| 1. | Covering differentiable manifold
覆盖微分流形 |
| 2. | In this papaer , a note about the proof of the chain rule in the book 《 an introduction to differentiable manifolds and riemannian geometry 》 is offered
给出了《微分流形与黎曼几何引论》一书中关于链法则证明的一个注记 |
| 3. | According to topological structure of control mesh , we generate generalized rational parametric surfaces in two manners . for a control mesh with consistent parameterization , the surface can be easily constructed because it can be mapped to differential manifold directly
对于具有简单拓扑结构，存在着一致的全局参数化的控制网格，我们将把控制网格直接映射到微分流形上，因此可以很容易地对曲面进行构造和控制。 |
| 4. | In the framework , a control mesh may be arbitrary one - dimensional or two - dimensional orientated topological manifold , and the curve or surface is defined on differential manifold homeomorphic to the control mesh with a potential function as its basis functions . this method is an extension of nurbs , which efficiently overcomes the limitations of nurbs
广义有理参数曲线曲面定义在与控制网格拓扑同胚的微分流形上，以高度一般的势函数为基函数，其控制网格可以是任意的一维拓扑流形和二维可定向拓扑流形。 |
| 5. | By employing the potential function , we construct the unit partition on differential manifold . and then , by regarding the curve and surface as a map from differential manifold to topological manifold , we present the framework of grpcs and discussed its basic properties in detail
然后通过势函数来构造微分流形上的单位分解，将曲线曲面看作微分流形到拓扑流形的映射，给出了广义有理参数曲线曲面的整体理论框架，并对广义有理参数曲线曲面的基本性质进行了讨论。 |
| 6. | Then a probability distribution of the states at the equilibrium corresponding a self - assembly model . all the possible can form a manifold called s , and the probability distribution of the states self - assembly system reached form a submanifold of 5 " , called a . so the difference of two self - assembly model is the division of two probability distribution at the manifold
这里，我们利用信息几何的知识给出了dna自装配的一个形式化模型，以分子两两构成的组合的个数为分量组成的向量表示自装配过程中的一个状态，那么，当每自装配系统达到平衡时，就有一个关于这些状态的一个概率分布，所有可能的概率分布形成了一个微分流形s 。 |
| 7. | Based on the analysis of topology structure of parallel mechanisms and using differential topology and differential manifolds as mathematical tools , we propose a new classification method . this method classifies singularities of parallel mechanisms into two basic types , i . e . topology singularity and parameterization singularity . this kind of classification has clear physical and mathematical meaning and fully reveals the characteristic of configuration space of parallel mechanisms
采用微分拓扑和微分流形等现代数学工具，在对并联机构位形空间的拓扑结构进行分析的基础上，提出了一种新的奇异位形的分类方法，即把奇异位形分为拓扑奇异位形、参数化奇异位形两种类型，这种分类方法充分体现了并联机构位形空间的特点，具有十分明确的物理和数学意义。 |
| 8. | The basic idea to construct grpcs is to establish object topology first , then use geometry to change the shape of differential manifold . in chapter 2 , we discuss the theoretical framework of grpcs that includes some relative idea about differential manifold . firstly , the definition of potential function on manifold is given
本文首先讨论了广义有理参数曲线曲面的理论基础，依次阐述了黎曼几何中关于流形、函数和映射的基本概念，并在此基础上提出了微分流形上势函数的定义。 |
| 9. | Its properties and design method is discussed in chapter 4 . for control meshes with arbitrary topology , we present a universal method in chapter 5 to construct parametric curves and surfaces . generalized rational parametric surface can be controlled precisely and flexible , and it is easy to model local features and 3d primitives
然后，在第五章中，我们将控制网格进一步推广到任意可定向二维拓扑流形，提出了一个通用的方法将控制网格映射到与之拓扑同胚的微分流形上，统一了广义有理参数曲线曲面的构造过程。 |
