【作者】 朱平；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 应用数学， 2009， 博士

【摘要】 CAD等几何造型系统希望建模工具能用更少的数据量表示几何模型,这有利于节省计算时间,提高工作效率,加快网络传输速度。Bezier曲线曲面和B-样条是CAD/CAM系统中广泛使用的造型工具,简化它们的建模技术对CAD系统有着实际的意义。本文对此展开深入研究,主要是两方面:Bezier曲线曲面的近似合并和变次数B-样条。主要研究成果及创新点如下:首先,传统的一般性的Bezier曲线近似合并只考虑曲线的参数连续。而我们考虑利用曲线的几何信息来研究合并问题,提出了Bezier曲线在L₂范数下保端点G²连续的最佳合并算法,即保持两条原曲线在左右端点的位置、切向和曲率大小。为了避免在两个端点处出现奇异点,我们还对误差定义作了修正,增加了正则项。与传统的保端点C²连续的方法相比,我们的方法能直接合并不同次数的两条曲线,显式地得到合并曲线的控制顶点,并且误差更小,因此逼近效果更好。如果要得到更高次的合并曲线,只需要提高合并次数,无需象以前的方法那样对原曲线进行升阶运算。其次,为了对CAD系统中的几何数据进行压缩,研究了两张相邻张量积Bezier曲面的合并问题。为了能更好地进行曲面合并逼近,利用张量积Bezier曲面细分后的矩阵表示给出了相邻张量积曲面可精确合并的充要条件,并在此基础上通过广义逆矩阵的方法求解出在L₂范数下合并逼近后的张量积Bezier曲面,得到了其控制顶点的显示表达式。与此同时,对带角点插值条件的曲面合并逼近也给出了结果。由于广义逆矩阵可以方便地求得最小二乘解,因此得到了能够显式表示,机时最省且逼近效果好的合并逼近算法。接着,我们研究了变次数B-样条。变次数B-样条是一种能够简化几何模型,压缩数据量的新的B-样条形式。本文对此进行了初步探讨,并第一次给出了最大变化次数小于3的变次数B-样条基函数的具体表达式,系统地给出了基函数的性质与曲线构造,并将之应用于样条曲线的升阶与合并,有效地简化了曲线模型。在利用变次数B-样条将代数双曲B-样条的升阶解释为几何割角之后,我们以此为基础得到了代数双曲B-样条的几何生成算法。由于代数双曲B-样条对造型系统有重要的意义,如能够精确表示双曲线、悬链线等常用的工程曲线,此算法有着实际的应用价值。同样地,利用变次数样条,也可以将其他样条曲线,如NUAT B-样条的升阶解释为几何割角过程,并得到它的几何构造。最后,鉴于以往的三角或双曲多项式样条模型定义在均匀节点上的缺陷,本文以代数双曲混合B-样条为例,将其推广到非均匀节点上去,并拥有B-基等很多良好的几何性质。更多还原

【Abstract】 It is better to represent geometric model with less amount of data by modeling tools in CAD modeling systems. This is good for reducing computing time, improving the efficiency of systems, speeding up the network transmission speed. Bézier curves, surfaces and B-spline are widely used as modeling tools in CAD/CAM. Their models’ simplification technology has practical significance on the CAD systems. In this paper, we have made in-depth studies on simplification technology, mainly two aspects: approximate merging of Bézier curves, surfaces and multi-degree B-spline. The main creative results are as follows.Firstly, in constrast to traditional methods, which only considered the components of the curves separately, we used geometric information about the curves to generate the merged curve and proposed optimal approximate merging of a pair of Bézier curves with G²-continuity in L₂ norm, where positions, tangents and curvatures were preserved at the two endpoints. For avoiding singular at the two endpoints, we amended the error definition and added one regularization term. Compared to traditional methods, our method could directly obtain control points of the merged curve, regardless of the degrees of the original curves and the approximation error was better. Furthermore, we obtained a higher degree merged curve through raising the merged Bézier curve’s degree instead of degree elevation of the original Bézier curves.Secondly, approximate merging of two adjacent tensor product Bézier surfaces was investigated to guarantee the compression of geometric data in CAD systems. Sufficient and necessary condition for precise merging of adjacent tensor product surfaces was obtained by using the matrix representation of subdivided Bézier surface, then merged tensor product Bézier surface was solved by generalized inverse matrices in L₂ norm based on precise merging condition and explicit representation of the merged surface’s control points was also obtained. At the same time, the result of approximate merging with corner interpolation was showed first time. Since the minimal least squares solution could be directly obtained by generalized inverse matrics, an approximate merging algorithm possessing explicit formula, less time consumption and better approximation result was found.After that, multi-degree B-spline（MD-spline） was investigated. Multi-degree B-spline is a new B-spline form to simplify geometric model and compress the amount of data. This paper made a preliminary study, and basis function formulae of MD-splines which maximal variational degree was lower than 3 was investigated first time. We gave basis function’s properties and curve’s construction completely, then applied them to degree elevation and mergence of spline curves for simplifing curve model. After degree elevation of algebraic hyperbolic B-spline can be interpreted as corner cutting using multi-degree B-spline successfully, we obtained geometric construction of algebraic hyperbolic B-spline based on above conclusion. Since algebraic hyperbolic B-spline has important meaning for modeling system, for example, it can represent hyperbola, catenary explicitly etc common engineering curve, this algorithm processes practical application value. Similarly, we can make use of multi-degree spline to obtain geometric construction of other splines, such as NUAT B-spline.Finally, in view of the previous triangle or hyperbolic polynomial spline model at uniform knots on the definition of the defect, taking a algebraic hyperbolic blending B-spline as an example, this paper extended to non-uniform knots, and the new spline holds a lot of good geometric properties such as B-basis.更多还原