几何形状

This paper first summarizes recent researches on this topic, and then proposes some methods for curve modeling with cubic algebraic curve based on pure geometric constraints that involve control points, tangent directions and curvatures.

然后提出了一种基于几何约束的三次代数曲线的插值方法，该方法完全通过几何量如控制顶点、切线和曲率来控制三次代数曲线的形状，使得对三次代数曲线的编辑与对三次 B-样条曲线的编辑一样灵活方便。

Chapter 1 briefs the relation between invariance and computer vision and summarizes the research and application of invariance in computer vision. Chapter 2 first derives the transformations of three camera models, then makes the correpondences between the models and three typical geometrical transformation groups by analysing the transformations respectively. The correspondences supply the theoretical basis for applying geometrical invariants to resolve the problems of computer vision. In Chapter 3, we describe the geometrical invariant theory and prove some geometrical invariants of coplanar points, lines or conics by algebraic method. In order to use the invariants of conic pairs to describe general 2D shapes, we discuss the perspectively invariant representation of planar curves using conies in detail. A system consisted of two TMS320C25 and based on moment invariants is introduced in Chapter 5. The system can recognize more than 30 different shapes of object model or more than 10 plane models with similar shape in real time.

第一章简述了不变性与计算机视觉的关系，以及计算机视觉中的不变性研究和应用概况；第二章推导了计算机视觉中常用三种投影模型的变换关系，通过对这三种变换关系的分析，分别建立了这三种投影模型和几何学中的三种变换群之间的一一对应关系，为几何不变性在计算机视觉中的应用提供了理论基础；在第三章中，我们介绍了几何不变性的理论，并且用代数方法证明了共面点、直线、二次曲线的几何不变量和射影不变量；为了把二次曲线的不变量用于一般二维形状描述，在第四章中我们详细地讨论了用二次曲线实现一般平面曲线的透视不变性表示的方法；第五章介绍了用两片TMS320C25构成的、基于不变矩形特征的运动目标实时识别系统。

After briefly reviewing the classification and history of curve and surface modeling, we discuss the characters, applications and representative work on three types of geometric approximation. And we present a survey of the vertex correspondence problem and the vertex interpolation problem in 2D and 3D shape blending modeling and key frame animation.

作者简要回顾了曲线曲面造型的分类和发展历史，论述了曲线曲面造型中所遇到的三类几何逼近问题的特点、应用和代表性工作，并就用于二维和三维形状混合造型的几何插值方法从顶点对应和顶点插值两方面做了一个系统的综述。

Based on the source investigation and the comparative approach of history of mathematics,a historical background of Gauss' competition essay and its contribution to intrinsic differential geometry are discussed.

高斯(C.F.Gauss，1777-1855)研究微分几何的出发点是"我们是否可以从曲面本身的度量出发决定曲面在空间的形状"[1]，这就是所谓的内蕴微分几何。

The geometrical parameters of the wing and tail planform and few geometrical parameters of fuselage which have more on the configuration are optimized and those constant parameters are given values according to the designer's experience or the optimizations of the parts.

在优化时，将整机外形几何参数分为优化变量和固定参量；主要优化对总体外形影响较大的机翼、尾翼的平面形状几何参数以及少数几个机身的几何参数；对于固定参量，根据经验或部件层次上的优化结果给出固定值。

The first class of curve contains the quartic Wang-Ball and Said-Ball curve and many curves locating between them. The second class of curve contains the quartic Said-Ball and Bézier curve and many curves locating between them. By analyzing the relation between the new curves and the quartic B6zier curve, the geometric meaning of the shape parameters is obtained, and the geometrical drawing method of the new curves is given.

第一种曲线包含了四次Wang-Ball和Said-Ball曲线以及介于它们之间的无数曲线；第二种曲线包含了四次Said-Ball和Bézier曲线以及介于它们之闻的无数曲线；通过分析新曲线与四次Bézier曲线之问的关系，得出了形状参数的几何意义，并给出了它们的几何作图法。

Experiments show that C-B spline curve, uniform B-spline with shape parameter, hyperbolic polynomial uniform B-spline with shape parameter and trigonometric polynomial uniform B-spline with shape parameter can be used to produce some frequently used free form curves in the industrial field when the shape parameter is assigned with some specific values.

通过调整形状参数来修改曲线形状是计算机辅助几何设计中一个非常活跃的研究课题，研究形状参数的变化对样条曲线形状的影响，对于增加曲线造型的灵活性是非常必要的。

The problem of the determination of the final geometry of the single-layer reticulated shell with zero element internal force is also successfully solved by using this method. Based on the research on the shape determination theory a conclusion has been obtained: the distribution of the initial self-internal-force vector determines the initial shape of an assembly which has form finding problem.

通过对形状确定问题的研究，可得出如下的结论：体系自应力分布矢量的方向决定体系初始态几何；体系自应力分布矢量的大小只影响体系的零状态几何；体系各个单元的截面尺寸设计也将影响体系的零状态几何。

The generalized criterion for minimum zone in the determination of form error of surface is presented on the basis of linear minimax techniques, and the criteria of such surfaces as for cylindricity and straightness with arbitrary direction are derived from this generalized criterion.

本文讨论形状误差最小区城的判别准则和判别方法。根据线性极差极小化问题解的特征，给出形状误差最小条件的统一判别准则。推导出各种形状误差最小区域判别法(包括圆柱度误差和任意方向直线度误差)，并在几何判别的基础上得出代数判别的格式。

The basic principle is that basic geometry shape of openings in the photo is revised; the image can't be distortion by use of projection reconstruction; the corrected image is exerted by open operation and close operation of gray-image; the shape of openings is gained lastly. The perimeter borderline of openings shape is extracted. The quantity of overbreak as well as underbreak is gained by comparing to extracted perimeter and designed perimeter. The same method is applied to other opening sections. Evaluation of overbreak as well as underbreak in the openings can be easily solved in the way of quantitative.

其基本原理是：对野外摄取的洞室的基本剖面形状进行几何校正，然后应用图像学中的投影重建方法使图像不失真，对修正后的图像实行灰度图像的开操作和闭操作运算，得到洞室的基本形状，然后用图像学中的边界提取技术提取洞室的周边界，与洞室的设计形状对比分析，得到洞室在该断面上的超欠挖方量，然后对其它断面采用同样的技术就可评价某一段或整个洞室的超欠挖数量，从定量的角度解决了工程中的超欠挖评价问题。

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。