【作者】 姜娜；

【导师】 张彩明；

【作者基本信息】 山东大学 ， 计算机应用技术， 2009， 硕士

【摘要】 在CAD/CAM造型系统中,随着曲线曲面造型技术的发展,人们提出了许多相关的理论。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。Bezier曲线和曲面广泛应用于CAGD和计算机图形学中,并且在形状设计方面有很多好的性质。所以对Bezier曲线或者曲面的设计和形状修改是曲线曲面设计的一个重要的问题。当生成Bezier曲线或曲面时,我们往往因为设计的需要而对它做形状的修改。然而实际的Bezier曲线或曲面的形状修改问题往往可以归结为数学优化问题,即可以通过用解方程的方式解决这类问题。参数曲线曲面造型按用户提供的初始信息不同可分为两类:一类是自由设计方法,它只要求设计者根据构思给出一些控制点和控制参数来定义曲线和曲面,然后在设计过程中允许改变这些控制点和参数来调整曲线和曲面的形状,直至它们符合设计要求为止。另一类是插值或逼近法(工程上统称为拟合法),其特点是给定一组离散点,要求生成的曲线或曲面要么通过所有这些点(成为插值曲线或曲面),要么以一定的精度贴近这些点(称为逼近曲线和曲面)。这两类方法生成的曲线曲面的形状都受参数化的影响。参数化既决定了所表示曲线曲面的形状,也决定了该曲线曲面上的点与其参数域内的点(即参数值)之间的一种对应关系。由此可见,参数化方法和插值逼近技术是曲线曲面造型的基础问题,具有重要的理论价值和实际意义。另外,在参数曲线曲面的设计应用中,单独的一段Bezier曲线曲面表示能力有限,特别是对飞机、汽车等现实生活中千姿百态的自由曲线曲面形状,由于它们形状复杂,光顺性要求极高,需要分段、分片进行表示。用户首先经常遇到的一个问题就是延长以后的曲线不能满足精度要求,需要重新调整。针对以上的问题本文提出了一种构造G~2连续的四次Bezier插值曲线的新方法,主要以曲线的二阶几何连续和曲线的能量为约束条件,进行形状优化设计。根据CAGD中几何连续性能客观准确地度量参数样条曲线连接光滑度的特点,用在每相邻的两个型值点之间增加一个控制点和两个自由度构造一段Bezier曲线的方法,研究了两条Bezier曲线在G~2连续下的光滑连接。同时给出了相关的实例。与已有的构造G~2连续曲线的方法相比,新方法构造的曲线灵活性更强且具有局部可调整性。用新方法构造G~2连续曲线可提供更多的自由度,用于提高设计和构造曲线的灵活性并且很好的控制曲线的形状。更多还原

【Abstract】 In CAD/CAM modeling system,with the development of Surface modeling, more theories were proposed.Computer-Aided-Geometric-Design mainly researches on free curve and surface.And Bezier curve and surfaces are one kind of the most commonly used parametric curves in CAGD and Computer Graphics.It is an important problem to modify the shape of the Bezier curve and surface.Developing more convenient techniques for designing and modifying Bezier curve/surface is an important problem.When the Bezier curve/surface is generated,we always want to modify the shape of the curve/surface in order to satisfy the design.The solution of the practical problem often leads to solving optimization problem.According to the initial information,curves and surfaces modeling can be divided into two methods.One is free form designing,based on control points and parameters,designers can define curves or surfaces and modify interactively until the shapes satisfy the design objective.The other is the technology of interpolation and approximation.Curves and surfaces reconstructed from the given points by interpolation are called as interpolation curves and surfaces or by approximation called as approximation curves and surfaces.These two methods are all influenced by parameterization.So curves and surfaces parameterization and the technologies of interpolation and approximation are the foundation of geometric modeling and have important theory value and practical meaning.Besides,in the design of curves and surfaces one separate section of Bezier curves and surfaces has limited capacity,especially the various free curves and surfaces like automobile,airplane surface shapes,many sections are needed because of their complicated shapes and highly demanding smoothing.First of all a problem which users often encountered is that the extended curve unable to meet the accuracy requirements and it need to be re-adjusted.According to the issue,this paper presents a new method for constructing a G~2 quartic Bezier interpolation curve,referring to the curves’ G~2 continuity and energy constraint to optimize the shape.According to the fact that geometric continuity in CAGD can accurately and objectively measure the degree of smooth for the connection between two spline parameter curves,by adding a control point and two degree of freedom on each sub-interval to construct a segment of the Bezier curve, we research the smooth connection between Bezier curves in G~2 continuity and proposes some examples.Compared with the existing methods of constructing G~2 curves,the curve by the new method has the properties that it has better flexibility and is locally adjustable.Moreover,G~2 curve by the new method offers additional degrees of freedom which can be used to increase the flexibility of construction and design to control the shape of the curve and therefore make the shape more desirable.更多还原