几何的

Conformal Geometric Algebra is a new tool for geometric representation and computation.

共形几何代数是一个新的几何表示和计算工具。

Also, we can present the geometric representation of biological sequences and structures. So the genometric topology and group theory are important also.

同样我们可以用几何方法来表示生物序列和二级结构，几何拓扑和群论也发挥了很大的作用。

The surfaces of discrete form such as meshes, subdivision surfaces and point surfaces are becoming more and more important methods of geometric representation in the fields of computer graphics and geometric design.

离散曲面——网格曲面、细分曲面和点云曲面正逐渐成为计算机图形学和几何设计领域中的重要几何表示方法。

Collision detection is one of a number of geometrical applications where robustness is very important.

碰撞检测是许多几何应用的一种，容错性对几何应用非常重要。

Based on the summarizing of the exiting algorithms of the texture mapping and antialiasing, this paper goes deeply into the basic theory and processing of texture mapping. The texture compress algorithm which can satisfy the need of the memory and time is given for the stochastic sampling in the inverse texture mapping; the geometry transformation in the texture mapping are discussed and the concepts of the unitary parametrization and the patch parametrization is proposed, some kinds of the unitary parametrization methods of parametric surfaces and the geometry transformation in the patch parametrization are also presented; in order to solve the aliasing in the parametrization of intermediate surface-sphere in two-part texture mapping, an area-preserving transformation is presented also; to improve the quality of the graph the aliasing in the texture mapping is analyzed from the aspect of signal process, and the modified summed-area tables algorithms is proposed; the distortion of the scanline conversion algorithms under perspective transformation is analyzed and the proper interpolation algorithm is presented so the aliasing problem of the polyhedral object after the unitary parametrization is efficiently solved.

本文在总结现有的纹理映射和反走样算法的基础上，深入研究了纹理映射的基本原理与过程，根据反向纹理映射对纹理随机采样的特殊要求，提出了一种可满足存储空间和寻址时间的纹理压缩存储算法；通过对纹理映射中几何变换理论的进一步深入探讨，提出了整体参数化和面片参数化的概念，并给出了参数化曲面的各种整体参数化方法和面片参数化的几何变换形式；针对现有的两步映射算法所存在的中间曲面—球面的参数化问题，提出一种等积映射方法，较好地实现了由非参数化面片拼接而成的多面体表面的纹理映射；从信号采样的角度深入分析了纹理映射中走样产生的原因，提出了改进的区域求和表算法，使图形生成的效果有所提高；还分析了一般的扫描线算法在透视变换中所存在的问题，提出了一种正确的插值算法，从而解决了整体参数化后多面体表面纹理映射的几何变形问题。

Introducing the fundamental knowledge of modern differential geometry, we present the geometric descriptions of Lie symmetry of the system.

将现代微分几何知识引入机电动力系统，给出该系统Lie对称性的几何描述。

The geometrical frameworks for constrained mechanical systems are stud-ied with the help of modern differential geometry in this thesis.

本文用现代微分几何方法研究约束力学系统的几何结构。

After introducing the basic knowledge of geometric algebra such as outer product,inner product and geometric product,this paper focused on the CGA description and computation with graphic reflection,rotation,translation,rigid body motion and screw motion,and gave the experimental demonstrations.

在简单介绍外积、内积和几何积等基本概念之后，重点论述了共形几何代数在图形反射、旋转、平移等变换和刚体运动、螺旋运动等方面的描述和计算方法，并给出了实验示例。

Space curve is one of the basic elements in geometric modeling, and its effective representation is the basis of modeling. Parametric representation based on features is a kind of high-level representation including semantic information. In Chapter 4, para- metric representations based on features for free form space curve are investigated, and a parametric model for the loop of plain knitted fabric is established, in which 5 parameters with practical meanings are used to represent the shape of a loop and the relationship between loops. A new shrinkage prediction method for knitted fabric are proposed based on the parametric representation and energy minimizing. Experimental results on some plain knitted fabrics illustrate the feasibility of the proposed method.

空间曲线是造型中最基本的几何元素之一，对其有效的表示是造型的基础，而特征参数化表示是其中一种包含语义信息的高层次表示，本文对空间自由曲线的特征参数化表示进行研究，建立了一种针织物线圈的特征参数化几何模型，采用5个具有实际意义的参数表示织物线圈的几何形状及线圈之间的位置关系，并将该模型与能量极小化的方法相结合，得到了一种能较好地预测针织物缩水率的新模型，对随机选取的一些针织布缩水预测实验表明了算法的可行性。

"Among the principal branch es of mathematics are algebra, analysis, arithmetic, combinatorics, Euclidean and non-Euclidean geometries, game theory, number theory, numerical analysis, optimization, probability, set theory, statistics, topology, and trigonometry."

数学的主要分支有：代数学、分析、算术、组合学、欧几里德几何和非欧几里德几何、赛局论、数论、计算数学、最优化、机率论、集合论、统计学、拓扑学和三角学。

The labia have now been sutured together almost completely.The drains and the Foley catheter come out at the top.

此刻阴唇已经几乎完全的缝在一起了，排除多余淤血体液的管子和Foley导管从顶端冒出来。

To get the business done, I suggest we split the difference in price.

为了做成这笔生意，我建议我们在价格上大家各让一半。