钣金数值放样的一种新方法

本文提出了一种基于可展曲面到平面等距对应原理的钣金数值放样的实用方法,其特征在于:等距变换公式的参数设法选用了曲面的投影坐标,曲面展开的方法采用短程三角法.本方法从数值计算到展开图的绘制都已实现了微型计算机化.     

线性代数变换与仿射几何变换

本文介绍线性代数中的二维代数变换与解析几何中的平面仿射变换的关系,代数变换及其变换矩阵的几何意义,各种仿射变换的矩阵表示及其矩阵性质,梳理仿射变换下的不变性质与不变量.     

三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面

本文主要给出了三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面的奇点分类,并且建立了奇点和曲线几何不变量之间的联系,其中曲线几何不变量与曲线同螺线切触的阶数密切相关.     

高维Mobius群几何形式的Jφrgensen不等式

本文给出了两个n维Mobius变换的共轭不变量,利用所得的不变量和不动点的交比,得到了两个关于高维非初等离散Mobius群的几何形式的JФrgensen不等式.特别地,这两个不等式是共轭不变的.     

A元不变量及其复合

在一个变换群下有许多的变换不变量,同时也有任意元的不变量或称为A元不变量,本文提出基于A元不变量,使所有的A元不变量都可以则基本A元不变量复合而成,证明A元基本不变量是存在的;给同一个充分必要条件,用于判定不变量的基本性,还对欧氏空间中各种常见变换群下的基本不变量进行稳定。     

矩形域与三角域Bezier曲面间的PPG~n变换

1．引言曲线,曲面间的几何连续性问题近几年来引起人们的广泛注意和兴趣［1－10］．在1984年国际CAGD会议上R．E．Barnhill和G．Farm将它列为计算几何尚待解决的重大问题之一．一方面是由于在CAGD中总希望所构造的曲线和曲面有更多的自由度或者有更多的几何控制参数以便使设计人员能自由地控制所要设计的对象的几何形状;另一方面是由于需要研究自由曲线曲面间的基于几何连续的拼接和各种形式的连续过渡曲面来构造更为复杂而灵活的曲线,曲面和几何体山．几何连续性的定义在不同的论文中不尽相同,但其本质是非奇异参数变换下的不变…     

高维Mbius群几何形式的Jφrgensen不等式

本文给出了两个n维M(?)bius变换的共轭不变量,利用所得的不变量和不动点的交比,得到了两个关于高维非初等离散M(?)bius群的几何形式的Jφrgensen不等式.特别地,这两个不等式是共轭不变的.     

De Sitter空间中类空曲线的对偶曲面的不变量

主要从切触几何的视角考察3维de Sitter空间中类空曲线的第一光锥对偶曲面和双曲对偶曲面的不变量的几何性质.     

超曲面的共形数量不变量

对给定的共形流形及其中的超曲面,本文用Fefferman和Graham的辅助时空及其中齐次关联超曲面,引进了由齐次关联超曲面在辅助时空中的伪Riemann数量不变量诱导的原超曲面的共形数量不变量,提供了一套构建更多超曲面的共形数量不变量的计算方法,为寻找像Willmore方程一样关于超曲面的共形不变偏微分方程创造了路径.     

Crofton定理拓广与图象特征提取——积分几何在图象分析与识别中的应用之一

<正> 和一切统计模式识别一样,图象的几何分析与识别也包括两个过程:提取几何特征,分类判决.而提取几何特征则根据不同场合的不同需要表现为运动不变量、仿射不变量或者拓扑不变量,以及几何特征函数等等.积分几何(又称整体几何学)的基本成果和方法为图象分析与识别提供了丰富多彩的几何特征.本文作为积分几何在图象分析识别中     

An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.     

Invariants and Bonnet-type theorem for surfaces in ℝ4

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.     

A Bishop surface with a vanishing Bishop invariant

We give a solution to the equivalence problem for Bishop surfaces with the Bishop invariant λ=0. As a consequence, we answer, in the negative, a problem that Moser asked in 1985 after his work with Webster in 1983 and his own work in 1985. This will be done in two major steps: We first derive the formal normal form for such surfaces. We then show that two real analytic Bishop surfaces with λ=0 are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature. Our convergence proof is done through a new hyperbolic geometry associated with the surface. As an immediate consequence of the work in this paper, we will see that the modular space of Bishop surfaces with the Bishop invariant vanishing and with the Moser invariant s<∞ is of infinite dimension. This phenomenon is strikingly different from the celebrated theory of Moser–Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants where the surfaces only have two and one half invariants. Notice also that there are many real analytic hyperbolic Bishop surfaces, which have the same Moser–Webster formal normal form but are not holomorphically equivalent to each other as shown by Moser–Webster and Gong. Hence, Bishop surfaces with the Bishop invariant λ=0 behave very differently from hyperbolic Bishop surfaces and elliptic Bishop surfaces with non-vanishing Bishop invariants.     

Intrinsic invariants of cross caps

It is classically known that generic smooth maps of \(\varvec{R}^2\) into \(\varvec{R}^3\) admit only isolated cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap \(f_{\mathrm{std }}(u,v)=(u,uv,v^2)\) has non-trivial isometric deformations with infinite-dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.     

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.     

Refined elliptic tropical enumerative invariants

We suggest a new refined (i.e., depending on a parameter) tropical enumerative invariant of toric surfaces. This is the first known enumerative invariant that counts tropical curves of positive genus with marked vertices. Our invariant extends the refined rational broccoli invariant invented by L. Göttsche and the first author, though there is a serious difference between the invariants: our elliptic invariant counts weights assigned partly to individual tropical curves and partly to collections of tropical curves, and our invariant is not always multiplicative over the vertices of the counted tropical curves as was the case for other known tropical enumerative invariants of toric surfaces. As a consequence we define elliptic broccoli curves and elliptic broccoli invariants as well as elliptic tropical descendant invariants for any toric surface.     

Mutation invariant functions on cluster ensembles

We define the notion of a mutation invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We show that these invariants have geometric and number theoretic interpretations, and classify them for ensembles associated to affine Dynkin diagrams. The primary tool used in this classification is the relationship between cluster algebras and the Teichmüller theory of surfaces.     

On the behavior of quasi-local mass at the infinity along nearly round surfaces

In this article, we study the limiting behavior of the Brown–York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat three manifold can be approximated by some geometric invariants of a family of nearly round surfaces, which approach to infinity of the manifold.     

Differential geometry of curves in Lagrange Grassmannians with given Young diagram

Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associated with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.     

Joint Invariant Signatures

A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated. August 25, 1999. Final version received: May 3, 2000. Online publication: xxxx.