二次曲线

We give three constraints of absolute conic images and use these constraints to evaluate absolute conic images and then to recover Euclidean reconstruction from projective reconstruction.

总结了三种关于绝对二次曲线的像的约束，并利用这些约束求解绝对二次曲线的像，进而实现从仿射重构恢复欧氏重构。

With the Lagrange multiplier method , the minimum distance of the center of a circle and a quadric surface was provided and the tangency condition of curve and surface was given.

利用拉格朗日乘子法求解二次曲线和二次曲面之间的最小距离，给出了曲线与曲面相切的条件。

An image conic can be the projection of either a planar conic or silhouette of a conicoid. We show that at least three images, taken from the same position with different camera orientations, of two planar conics, or two conicoids, or one conic and one conicoid, are needed for the calibration, and a closed-form solution can be obtained. Our new conic based calibration technique requires no knowledge of camera's orientations, and relies solely on image correspondences.

在不同的方位拍摄三幅或三幅以上图像，每幅图像至少包含两个空间平面二次曲线、或两个二次曲面、或一个平面二次曲线与一个二次曲面的投影，利用图像之间的二次曲线对应关系，可以确定摄像机的内参数矩阵，同时可以获得摄像机不同方位之间的旋转矩阵。

We reseach the equation of conic section and obtain the equation which can give all plane conic section .

探讨二次曲线的方程，构造了一个可以表示平面上所有九种二次曲线的方程。

Secondly, in our methods, the essential geometry of the image single axis geometry may be specified by six parameters and this may be estimated from one conic and one fundamental matrix (a total of 12 parameters) or may be minimally estimated from two conics (a total of 10 parameters).

本文证明了单轴旋转运动的不变量可以通过一个基本矩阵和一条二次曲线来确定，在这种情况下，由于基本矩阵的自由度为7，二次曲线的自由度为5，所需确定的参量个数仅为12，大大减少了不变量的计算量；本文同时证明单轴旋转运动的不变量可以通过最少两条二次曲线来确定，在这种情况下所需确定的参量个数仅为10，该方法是目前同类算法中参数最少的；本文提出了用多条二次曲线求解单轴旋转运动的不变量的最大似然估计算法，其所需确定的参量个数为6+2n，其中n为二次曲线的个数，该公式更深刻地反映了二次曲线与不变量的参数关系。

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。

In order to overcome shortcomings that the circular arc and ellipse could not be accurately represented by CE-Bézier curves, this method used the continuity condition of CE-Bézier curves and C-Bézier curves to resolve the representation of circular arc and ellipse in CE-Bézier curve modeling.

该方法首先构造了一种具有优良形状可调性和更好逼近性的带3个形状参数α，β，γ的三次扩展Bézier曲线(CE-Bézier曲线)；并针对CE-Bézier曲线无法精确表示圆弧和椭圆弧等二次曲线的缺点，利用CE-Bézier曲线与C-Bézier曲线间的拼接技术，解决了CE-Bézier曲线造型中圆弧和椭圆弧的表示问题。

Inner point and exterior point of a quadric and its plane of the type of contact plane;2. The article gives a way by which a quadratic curve is processed by a linkage,and then a quadric can be formed by revolving the quadratic curve as a generatrix around the axis.

提出一种利用能够生成二次曲线的连杆机构作辅助装置，将生成的二次曲线作为母线绕轴线旋转而形成旋转二次曲面的方法，从理论上进行了推导，并通过实例说明了该方法的应用。

The computational methods of line element conic curve s tangent point decided by any line on projective plane is presented on the basis of this study.

从实际计算的角度出发，使用N矢量表示视平面上的点和直线，并由线素二次曲线的射影定义推导出线素二次曲线的N矢量方程；在此基础上，给出了射影平面上任意一条直线所确定的线素二次曲线切点的N矢量的计算方法。

Furthermore,it can represent the elliptic curves,parabola and other quadratic curves without using rational form.

给出了一种基于三角函数的类三次参数曲线，该曲线不仅具有类似于三次Bézier曲线的诸多性质，而且无需有理形式即可精确地表示椭圆、抛物线等二次曲线。

It has been put forward that there exists single Ball point and double Ball points on the symmetrical connecting-rod curves of equilateral mechanisms.

从鲍尔点的形成原理出发，分析对称连杆曲线上鲍尔点的产生条件，提出等边机构的对称连杆曲线上有单鲍尔点和双鲍尔点。

The factory affiliated to the Group primarily manufactures multiple-purpose pincers, baking kits, knives, scissors, kitchenware, gardening tools and beauty care kits as well as other hardware tools, the annual production value of which reaches US$ 30 million dollars.

集团所属工厂主要生产多用钳、烤具、刀具、剪刀、厨具、花园工具、美容套等五金产品，年生产总值3000万美元，产品价廉物美、选料上乘、质量保证，深受国内外客户的青睐