【作者】 张媛媛；

【导师】 朱春钢；

【作者基本信息】 大连理工大学 ， 计算数学， 2021， 硕士

【摘要】 计算机辅助几何设计(简称CAGD)的重点研究内容之一是曲线曲面的表示与逼近,CAGD十分重视CAD/CAM的数学理论和几何体的构造,利用数学理论描绘曲线、曲面、零部件、装配件等几何形状间的配合、包含、约束等关系,利用计算机手段对这些几何形状进行分析、调整、优化进而达到对产品设计等预期的目标。由于Bézier曲线具有变差缩减、细分等优良性质和良好的形状控制能力,所以在曲线造型中得到了广泛应用,在CAGD领域中占据非常重要的位置。当利用曲线进行渲染、交叉测试或设计等实际应用时,就出现了一个问题,即折线多边形在多大程度上近似于精确的曲线几何形状,因此利用折线多边形逼近Bézier曲线并给出其逼近误差界是CAGD长期以来关注的研究问题之一。最简单的折线多边形逼近方式是采用连接Bézier多项式曲线的控制顶点所构成的控制多边形来逼近曲线。1999年,Nairn等人利用控制顶点序列的二阶差分和一个仅依赖于多项式次数的常数,给出控制多边形来逼近Bézier多项式曲线时的误差上确界。在此基础上,2005年,Zhang和Wang采用相邻三个控制点的线性组合来定义拟控制顶点,利用连接这些拟控制顶点得到的拟控制多边形来逼近Bézier多项式曲线,并给出误差上确界,拟控制多边形的逼近曲线效果要优于用控制多边形逼近。在本文中,基于Zhang和Wang的工作,对拟控制顶点的定义进行推广,将最初每相邻三个控制顶点的线性组合系数由原来的1/4,1/2,1/4推广为更具有一般性的系数l,1-2l,l,从而定义了一类新的拟控制顶点,称之为广义拟控制顶点,连接这些广义拟控制顶点得到了一类新的逼近折线多边形,称之为广义拟控制多边形。本文利用广义拟控制多边形来逼近Bézier多项式曲线,其中广义拟控制多边形的选取与Bézier多项式曲线次数相关,针对性更强。我们给出了用广义拟控制多边形逼近Bézier多项式曲线的逼近定理与误差界,并给出了该定理的详细证明。显然,与上述两种方法相比,本文方法同样简单易行并且逼近曲线的误差更好。最后,本文给出典型实例来对比分析用控制多边形、拟控制多边形和广义拟控制多边形逼近Bézier多项式曲线的效果,实例结果表明用广义拟控制多边形逼近Bézier多项式曲线的效果更好。更多还原

【Abstract】 The representation and approximation of curves and surfaces is the key research content of Computer Aided Geometric Design(CAGD).CAGD attaches great importance to the mathematical theory and the structure of geometric bodies of CAD/CAM.It uses mathematical theories to describe the relationship between fits,inclusions,and constraints among geometric shapes such as curves,surfaces,parts,assem blies,etc.And using computer means analyze,adjust and optimize these geometric shapes to achieve the expected goals of product design.Bézier curve has excellent properties such as variation reduction,subdivision,and good shape control ability so it has been widely used in curve modeling and occupies a very important position in the CAGD field.When we use curves for practical applications such as rendering,cross-testing or design,a question arises,that is,how close the polyline polygon is to the exact curve geometry.Therefore,the use of polyline polygons to approximate the curve and give its approximation error bound is one of the research issues that CAGD has been paying attention to for a long time.The simplest way a broken line polygon approximates Bézier curve is using the Bézier control polygon which is produced by connecting the Bézier control points.In 1999,Nairn et al.obtained a sharp,quantitative bound on the distance between a Bézier polynomial curve and its Bézier control polygon.The bound depends on the maximal absolute difference of control point and the degree of Bézier polynomial curve.In 2005,Zhang and Wang defined a quasi-control polygon by connecting the points which are the kind of linear combination of Bézier control points.Zhang and Wang used it to approximate Bézier polynomial curve and the effect of using quasi-control polygon to approximate Bézier curve is better than that of using control polygon to approximate.In this paper,based on the work of Zhang and Wang,the definition of quasi-control points is generalized.The initial linear combination coefficients of each three adjacent control points are extended from the original 1/4,1/2,and 1/4 to more general coefficients l,1-2l,and l.Therefore,a new type of quasi-control points is defined,which is called generalized quasi-control points.Connecting these generalized quasi-control points,a new type of approximating broken line polygon is obtained,which is called generalized quasi-control polygons.In this paper,the generalized quasi-control polygon is used to approximate the polynomial curve.The selection of the generalized quasi-control polygon is related to the degree of the Bézier polynomial curve and it’s more targeted.We give the approximation theorem and error bounds for approximating Bézier polynomial curves with generalized quasi-control polygons,and give a detailed proof of this theorem.Obviously,compared with the above two methods,the method proposed in this paper is equally simple and accessible to complete and the error of approximating the curve is better.Finally,this paper gives typical examples to compare and analyze the effect of approximating polynomial curve with using the control polygon,the quasi-control and the generalized quasi-control polygon to approximate Bézier polynomial curve.And the result of the examples shows that the effect of approximating Bézier polynomial curve with generalized quasi-control polygon is better.更多还原