曲面

In this paper, firstly, the embeddability of near-triangular graphs on the orientable surface is studied. By using Petersen's Theorem about 1-factor, we first prove that for a near-triangulation on the orientable surface, its geometry dual graph has a 1-factor; then after a procedure of operation leaded by the 1-factor, we show that if a graph G triangulates some orientable surface S_g, then G has a near-triangular embedding in S_h for h = g,g + 1,...,β(G/2」, where β is the Betti number of G. Hence we obtain a conclusion: A near-triangulation of the orientable surface is upper-embeddable. As a generalization, a class of near-quadrangulation is studied, and similar results are obtained.

本文首先研究了近三角剖分图的可定向曲面嵌入性质，通过运用Petersen关于1-因子的定理，首先证明了对于可定向曲面上的三角剖分图，其几何对偶图具有1-因子；然后在1-因子的导向下，通过做一系列增加亏格的手术，证明了如果一个图G三角剖分可定向曲面S_g，那么G可以近三角剖分可定向曲面S_h，这里h=g，g+1，…，「β(G／2」，β是指图G的Betti数，从而得出推论：可定向曲面上的三角剖分图是上可嵌入的，作为推广，又研究了一类近四角剖分图的可定向曲面嵌入性质，并得到类似的结论。

In the particular ease where a and γ are equal to 1, the developable CE-Bézier surface is a developable Bézier surface. Finally, some applications in developable surfaces design are discussed, The modeling examples illustrate that the developable CE-Bézier surfaces provide two valuable ways for the design of developable surfaces.

由新方法生成的可展曲面不仅具有灵活的局部形状可调性和更强的描述能力，而且保留了CE-Bézier曲面的特性，特别是当α、γ都取值为1时，所生成的可展曲面即为CE-Bézier可展曲面。2种可展曲面设计方法的应用实例表明，该设计方法不仅简单、有效，而且易于控制曲面形状，从而为可展曲面的设计提供了一种新途径。

The surface representation in discrete form and its differential computation are important issues in computer graphics and geometric design.

0引言一般来说，曲面的一阶微分量是指曲面的切平面方向和法向量，二阶微分量是指曲面的曲率等有关量·它们作为重要的曲面信息度量指标，在计算机图形学，机器人视觉和计算机辅助设计等领域发挥了重要的作用·传统的曲面是连续形式的参数曲面和隐式曲面，其微分量的计算已经有了较完备的方法·随着激光测距扫描等三维数据采样技术和硬件设备的长足进步，以及图形工业对任意拓扑结构光滑曲面造型的需求日益迫切，离散形式的曲面———细分曲面、网格曲面和点云曲面正在逐渐成为计算机图形学和几何设计领域的新宠·于是，对这种离散形式的曲面如何估

In the case of kod≥0,we prove that s≥6,and construct a series of examples of s=6,where S is rational surface,elliptic ruled surface,Abel surface,K3 surface,Enriques surface,biellipticsurface respectively.

当〓时，证明了〓。并且找出了s=6的一系列例子，曲面类型分别为有理曲面，椭圆直纹面，Abel曲面，K3曲面，Enriques曲面，双椭圆曲面。

The representative minority cultural relic-pottery relic of Buyi minority and the two wooden masks is taken as the research object. Firstly, Hiscanner laser scan detector was adapted for data collection, and then obtain point cloud data which could be recorded on disc for permanent saving was obtianed; Furthermore, the new popular reverse engineering software Geomagic for some disposals of point cloud data was adopted in data condensing, data mending, data filtering, feature line picking up to obtain NURBS surfaces; and then the error between the reconstructed CAD model and the point cloud data was evaluated to guarantee the precision of surface reconstruction to keep the relics keep its original appearance. In the end, the NURBS surfaces are imported to 3D software Solidworks and entities of CAD are generated by surface sewing, mold loft and ablating and other instructions.

选用有代表性的贵州少数民族文物―布依族酒壶、水壶以及木制傩戏面具作为研究对象，采用Hiscanner激光扫描测头对其进行数据采集，扫描得到的点云数据可刻录到光盘上作为文物数据永久保存；更进一步地，采用目前流行的逆向工程软件Geomagic studio对点云数据进行数据精简、数据滤波、数据修补、提取特征线等处理后得到NURBS曲面；为保证曲面模型重建的精度，在曲面重建模型后对其误差评价，通过控制误差的方式指导曲面模型重建，使重建的文物曲面模型能真实完整地反映原状；最后，在功能强大的三维造型软件Solidworks中导人重构的NURBS曲面进行曲面缝合、放样等，将其生成实体CAD模型，并对模型的局部特征进行适当的再设计以实现文物实体模型的优化。

The interpolate algorithms of three-order NURBS curve and double three-orders NURBS surface are presented. The expressions of partial derivative and normal curvature of NURBS surface are derived by the theory of differential geometry. The geometric characteristics of different curvature area of the free-form surfaces are analyzed in detail. All these work lays a theoretical foundation for constructing the cutter location calculating model of free-form surfaces NC machining. 2 On the basis of analyzing the methods of cutter path generation, the standard of evaluating the various methods of cutter path generation is presented. The method of generating cutter path with constant scallop height is presented for the first time.

本文以提高复杂曲面多坐标数控加工精度和加工效率为目的，在对国内外研究现状进行深入分析的基础上，对数控编程中各环节的关键技术问题进行了深入研究，主要内容包括以下几方面： 1 分析了复杂曲面各种造型方法的优缺点，给出了根据各种边界条件所确定自由曲面的NURBS表示方法以及三次NURBS曲线曲面反算算法；用微分几何理论推导了NURBS曲面的偏导矢及法曲率计算表达式；对曲面不同曲率特性区域的几何特征进行了深入分析，为建立复杂曲面的数控加工刀位计算模型奠定了理论基础。

By the definition of Darboux type vector and rectifying Darboux type vector ,we constructed the rectifying type developable surfaces and Darboux type developable surfaces and tangential Darboux type developable surfaces ,and study these three developable surfaces relationship with Minkowski slant helices and Minkowski conical geodesic curves. Bying applying the singularity theoerical knowledge, We give a classification of special developable surfaces under the condition of the existence of such a special curve as geodesic.

通过定义三维Minkowski空间中的达布型向量以及修正达布型向量，构造出三维Minkowski空间中的三类可展曲面，即修正型可展曲面、达布型可展曲面和切达布型可展曲面，研究Minkowski斜螺线和Minkowski锥面测地线这两种特殊曲线和这些曲面的关系，并研究Minkowski斜螺线和Minkowski锥面测地线作为这些曲面的测地线的条件下，运用奇点理论的知识，对这三种可展曲面进行奇点分类。

Two above methods are generalized to tensor product Bézier surfaces and triangular Bézier surfaces, we give two approaches of degree reduction for tensor product Bézier surfaces, each of which reduces a given degree n×m tensor product Bézier surface to a n〓×m〓 one, and two methods of degree reduction for triangular Bézier surfaces, which reduce a given degree n triangular Bézier surface to a degree m one. The explicit formulas of control points for reduced tensor product and triangular Bézier surfaces are obtained respectively.

张量积Bézier曲面的两种降阶方法均可把n×m阶张量积Bézier曲面一次降阶为n〓×m〓阶张量积Bézier曲面；三角Bézier曲面的两种降阶方法均可把n阶三角Bézier曲面一次降阶为m阶三角Bézier曲面，并给出降阶张量积Bézier曲面和降阶三角Bézier曲面控制顶点的显示表示式。

For the first time, a screw pile and pile driver system for preventing flood and rushing to deal with an emergency was manufactured. The screw piles of the system is different from screw drills used to sap, and it doesn't push soils out. Analyses, including the screw pile's structural strength analysis and the screw blade's welding analysis, were done by ANSYS, which showed that the screw pile's structural strength is enough and that the distort of the screw blade is small. Testing on the screw pile and pile driver system showed that the system performed stably and could meet the need of preventing flood and rushing to deal with an emergency. A mechanics model of interaction of a screw pile and soil was built, and expressions of torque of screw pile's cylindrical part on the condition of screw pile's crew part being driven into soil fully were gained, then the relation of the torque with screw pitch and out diameter of cylinder, which shows the torque decreases with the screw pitch increasing, and that the torque increases with the out diameter of cylinder increasing, which was according with the experiments of the screw pile's being driven into soils, and thus a screw pile with the best structure was chosen. Experiments of bionic surface showed that, on the condition of certain pressure on the surface, the protruding bionic surface can not reduce adhesion and resistance and that the corrugated bionic surface can reduce adhesion and resistance. An orthogonal experiment of the corrugated bionic surface was done with the factors including the width of protrusion, the space between two protrusions, the trolley's velocity and the vertical surface pressure and with the surface frictional coefficient as the experimental target, and some inclusions were drawn that the best combination of the factors was acquired and that the width of protrusion affected the surface frictional coefficient remarkably.

研制了防汛抢险螺旋桩机系统，提高了沉桩作业效率；利用ANSYS对桩体进行整体结构强度分析以及螺旋叶片的焊接分析，表明桩体结构强度足够，焊接过程中，螺旋叶片的变形小；对整个螺旋桩机系统进行试验，表明该系统性能可靠，工作稳定，作业效率高，能够满足防汛抢险沉桩作业基本需要；建立了螺旋桩桩土作用力学模型，得出螺旋桩沉桩过程中螺旋桩螺旋部分完全进入土体时螺旋桩圆柱部分所受扭矩以及扭矩与防汛抢险螺旋桩的螺旋导程和圆柱外径的关系曲线，表明螺旋桩螺旋部分完全进入土体时螺旋桩圆柱部分所受扭矩随着螺旋导程的增大而减小，随着圆柱外径的增大而增大，这与螺旋桩沉桩试验结论相符合，据此优选出最佳结构的螺旋桩；仿生曲面试验表明，尺寸较小的球冠突起型仿生曲面在一定压力作用下未能起到减粘降阻的作用，而尺寸较大的波纹型仿生曲面相对平板具有一定的减粘降阻作用；以波纹型仿生曲面突起宽度、波纹型仿生曲面两突起之间间距、台车的牵引速度以及叶片表面垂直载荷为因素，以曲面摩擦系数试验指标进行正交试验，寻求出在一定试验条件下最佳因素水平组合；波纹型仿生曲面突起宽度在本试验条件下对曲面摩擦系数影响显著，而其它几个因素对曲面摩擦系数影响均不显著；通过波纹型仿生曲面突起宽度和台车牵引速度的单因素试验，表明曲面摩擦系数随着仿生曲面突起宽度的增大而减小，而随着台车牵引速度的变化基本保持不变，和正交试验结果保持一致。

Single factor experiments were done with the factors including the width of protrusion and the trolley's velocity, which showed that the surface frictional coefficient decreases with the width of protrusion increasing, and doesn't change with the trolley's velocity changing, which was consistent with the results of the orthogonal experiment. A theoretical model of the horizontal projection of a corrugated bionic surface's contact area was built, which showed that the contact area's horizontal projection decreases with width and interval increasing respectively and that the ratio of the contact area's horizontal projection and the vertical load decreases with the vertical load increasing. It indicated that the surface frictional coefficient decreases with the above three factors increasing respectively. Finite element modeling and analysis of interactions of a plate and soils was done using ANSYS, which showed that the plate frictional coefficient is little relative to soil elastic modulus and decreases with vertical surface pressure and that trolley's pull and vertical displacement of bionic surface's control pilot keep constant almost during the plate being pulled. Finite element modeling and analysis of interactions of a corrugated bionic surface and soils was done using ANSYS, which showed that surface frictional coefficient is little relative to soil elastic modulus and decreases with bionic surface's structural parameters and vertical pressure increasing. However, the three factors all didn't affect the surface frictional coefficient remarkably on the condition of 90% dependability. For the first time, bionic surfaces were applied to screw piles, which formed bionic screw piles, and the contrast experiments showed the bionic screw pile had certain resistance-decreasing effects.

建立了波纹型仿生曲面与土壤的接触面积的水平投影的理论模型，表明接触面积的水平投影A随着波纹型仿生曲面突起宽度、两相邻突起间距的增大而减小，接触面积的水平投影与波纹型仿生曲面表面垂直载荷之比随着波纹型仿生曲面表面垂直载荷的增大而减小，进而得出仿生曲面摩擦系数随着波纹型仿生曲面突起宽度、两相邻突起间距以及表面垂直载荷的增大而减小；利用ANSYS软件对平板与土壤的相互作用进行了有限元建模与分析，表明平板摩擦系数与土壤弹性模量关系不大，而随着表面垂直载荷的增大而减小；平板牵引过程中台车牵引力和仿生曲面控制节点垂直方向位移基本保持不变；利用ANSYS软件对波纹型仿生曲面与土壤的相互作用进行了有限元建模与分析，表明波纹型仿生曲面系数在一定范围内基本不受土壤弹性模量的影响，随着波纹型仿生曲面突起宽度、两相邻突起间距以及表面垂直载荷的增大而减小，但在可信度为90％的前提下，三个因素影响都不显著；首次将波纹型仿生曲面应用于与土壤作用方式以挤压为主的螺旋桩，形成仿生螺旋桩，并与普通螺旋桩进行了沉桩对比试验，表明波纹型仿生曲面应用于螺旋桩桩体表面具有明显的降阻作用。

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。