的典型优化准则—最小内角最大准则提出了改进，按照改进后的优化准则实现了采样点的局部三角划分，并进一步求解得到采样点的法向量，依据法向量及邻域拓扑关系在二次曲面的基础上实现了散乱数据曲面重构中的数据点的自动分类。

The systems of machining complex curved surface by means of WEDM are classified to establish the main kinematics theories of machining complex curved surface by means of WEDM By analyzing various systems of machining complex curved surface by means of WEDM, the combination, characteristics, motion parameters and motion types of them are summarized And a model system of the is established according to the basic principles of the forming of ruled surface in spatial analytic geometry.

主要内容及取得的成果体现在以下几个方面：为了建立复杂曲面电火花线切割运动学的主要理论，本文对复杂曲面电火花线切割加工系统进行了分类，并分析总结出各种复杂曲面电火花线切割加工系统的组成、特点、运动参数及运动形式。

Because the free form surface expressed by discret points has difficulties in aberration correction, manufacture and test, the Zernike polynomial was adopted to express the free form surface. The performance of this system was optimized on the basis of the relationship between Zernike polynomial and Seidel aberration.

由于离散点表述的自由曲面在像差校正、加工及检测等方面存在困难，提出了采用Zernike多项式表征自由曲面的面形，并利用Zernike多项式与Seidel像差之间的关系，优化自由曲面成像系统性能。

To fulfill the need of the application of RP technique in medical domain, reverse CAD modeling from medical cross sections is systematically studied in this dissertation. Firstly, Some algorithms related to the research of this dissertation are studied. New algorithms for orientation and inclusion test for simple polygon, an error constrained automatic faring algorithm for B-spline curve and a theorem regarding the termination criterion for subdivision of triangular Bézier patch are proposed. Then based on similarity between contours of adjacent sections, a methodology for surface reconstruction from cross sections is presented. Finally, a virtual measuring method is proposed.

本文针对RP技术在医学领域应用的需要，系统研究了基于医学断层轮廓数据的反求CAD建模理论和方法：(1)在基础算法研究部分，提出了简单多边形方向及点在多边形内外判断的新方法、三角Bézier曲面片离散的误差控制定理和一种带误差约束的B样条曲线的自动光顺方法；(2)提出了一种基于相邻层轮廓相似性的医学断层轮廓数据曲面重构方法；(3)提出了一种称为"虚拟测量"的曲面模型处理方法。

"We can perhaps forgive those who said the moons of Jupiter were produced by Galileo's spy-glass if we recall that in his day, as for centuries before, curved glass was the popular contrivance for producing not truth but illusion, untruth; and if a single curved glass would distort nature, how much more would a pair of them?"

如果我们回忆一下，Galileo时代如同它的几个世纪以前一样，曲面镜不是用来产生真像而是产生幻影或假象的常用器械，那么，对于那些把木星的卫星说成是用Galileo小望远镜产生出来的人们，我们或许会予以原谅的，如果一面曲面镜会歪曲大自然，那么一对曲面镜将会造成多大的歪曲呢？

I didn't watch TV last night, because it .

昨晚我没有看电视，因为电视机坏了。

Since this year, in a lot of villages of Beijing, TV of elevator liquid crystal was removed.

今年以来，在北京的很多小区里，电梯液晶电视被撤了下来。