【作者】 陈雪娟；

【导师】 曾晓明；

【作者基本信息】 厦门大学 ， 基础数学， 2007， 博士

【摘要】 计算机辅助几何设计(Computer Aided Geometric Design,简称CAGD)是随着航空、汽车等现代工业发展与计算机的出现而产生与发展起来的一门学科。而自由曲线、曲面造型是CAGD的重要内容。本篇论文主要研究曲线、曲面造型中的几何逼近和收敛性问题,在如下几个方面取得一些进展。1.曲线曲面的等距计算在几何造型、NC(Numerical Control)加工和机构运动学等领域具有广泛应用。一般来说,由于单位法向量的表达式中含有根号,所以等距曲线、曲面的函数表达式比原曲线、曲面的表达式更加复杂。因此常用低次有理参数曲线、曲面来逼近等距曲线、曲面。本文提出两种新的等距曲线的逼近方法:(1)等距曲线的Bézier逼近算法。此算法先将任意形式的参数曲线转化成分段三次Bézier曲线,利用Bézier曲线的性质容易得到逼近曲线的切向量和法向量,从而计算出其等距逼近曲线。(2)利用样条曲线插值的等距曲线逼近方法。利用样条曲线和原曲线加权组合构造一条新的有理曲线,该曲线通过插值原曲线的等距曲线上的采样点,从而逼近等距曲线。文中分析和比较了这两种算法的优缺点,并将第二种方法推广应用于求张量积等距曲面的逼近。2.在工程应用中,由于实际问题的需要,我们必须拓广古典意义下的等距曲线、曲面的定义,即改变古典等距定义中的等距方向和等距距离,我们称之为广义等距曲线、曲面。本文利用曲线上各点的切向量和法向量所形成的局部坐标系来确定等距方向,从而给出一种广义偏距曲线的新定义。并在此基础上做相应的讨论,研究其正则性、曲率和积分性质,推广了J.steiner关于卵形线的外等距线所围面积的一个著名定理。3.Bézier曲线、曲面的升阶和子划分算法在几何造型中是很重要的方法。将升阶和子划分算法分别应用于Bézier张量积曲面,会产生一系列的控制网格,得到原曲面的分片双线性逼近。本文将曲面的控制网格进行分片均匀参数化,给出了它们的任意阶离散偏导数的定义,证明这两种算法的光滑收敛性,即控制网格的离散偏导数都收敛于原Bézier曲面的相应的连续偏导数。Ron Goldman利用负二项分布作为基函数,定义了一种新的有理Bézier曲线。本文也证明了这类有理Bézier曲线的光滑升阶收敛性。这对于研究逼近曲线曲面的光滑性是很有意义的。4.细分曲面是自由曲面造型的强有力的工具。Catmull-Clark细分曲面将双三次B样条曲面推广到任意拓扑网格上。我们通过介绍相邻顶点的概念,利用控制点的一阶差分,得到Catmull-Clark曲面的控制网格的收敛率。而且,推导出一个计算细分后控制网格到Catmull-Clark曲面的距离公式。更多还原

【Abstract】 Computer Aided Geometric Design (CAGD) is a subject which emerged with the development of modern industry and computer science. Free-form curves and surfaces modeling is one of the most important tasks in CAGD. This dissertation focuses on solving geometric approximation and convergence problems in curves and surfaces modeling. The major contributions of this dissertation are summarized as follows.1. Constant radius offsetting for curves and surfaces is one of the most important geometric operations in CAD/CAM due to its immediate application to NC machining. Due to the square root function in the denominator of unit normal vectors generally, the exact offset curves and surfaces are not rational. Therefore approximations are needed, often by using rational parameter curves and surfaces with low degree. This dissertation presents two new methods of offset approximation: (1) Bezier Approximation algorithm of offset curves. This algorithm firstly translates arbitrary parameter curves into the piecewise cubic-degree Bezier curves, then using the properties of Bezier curve we can get the tangent vector and the normal vector of every point on the approximation curve, and calculate the approximating offset curve. (2) The approximating offset curve by interpolatory using spline curve. Spline curve and base curve are combined to generate a new rational curve by adding the weight. This curve approximates offset curve by interpolating some sample nodes on the offset curve. We analyze and compare the advantage and the weakness of these two algorithms, and apply the second method to the approximation of tensor product offset surface.2. In order to solve practical problems in real engineering applications, the classic definition of offset curve should be extended, which means the fixed distance and direction in the classic definition will not be a necessary request. This dissertation presents a definition of general offset curve, which has fixed offset distance, but variable offset direction. The offset direction is defined by the local coordinate system, which is formed of the tangent vector and the normal vector of every point on the curve. Based on this definition, the curvature、regular and integral properties of the general offset curve are discussed. As a result, J.steiner’s celebrated theorem of the oval is developed.3. The algorithms of degree elevation and subdivision for Bezier curves、surfaces play an important role in geometric modeling. For the Bezier tensor product surface, recursive degree elevation and subdivision both generate a sequence of control meshes that converge to the underlying Bezier surface, and get the piecewise bilinear approximation of the original surface. This dissertation uniformly parameterizes the control nets, provides the definition of its discrete partial derivatives for arbitrary order and proves the smooth convergence property of these two algorithms. That is, the discrete partial derivatives of control nets convergent to its corresponding continuous partial derivatives. Ron Goldman introduced an alternative notion of rational Bezier curves defined in terms of the negative degree Bernstein blending functions. This dissertation proves the smooth convergence property of degree elevation for this kind of rational Bezier curve as well. It is significative for the smooth property of approximating curves and surfaces.4. Subdivision surfaces are powerful and useful technique in modeling free-form surfaces. The Catmull-Clark subdivision surface was designed to generalize the bi-cubic B-spline surface to the meshes of arbitrary topology. By introducing the concept of neighbor points and using the first-order difference of control points of Catmull-Clark surfaces, we obtain the rate of convergence of control meshes of Catmull-Clark surface. With the result of convergence we derive a computational formula of subdivision depth for Catmull-Clark surface.更多还原