仿射空间

In this paper,the area ratio of plane figure affine transformation in space is investigated,and the transformation equation of λ multiple of area is got , the area change of plane figure under affine transformation in space is studied, a important property of three axis ellipsoid is given and proved.

探讨了空间场的仿射变换下平面图形的面积比，推导出λ倍积变换方程式，进而探讨了空间场的仿射变换下平面图形面积的变化规律，给出并证明了三轴椭球体的一个重要性质。

Proves by matrix technique and ways that affine Unitary groups have double transition on the affine maximal and deseribes the total isotropic subspaces over affine Unitary geometry and their orbits and the use of affine maximal and total isotropic subspaces to construct several classes association scheme and PBIB designs, and the calculation of all parameters as well.

利用矩阵的技巧和方法，证明了仿射酉群双可迁地作用在仿射酉几何中的仿射极大全迷向子空间上，并且给出了轨道的明确表示。作为该理论的一个应用，利用仿射酉几何中的极大全迷向子空间的全体作为处理集构作了一个多个结合类的结合方案和PBIB设计，并计算了所有的参数。

Considering the limitation of traditional Bill of Materials models in enterprise applications, a novel BOM model based on n-dimensional affine space was proposed.

考虑到传统物料清单模型在企业应用系统中的局限性及运行效率等问题，提出一种基于 n维仿射空间的产品结构模型。

In conic optimization one minimizes a linear objective function over the intersection of an affine space with a regular cone in finite dimension.

它指的是在一个仿射空间与一个正则锥的交集上，求线性目标函数的极小或极大值。

This method represents the virtual objects in Euclidean space instead of in affine space, which is consistent with the popular graphics rendering tools such as OpenGL and is more common in practice. A new classification based local and global tracking strategy to match features between the reference image and the currently frame directly is presented, which enables the augmented reality system working effectively even under large changes of viewing angles. A novel affine coordinate system definition method based on a collection of feature points is also proposed, which makes the tracking method effective even when not all the reference points are detected during the whole process, as long as there are at least four pairs of reference points correspondences.

该方法以仿射变换为桥梁来建立欧氏空间下的摄像机坐标系与世界坐标系之间的变换关系，它与OpenGL等当前流行的图形渲染工具一致，具有更强的通用性；采用基于分类的自然特征匹配策略，能够在视角和视点发生大幅度变化情况下依然正常工作；提出基于特征点集合的仿射坐标系定义方法，使增强现实系统在最小配置下只需4点追踪成功即可完成注册，比需要6点的透视投影方法具有更好的健壮性和精确性。

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按"序"最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

In this paper we define the concept of Projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds. We prved that if M be a complete projective Blaschke n-sphere and its universal covering manifold is isometric to a complete (n+1) dimensional parabolic, elliptic or hyperbolic affine hypersphere, then M is a quotient space of E^n, S^n or D^n by a isometric subgroup of its corresponding spaces.

在这篇文章中我们定义了射影Blaschke流形的概念，将等仿射微分几何的理论推广到了射影Blaschke流形，并证明：如果n维完备射影Blaschke 超球面 M 的通用覆盖流形分别是完备的抛物型、椭圆型或双曲型仿射球，则M分别是n维欧氏空间、n维超球面或n维单位圆盘关于各自空间的一个等距离散子群的商，从而对完备射影Blaschke 超球面进行了分类。

Briefly speaking, a space curve is called affinely infinite if it is intersected by every plane in the 3D real projective space; otherwise, it is called affinely finite.

简单地说，在三维实射影空间，如果一条空间曲线与射影空间中任意一个平面都相交，那么称这条曲线是仿射无限的；反之是仿射有限的。

Firstly, affine covariant neighborhoods of visual words are used to verify the spatial consistency. Secondly, a set of affine transformation matrices between the corresponding covariant regions is calculated, and the matrix with the most inliers is chosen as the best matching affine transformation matrix. In this step, only a single pair of corresponding affine covariant regions is used to calculate the 6 degree of freedom affine transformation for acceleration. In the last step, the spatial consistency is verified by the BMATM.

首先通过视觉关键词仿射协变的空间邻近特征集进行初步空间关系一致性匹配；然后计算对应仿射协变区块的多组可能的仿射变换矩阵，并选择满足区块匹配最多的变换矩阵作为对象与图像之间最终仿射变换矩阵，其中，仅利用一对协变区块来计算自由度为6的仿射变换的算法有效地提高了计算速度和鲁棒性；最后通过计算出的仿射变换矩阵来精确验证对象与图像之间的空间位置关系是否一致。

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按&序&最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了