Because this algorithm makes the small triangular face normal to be a byproduct of calculating point of intersection , it does not need other time to calculate it . so it is possible to display high quality realistic pictures smoothly on microcomputer 尤其是该算法计算切削后生成的小三角片的法向量成为计算交点的一个副产品，几乎不再占用额外的计算时间，为做到平滑地动态显示高质量的真实感图形提供了可能。
Then a new system of nonlinear equations can be formed from the variation gray of same point in multiple images and the optimum solution of the system can be obtained from gauss - newton and levenberg - marquardt algorithms , so that the normal vector at that point of the surface can be defined 再根据多幅图像上固定位置一点的灰度值变化列多个非线性方程，以gauss - newton算法和levenberg - marquardt算法为基础求非线性方程组的最优近似解，将选取点的法向量确定下来。
In this thesis , used thermocouple to sense the temperature in the space of the hearth , attributed the thermocouple based on calorifics and scientific arithmetic , acquired data by data acquiring card , processed data according to the demand of setting up temperature model and drawing graphics and then caculated the coordinates of isothermal points and normals of isothermal facets , ascertained the order of connecting points and the model of construct , drawed the isothermal facet of temperature field with opengl at last 通过数据采集板卡进行数据采集并与计算机接口进行数据处理。然后，将处理后特殊点的温度数据依据marchingcubes算法建立数学模型求取等温面上的点的坐标及各点的法向量等。最后，确定计算的三维坐标点的连接顺序和拓扑模型，利用opengl进行等温面的绘制、消隐和模式变换等工作。
A new system of nonlinear equations can be formed from the variation gray of same point in multiple images and the optimum solution of the system can be obtained , so that the normal vector at that point of the surface can be defined . then we can get the surface height at the point by applying composite numerical integration . according to variational calculus and finite difference method , the fitted surface is further iterated and modified , so the reconstruction error can be reduced 根据多幅图像上固定位置一点灰度值的变化列多个非线性方程，通过求解该非线性方程组，确定出各选取点的法向量；然后通过复化积分确定选取点的高度值，并利用变分和有限差分思想对所得表面进行进一步的迭代和修正，以减小重构误差。
As it can be expressed easily by argument equations , we use the conicoid as the criteria of points " classification . based on the point ' s normal and the adjacent field , corresponding to the conicoid equation such as plane , ball and cylinder , we search the points that fit the equation along the adjacent field . using this method , we realized the automatic classification of unorganized points 本文采用了可以比较方便地用参数方程表示的二次曲面作为数据点的分类标准，根据点的法向量以及其邻域结点的各种属性，对应于平面、球面、柱面等二次曲面，依据曲面方程的特点，从起始点开始沿邻域深度优先递归寻找符合方程的数据点，实现了散乱数据点的自动分类。
Since the varies of gun attitude angle and the vertical vector of the homograph plane are one - to - one , all the elements of modulation gun can be transformed into the calculation of the vertical vector angle , which avoids the calculation accuracy loss of the traditional approach under the condition of the tiny attitude angle disturbance 由于火炮姿态角变化与相应平面的法向量是一一对应的，从而可将调炮诸元转化为法向量夹角来计算，这样可避免传统方案在微小姿态角扰动情形下导致的计算精度可能的损失。
It presents the angle of the two normal vectors formed by the new and the old triangles , the threshold distance and the principle of maximal - minimal angle in the paper to constrain the growing of the triangles , so as to reduce the searching area , improve the correctness and confirm the quality of the mesh 该方法通过引入三角形法向量夹角、阈值距离以及最小内角等几个参数对三角形的生长进行约束,同时利用生长法,对空间中的离散点集进行逐步三角划分,直至所有三角形扩展完毕。
Based on data cloud , which is measured from 3 - coordinate measuring machine ( cmm ) or so , an algorithm to search lopological structure from 3d unorganized points using envelopment - box technology is proposed in this thesis . based on this method , we searching neighbor points of sampling point . we also improve the max - min angle criteria to realize local triangulation , and then get the normal of sampling point from the triangulation 本文以测量得到的曲面数据点点云为基础，提出了一种基于包围盒的自动寻找三维散乱数据点之间拓扑结构的方法，采用该方法寻找采样点的邻域结点，并对三角剖分中的典型优化准则?最小内角最大准则提出了改进，按照改进后的优化准则实现了采样点的局部三角划分，并进一步求解得到采样点的法向量，依据法向量及邻域拓扑关系在二次曲面的基础上实现了散乱数据曲面重构中的数据点的自动分类。
To solve the overflow or underflow of the components of appearance attributes caused by watermarking , we clip the error ones to the correct ranges by adjusting the corresponding basis function elements . the normalization operation would destroy the watermark embedded in the normal vector . the author presents a novel approach by rotating the normal vector around a fixed axis , which will modify its orientation , not magnitude 其中，针对在外观属性分量上植入水印的值溢出问题，作者提出了动态修改基函数的方法；考虑到直接修改法向量分量的方法会受到法向量单位化操作影响导致水印提取困难，作者提出一种绕固定轴旋转法向量的方法，有效地避免了单位化对已植入水印信息的影响。
In the second part , we investigate the compact submanifolds m with the parallel isoperimetric section in the real space forms rm ( c ) and prove that if there exists a parallel isoperimetric section on m , and the sectional curvature of m is always greater than zero , then m is contained in a hyper - sphere ; and get that the gauss curvature of the compact surfaces m with constant mean curvature in constant curvature space r4 ( c ) is always greater than zero , then m is a totally geodesic surface or a sphere , where an isoperimetric on m means a unit normal vector field defined globally on m with m1 ( ) = constant ( 2 )研究了实空间形式r ~ m ( c )中具有平行等参截面的紧致子流形m ，证明了具有一平行等参截面的子流形m ，如果m的截面曲率恒正，则m包含在r ~ m ( c )的一个超球面内；对于常曲率空间及r ~ 4 ( c )中具有常平均曲率的紧致曲面m ，如果m的高斯曲率处处大于零，则m或为r ~ m ( c )中的全测地曲面或为一球面。这里m上的等参截面是m上整体定义的单位法向量场，使得m关于它的平均曲率m _ 1 ( )是常数。
