曲面

Bezier and NURBS are chosen as the fundamental methods to achieve surface reconstruction after discussing the fundamental theory of surface reconstruction. The two methods are proved to be effective by some examples.

分析了自由曲面重构的基本理论，采用Bezier和NURBS方法实现了曲面重构，并通过实例验证了两种方法对小规模点云数据曲面拟合的合理性。

The fundamental theory of surface reconstruction is further discussed which leads to the finding that there are some disadvantages in Bezier and NURBS. So both Bezier and NURBS are chosen as the fundamental methods to achieve surface reconstruction.

对曲面重建基本原理进行深入分析，针对Bézier方法和NURBS方法的各自特点，在系统中选择Bézier和NURBS两种方法作为系统软件曲线曲面重构的基本方法，并实现曲面重构。

A systematic study on computing the integral curvature line on NURBS surface is presented. Discrete normal is used to simplify calculation of the second fundamental form of the surface, resulting in accelerating the Euler iteration process in solving the differential equation of curvature line. Discrete points on the curvature line are then obtained and approximated by an explicitly presented B-spline curve on the NURBS surface using singular mixed interpolation and capable of controlling the error within a given tolerance.

对NURBS曲面的曲率线的积分进行了系统的公式推导，并利用NURBS曲面的离散法向量有效地简化了曲面第二基本量的计算，加速了Euler法迭代求解曲率线微分方程的过程；在求得曲率线上的离散点集以后，应用奇异混合插值技术，在可控精度内把曲率线用显式直接表示为位于NURBS曲面上的B样条曲线。

For the timelike surfaces with constant positive Gaussian curvature (K=1) in R〓, we construct the correspondence between R〓 and sl (2, R), and deduce the fundamental equations for a special choice of the velocity of motion of time-like surfaces in asymptotic coordinates under suitable conditions, meanwhile, we give the B〓cklund transformation and one soliton solution of the evolution equations, then in terms of a gauge transformation we obtain the explict expression of the motion of surfaces.

对于三维Minkowski空间R〓中具有正常高斯曲率（K=1）的类时曲面，取定特定速度，在以渐近线作为坐标曲线的情况下，曲面运动满足两个基本方程，即sinh-Gordon方程和θ随时间演化的方程（其中θ表示两条渐近线之间的夹角），同时给出两个基本方程的B〓cklund变换和单孤子解，并通过规范变换给出相应的正常高斯曲率曲面的显式表达式。

The production of globoid indexing cam and gas engine burningchamber are employed as refrences to analyze the method to mathematical modeling of complexed functional feature, furthermore, the research work of the character of surface feature production is deeply done .Advancing and optimizing the producting process, as well as the whole simulated machining by making UG NX2. 0 as the supporting system. Building the Fixture Library CAD/CAM Document system and producting template of DMU70V. Exploring the structure and flow chart of the Document System;Developing the MDF File by means of GPM module. All the above provide profound preparation to establish a further integrated manufacturing environment around 5-Axis NC production center.

在本文中，综合论述了特征建模的方法，以弧面分度凸轮和内燃机燃烧室作为实例，重点研究了复杂功能曲面的建模及曲面分析方法；并从五坐标加工角度出发，详细研究了曲面零件的加工特性，提出并优化了具体的工艺过程；利用UnigraohicsNX2.0作为开发工具，对建模和加工过程进行了全程仿真；围绕DMU70V建立了专属夹具库、文件体系及工艺模板；综合规划了五坐标数控加工的CAD/CAM文件体系的结构和运作流程；开发了GPM五坐标后处理DMF文件，为建立以五坐标数控加工中心为主的加工体系打下了良好的基础。

We generalize the concept of the set of variability defined in the unit diskby K.Strebel to general Riemann surfaces and prove that the set of variablity withrespect to the modulo homotopic class of a quasiconformal mapping between twoRiemann surfaces is a compact and connected subset.Consequently the num-ber of the extremal mappings in the modulo homotopic class of a quasiconformalmapping between two general Riemann surfaces is either one or infinity.

Strebel的定义在单位圆上的可变性集合的概念推广到一般Riemann曲面上去，并且证明了Riemann曲面之间的拟共形映射的模边同伦类所确定的可变性集合是一个连通的紧子集，从而得出一般Riemann曲面之间拟共形映射的模边同伦类中的极值映射要么唯一，要么有无穷多个。

It is proved that a complete hypersurface in the Eucligean space with constant mean curvature and finite total curvature must be either minimal or compact.

我们证明了：欧氏空间中具有常平均曲率和有限全曲率的完备超曲面，或是极小超曲面，或是紧致超曲面。

By anchoring an imaginary spring between each data point and deformable curve or surface, the fairing model is established. Then FEM is used to solve the deformation problem of the model caused by imaginary springs and a bicubic B-spline surface patch is taken as one element. Lastly the B-spline active surface model is applied to surface reconstruction from layer data and the problem resulted from fitting for uneven cross-sectional curves can be solved.

通过在数据点与模型之间连接虚拟弹簧，建立光顺模型，并以双三次B样条曲面片作为单元，运用有限元法对模型在虚拟弹簧作用下的变形问题进行求解，得到光顺后的B样条曲面；并在此基础上，提出了一种新的基于断层数据的表面重建方法，可很好地解决不均匀截面族的曲面生成问题。

The main difference between the proposed method and the existing methods is that the new one embeds objects into the parametric space of the base surface by isometric or locally isometric mapping using the planar development of the base surface as the intermediate medium, thus the object to be deformed can follow the shape of the base surface homogeneously.

新方法中，变形控制曲面首先以等距或局部等距的方式展开为平面，然后经由平面的线性映射，将待变形物体嵌入控制曲面的参数空间中，从而产生符合控制曲面形状的局部均匀变形。

A sufficient and necessary condition of an isothermal surface being a minimal surface is obtained.

除平面外没有其它二次多项式极小曲面，Enneper曲面是目前微分几何文献中唯一出现的三次参数多项式极小曲面，还没有出现高次多项式极小曲面的例子。

Plunder melds and run with this jewel!

掠夺melds和运行与此宝石！

My dream is to be a crazy growing tree and extend at the edge between the city and the forest.

此刻，也许正是在通往天国的路上，我体验着这白色的晕旋。