单位球面

This thesis contributes to geometry compression in the following aspects: Proposes a method of progressive spherical mesh parameterization based on local parameterization by directly projecting the deleted vertices to its neighborhood. Based on spherical mesh paramterization, an adaptive subdivision remeshing method by progressivly redistributing of sampling points is proposed, by this means, the main geometry information of the original mesh can be captured with only a small amount of samples, it can also be used in mesh simplification. By regarding attributes binded on mesh vertices as geometry signals and transferring them onto a uint sphere by means of spherical mesh parameterization, meshes with different attributes are compressed with spherical wavelet successfully with high compression rate. As a kind of lossy geometry compression technique, two mesh simplification algorthimes are also presented in this thesis.

归纳起来，本文的主要贡献有：提出了以直接投影局部参数化方法为基础的网格累进球面参数化方法；在网格球面参数化方法的基础之上，提出了一种基于累进优化布点方案的网格细分采样方法，该方法可以用很少的采样点就可以捕获到网格的主要特征，因此我们也将其用于后面的网格简化工作当中；通过将网格顶点上定义的各种属性看成几何信号，并以网格球面参数化为手段将几何信号迁移到单位球面上，从而成功地实现了用球面小波对各种几何信号的压缩；作为一种有损压缩方法，本文还提出了基于统一距离的网格简化算法和基于重采样的网格简化算法。

This paper concerns submanifolds with constant mean curvature in Euclideanspaces or unit spheres.

本文考虑欧氏空间或单位球面中平均曲率为常数的子流形，主要考虑极小的，即平均曲率为零的子流形。

We consider Grassmann manifold G(2,8) as a submanifold of the unit sphere of /\2 which is the exterior vector space on Rn. Thus we have the induced metric and connection on G(2, n).

把Grassmann流形G（2，n）看作等距地嵌入在欧氏空间R~n上2-外向量空间∧~2中的单位球面上的子流形，得到它上面的度量和联络。

In this paper by use of a new differential operator similar as one introduced by Cheng and Yau,we obtain a rigidity result for a compact submanifold of constant scalar curvature in a unit sphere.

宣满友 ，盛为民本文利用一个类似于Cheng和Yau引进的微分算子的新微分算子，得到了单位球面中常数量率的紧致子流形的一个刚性结果。

In section four, we study the problem of isometric extension from the unit sphere of Banach space, which we also called Tingley's problem.

第四节介绍单位球面上的等距延拓问题，这是一个准线性问题。

Convex modulus depicted overall degree of convexity of the unit sphere of spaces. It was very different for degree of convexity of each point that it would produce very great influence on the whole feature of the spaces.

凸性模刻划了空间单位球面的总体凸性程度，但是每一点的凸性程度都有很大的不同，这将会对空间的整体性质产生很大影响。

The numerical results based on this algorithm show that it is possible to obtain a reasonable recovery of the shape of the three-dimensional obstacle from the incomplete far-field measurements for low wavenumbers even when the observation range is as small as the quarter of a unit sphere.

基于这个算法的数值模拟表明了，对较小的波数，即使只用测量范围为四分之一单位球面范围中少量测量方向上的信息，得到三维障碍物的一个合理的重构也是可能的。

Given a manifold triangle mesh with zero genus and arbitrary topology, it is globally parameterized over the unit sphere S2 in E3 firstly.

对于给定的具有任意拓扑结构的零亏格三角形网格，算法首先将其在单位球面上进行全局参数化得到一个参数化网格。

Normal vectors, Gaussian and mean curvatures estimation are obtained by different curvatures estimation methods, and the error is computed by comparing estimation results with real ones on the sphere of unit radius and toms.

在单位球面和环面上，利用不同的曲率估计方法得到法向量以及高斯曲率，平均曲率的估算值，再将估算值与实际值做比较求误差。

The algorithm is applied in the segmentation of Magnetic Resonance Imaging of human brain for the 3D reconstruction on dependent of different human tissue structures, in which the problem on wraps between different tissues and fuzzy edges between the different regions in medical volume data may better be settled, which can not be well done by the typical"hard"segmentation using binary logical representation.(3) An accelerated algorithm of volume rendering with X-ray perspective and Phong shading effect based on textures mapping and blending is presented for realizing 3D reconstruction of medical volume data by 2D texture mapping and blending, in which the accelerated algorithm of window transformation is utilized to computing voxels' gray scales and opacities to produce X-ray perspective images, and Phone illumination model is used to import world light sources and an unit spherical surface under the same shading environment as the actual volume rendering is employed to simulate the real reflectional light of per-voxel so that 3D reconstruction possesses illumination effect in the world. Furthermore, 3D matrix transformations of degree rotation and other operations of 3D object are studied, and a fast equivalent method of corresponding shading correction is brought forward to improve its interaction speed in volume rendering process.

该方法利用体数据的窗口变换加速算法将医学体数据转换为计算机显示值，可以得到具有X光透视效果三维图像；该方法提出的Phong明暗处理加速算法利用Phong光照模型引入外部光源，使用一单位球面体来仿真体数据的法线矢量集所产生的反射光强，以快速得到每一个体素的反射光强度，使物体的三维重建达到现实世界中人眼直接观察的逼真效果；此外，还研究了三维重建物体的三维旋转等矩阵变换，提出了一种Phong明暗修正的快速算法来保证了体绘制中三维几何变换的多视角观察的交互速度。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。