| 1. | We find out the ball basis of the space kn and define the h - ball curves with this ball basis
由这组基通过控制多边形定义的曲线即是空间k _ n中的ball曲线 |
| 2. | By changing the value of the shape parameter , we can adjust the approaching degree of the curves to their control polygon
通过改变局部形状参数的取值，可以调整曲线接近其控制多边形的程度。 |
| 3. | This paper present a sufficient and necessary condition of constructing quintic ph curves and analyze the geometric meaning of the control polygon
本文给出了构成五次ph曲线的充要条件，分析了其控制多边形的几何意义。 |
| 4. | The main innovation of our method is that we only need construct polygonal mesh possessing simple symmetric properties on both sides of control polygon edges of interpolated curves , and do n ' t need modify the subdivision rules near the interpolation curves during the process of subdivision . thus the subdivision rules are simple . the process is convergent and the limit surface is c everywhere except a finite number of points
该方法的主要创新思想是，在被插值曲线的控制多边形两侧构造具有简单对称性质的多边形网格，而在细分过程中，则无须修改被插值曲线附近的细分规则，凶此细分算法是简单的，细分过程是收敛的，且最终的插值曲面除有限个点外是c ~ 2连续的。 |
| 5. | Its accomplishment is simple and intuitional . the main innovation of this method is that we only need construct a symmetric zonal mesh for each interpolated curve respectively and write appropriate knot spacings for mesh edges . then applying nurcc rules to the initial control polygonal mesh consisting of those symmetric zonal meshes can generate a surface that satisfies interpolation constrains
该方法的主要创新思想是在被插值曲线的控制多边形两侧构造具有简单对称性质的四边形网格，称为对称网格带，但关键还在于对网格边赋予适当的节点距，并对由这些对称网格带组成的初始控制网格应用非均匀有理catmull一clark细分规则，来生成满足插值要求的曲面 |
| 6. | The parametric speed of the curve is firstly approximated by the bezier polynomial which takes the lengths of control polygon ' s edges of the direction curve of normal as bezier coordinates . then the corresponding geometric offset approximation algorithm is given . moreover , an offset approximation with high precision is obtained by degree elevation of the direction curve of normal
首先利用以法矢方向曲线的控制多边形边长为b zier纵标的b zier多项式来逼近曲线的参数速度，给出了相应的几何等距逼近算法，进一步结合法矢方向曲线的升阶获得了高精度逼近 |
| 7. | Hence designers can adjust the shape of curves by changing not only control points but also shape factor . our experiments show that h - bezier model approximate to the control polygon more closely than bezier model . so they are suitable to shape design and modeling in cad systems
而且h - b zier曲线还引入了一个称为形状因子的参数，形状设计者不仅可以像b zier曲线一样通过调节控制多边形来控制曲线形状，而且还可以调节形状因子来调整曲线对控制多边形的逼近程度 |
| 8. | This process is as follows : firstly , the data of different section contour and of section line have been obtained by reverse - engineering ; secondly , non - uniform b - spline approximation algorithm is used to fit the discrete data ; then the data have been optimized and reduced by adopting adaptive sampling of key points of the fitted curve based on vertical distance , sequently the adaptive sampling data is transformed into the format of the . ibl file of the famous 3 - d design software pro / e . therefore we reconstruct the surface and a model is generated ; lastly , the effectiveness of the adopted algorithm and modeling approach are demonstrated by the experiments
建模型的过程中，首先运用逆向工程测量方法，获得曲面物体上不同截面轮廓线的测量数据点列，接着对测量数据点进行平滑处理，用最小二乘法求解基于数据点列的控制多边形，以非均匀三次b样条曲线拟合截面轮廓曲线；然后利用基于垂距的数据采样算法，对拟合曲线上的数据点进行重新采样，达到优化数据点的分布及减少描述曲线数据量的目的。 |
| 9. | This new basis provides properties analogous to berstein polynomials , including symmetry , zeros of the basis functions , positivity , normalization , etc . based on this new basis , we define a new kind of curves , to be called h - bezier curves , with control polygon
本文利用多项式混合双曲形式在空间中构造了一组新的基，称为h - b zier基，它具有类似于bernstein基的端点性质，零点阶数，正性，正规性质，对称性等性质进一步，文章通过控制多边形的方式定义了h - b zier曲线 |
| 10. | During the past decades , some researchers , such as : bezier , kj . versprile , deboor and cox , etc . had made great progress in the filed of constrained b - spline curve and surface fitting . in this article the method of constrained b - spline curve was introduced , which is used to figure out the control vertexes . using this interpolation method we can calculate every points of a uniform b - spline
并且分析了在各种端点情况下，在重节点的情况下，如何反算控制多边形的顶点；如何在求出控制多边形顶点之后，插值计算b样条拟合曲线上的每一点；并通过结合双尾船的型线，采用visualc + +和autocad2002为平台，分别编制了相应的软件，对提供的型值做出绘图处理，取得了良好的效果。 |
