曲线的挠率

Properties of Curves, Surface in Three Dimensions, The First Fundamental Form

课程内容：平面与空间中曲线，曲率和挠率，全曲率，同周边不等式，曲面的基本式，平

For the application of Mathematica in the differential geometry,this paper introduces how to processe the curve curvature and the torsion questions,the graphic expression of the curvature and the torsions,and draw curves by the curvature and the torsions. Curvature ; torsion ; solution curve ; osculating plane ; natural parameter ; natural equation

利用Mathematica，只要输入有关系统命令就可以得到期望的结果。1曲率和挠率及其图形意义假设R3上的曲线一般参数表示为r=r它的曲率和挠率计算公式是：κ=‖r′×r"‖‖r′‖3τ=‖r′×r‖我们知道，空间曲线r=r在一点(s=s0)邻近的近似方程为ξ=sη=120κs2ζ=160κ0τs3通过近似方程可以确定曲线在其基本三棱形的三个平面(即密切平面，法平面，从切平面)上的投

In this paper a framework is built for differential quantity calculation on spatial curves of point cloud and a new matching method for given point curves is proposed.

建立了点云几何分析的相关理论框架，即定义和计算点云潜在曲线的几何微分量，包括Frenet标架、曲率、挠率等；在此基础上提出一种新的点云空间曲线匹配方法。

Finally, a method is developed for detecting feature points of the fragment con-tour based on total curvature.

将曲率和挠率相结合得出轮廓曲线上各点的全曲率，然后由全曲率值来检测碎片轮廓的特征点。

In order to reduce the cost of matching, we used multiple scale technique. 4. We presented a novel method of searching for similar fragments of 3D curves. with this method, a hash vector is associated with each fixed-length fragments of 3D sherd. Each vector consists of low frequence component of Fourier-Like spectrum for the distance between profile curve and the centroid.

依据样条曲线的基本理论，研究了基于B样条的轮廓曲线的匹配方法，给出了由轮廓曲线不变量曲率和挠率构造的相似不变量的选取以及基于相似矩阵的匹配算法，并对该算法的时间复杂度作了估计，同时，将多尺度技术引入到物体轮廓的匹配问题中。

Theory; the spatial meshing theory includes comparative motion, comparative differential and conjugative curved surface; and the application of meshing theory includes gear drive and worm drive. The spatial meshing theory is the main part of the course.

课程的重点讲解内容有曲线的参数方程、切线、法面、弧长、曲率、空间曲线的基本公式、挠率及平面曲线的基本公式等；曲面第一和第二基本公式，法曲率、主方向和主曲率、欧拉公式、短程挠率、欧拉公式和贝特朗公式推广、相对法曲率和相对短程挠率等；刚体的绝对和相对运动速度、相对微分和绝对微分、相对速度和相对微分、轨迹曲面的法曲率和短程挠率等；空间共轭曲面的啮合条件、诱导法曲率、两类界限点、等距共轭曲面、空间啮合的二次接触原理等；蜗杆传动的数学模型建立及啮合特性分析方法等。

The estimation of forming part"s sideline The estimation of forming part"s sideline is significant in weld and assembly process, the paper puts forth two kinds of method to realize quantitative analysis for the deformed sideline, one is using line contour error to estimate deformed sideline combined with GA and taking the value of line contour error as target to check whether it is in the range of precision; The other is comparing curvature and torsion of design model with curvature and torsion of deformed model to check the sidelines precision.

冲压件边线变形的评价。冲压件的边线变形的评价在焊装工艺中有着重要意义，本文提出两种方法来实现冲压件边线的变形评价：一是借鉴线轮廓度评定的方法，结合遗传算法计算出边线的线轮廓度，并以线轮廓度值作为边线是否满足要求的评价指标：二是通过比较变形前后曲线的曲率和挠率来评价边线的变形情况。

According to the objective of third-order derivative related to torsions, the formula to calculate the cubic optimized Hermite interpolation curves is given. And then we propose a new method to construct curves.

如果以与挠率有关的三阶导数为目标，给出优化三次Hermite曲线的计算公式，从而提出一种新的曲线构造方法。

For the application of Mathematica in the differential geometry,this paper introduces how to processe the curve curvature and the torsion questions,the graphic expression of the curvature and the torsions,and draw curves by the curvature and the torsions. Curvature ; torsion ; solution curve ; osculating plane ; natural parameter ; natural equation

利用Mathematica，只要输入有关系统命令就可以得到期望的结果。1曲率和挠率及其图形意义假设R3上的曲线一般参数表示为r=r它的曲率和挠率计算公式是：κ=‖r′×r&‖‖r′‖3τ=‖r′×r‖我们知道，空间曲线r=r在一点(s=s0)邻近的近似方程为ξ=sη=120κs2ζ=160κ0τs3通过近似方程可以确定曲线在其基本三棱形的三个平面(即密切平面，法平面，从切平面)上的投

Therefore, the contour description combines the simple global information with lo-cal feature points.

轮廓曲线的描述就由其上每一点的曲率和挠率以及特征点来表征。

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