【作者】 任叶庆；

【导师】 韩旭里；

【作者基本信息】 中南大学 ， 应用数学， 2011， 博士

【摘要】 计算机辅助几何设计(CAGD)主要研究以复杂方式自由变化的曲线曲面,即所谓的自由型曲线曲面,其中参数曲线曲面造型与形状调整是CAGD的一个重要内容。本文主要研究带几何约束的Bezier曲线造型方法；带可调整插值点的多项式样条的设计；扩展的二次B样条的构造与应用；基于函数高阶逼近的带局部形状参数的广义Bezier曲线、广义张量积Bezier曲面的造型与形状修改；利用对称混合函数生成网格的方法。并分别讨论了所构造的曲线曲面的性质,形状参数对曲线曲面形状调整的影响及连续拼接问题。全文共由七章组成。第一章简要介绍计算机辅助几何设计的来源及自由曲线曲面特别是Bezier曲线、曲面,B样条曲线曲面以及NURBS曲线曲面的发展历史,对各种曲线曲面形状调整方法的分类、特点、性质等进行了综述,并对本文主要的研究内容进行介绍。第二章是基于约束优化的Bezier曲线形状调整问题的研究。通过构造带修正向量的Bezier曲线,利用约束优化方法,对控制点进行修正,使形状修改和变形具有更大的灵活性。并进一步讨论了多个几何约束的情况下Bezier曲线的修正。第三章是对三次样条逼近曲线和插值曲线的统一表达式所生成曲线方法的研究。该方法只需调整形状参数的值,就可分别得到B样条曲线和插值曲线。还引入张力参数对曲线进行局部形状修改。同时为了提高曲线的连续阶数将曲线次数提高到四次,得到了与三次多项式样条结构类似的统一表达式。第四章是研究扩展的二次B样条曲线的构造与应用。在对圆锥型曲线的逼近问题上,利用所构造的曲线能较好地反映圆锥型曲线的性质。通过改变形状参数的值可以对曲线进行局部形状调整。并将扩展的二次B样条曲线用于带面积约束的直方图逼近中,由此将二次非均匀B样条曲线的应用范围进一步扩大。第五章是对Bezier曲线曲面的拓展研究,基于一种多节点函数的高阶逼近式,选择形状参数,分别定义带局部目标一、二阶导矢的广义Bezier曲线和在矩形域上定义带局部方向偏导矢的广义张量积Bezier曲面。通过形状参数的调整,能对较高次的Bezier曲线曲面进行有效地修改。还给出广义Bezier曲线曲面的拼接条件及应用。第六章定义并利用对称混合函数,提出带有两个形状参数的u-方向、v-方向的网格曲线生成方法,降低了用优化方法生成网格的复杂度,生成的网格有满意的形状。第七章是对全文工作的总结及对今后将要开展的工作提出我们初步的看法。更多还原

【Abstract】 Computer aided geometric design(CAGD)mainly researches curves and surfaces which vary in the free and complex forms, that is to say, curves and surfaces of freedom forms. Parametric curves and surfaces modeling and shape modification is one of the most important issue in CAGD.This thesis is devoted to construct Bezier curve with geometric constraints; to design piecewise cubic and quartic polynomial spline with adjusted interpolation points; to construct an extended quadratic B-spline curves; to design the general Bezier curves and surfaces based on a higher order approximation of function with local shape parameters; to study grid generation based on symmetric blend functions of boundary information. It is composed of seven chapters.In chapter 1, we briefly introduce the historical background and the present progress of problems that concern with curves and surfaces shape modification. The main works of this paper are concluded as well.In chapter 2, a new modeling method is used to construct Bezier curve with modified vectors based on constrained optimization. The control points are modified by using the optimization with signal and multiple constraints.In chapter 3, piecewise cubic and quartic polynomial curves with adjusted interpolation points are presented. The given representation is intergrated of cubic approximation and interpolating curve. By changing the values of local shape parameters, local approximating curves and local interpolating curves can be generated, respectively. Tension parameters are also considered to locally modify the shape of the curves.In chapter 4, presents a construction of an extened quadratic B-spline curves and its application. The properties of approximation conics are given based on the extended spline curves. The shape of the extended quadratic B-spline curves can be adjusted locally by changing the value of the shape parameters. Approximation for histograms also discussed by using the extended curves and the applications of the quadratic non-uniform B-spline curves can be extended as well.In chapter 5, generalized Bezier curves and surfaces with local shape parameters are presented. Based on a kind of higher approximate polynomials of a function, satisfying curves and surfaces can be generated by changing the given object derivative/partial derivatives vectors. Connecting of surface patches also considered.Chapter 6 presents a method to generate grid curves by defining and using a kind of symmetric blend functions. The expression of u-directional grid curves and v-directional grid curves with shape parameters are constructed.更多还原