【作者】 吴荣军；

【导师】 彭国华；

【作者基本信息】 西北工业大学 ， 应用数学， 2015， 博士

【摘要】 在计算机辅助几何设计的许多应用中,良好的外形设计应该消除不必要的奇点和拐点;凸性也是外形设计中不可或缺的要素.这些几何性质都直接影响外形设计的动力学性能、算法的复杂性和加工的可操作性.因此,对于作为形状设计基本工具之一的参数曲线,特别需要分析与预知它们各种可能的几何形态,以规避可能出现奇异形态的设计风险.参数曲线的形状分析方法主要有苏步青与刘鼎元提出的仿射不变量方法(简记为AIM)、Maureen C. Stone与Tony D. DeRose提出的基于控制顶点轨迹的几何刻画方法(简记为GCM)和叶正麟提出的基于包络和拓扑映射的方法(简称叶方法).关于三次多项式及其有理类的平面参数曲线情形,形状分析研究已有比较成熟的成果.然而对于高次的和非代数多项式类的参数曲线,问题较为复杂,便于实用的相关研究较少.本文旨在研究参数曲线不同形状的几何特征,局部凸和全局凸的区分条件,若干四次多项式类和三角类的平面参数曲线的形状分析.主要结论如下:1.将叶方法所得的三次平面参数曲线的形状条件分布图(简称形状图)归纳为三种Bézier型和三种B样条型形状图,进而给出了它们和AIM及GCM所得形状图之间的关系.并在相关形状图中划分出局部凸和全局凸区域,使AIM、GCM和叶方法的形状图更加完善.2.针对前述方法的不足,提出了一种新的特征空间方法(简记为CSM).方法的核心思想是利用参数曲线几何特征表示式中边向量的系数构造出相应的特征锥面:尖点条件锥面和重结点边界条件锥面;这些特征锥面及其切平面将三维空间划分为不同的空间区域,分别对应于参数曲线的不同形状特征.利用CSM分别得到了三次Bézier曲线和三次B样条曲线的特征空间.用垂直于坐标轴的平面切割特征空间,可得到叶方法和GCM的所有形状图,而用非平面切片切割特征空间可获得AIM的形状图.CSM的优点包括三个方面:首先,判定曲线形状特征时不必假设某两条控制边向量不平行;其次,曲线的形状特征完全由特征点在特征空间的位置所决定,特征空间包含了除四个控制顶点共线的平凡情况外的所有退化情况;第三,特征点的计算仅需用控制边向量表示的三个二阶行列式,位置判定仅涉及特征点与平面和三个锥面的位置关系.因此, CSM更适合于计算机程序自动判定曲线的形状特征.3.我们用叶方法给出了有理三次Bézier曲线、带形状参数的三角Bézier曲线、四次带参Bézier曲线、C-B样条曲线以及一类四次带参B样条曲线的形状条件.我们发现这些形状条件对应的形状图都与三次曲线情况非常类似.进而分别讨论了每类曲线中形状参数的变化对形状条件分布图的影响,相关结果可使设计者明确如何配置控制顶点或者调节形状参数,使得生成曲线为全局凸或局部凸曲线,或具有所需要的奇点与拐点,或将当前曲线形状调节为另一种所需的形状.另外,对形状参数超界问题也作了一些探讨.4.对于有理三次Bézier曲线和四次带参Bézier曲线、C-B样条曲线、特殊形状参数下的四次带参样条曲线和带参三角Bézier曲线,分别给出了它们的尖点条件锥和重结点边界条件锥,并进一步讨论了形状参数变化对相应特征空间的影响.更多还原

【Abstract】 In many applications of computer aided geometric design, the curves with satisfyingshape should not constitute any singularities and undesirable inflection points. In addition,convexity property plays indispensable role in geometric modeling. These geometricproperties have direct impacts on various aspects of shape design such as the dynamicsperformance, algorithm complexity and feasibility in manufacture. Therefore, to avoidpotential risk in shape design, it is of vital importance to analyze and predict the shape ofparametric curves, which can be seen as one of most fundamental tools for geometricmodeling. In the literature, there are three kinds of classical methods for shape analysis ofparametric curves, including the Affine Invariant Method (abbreviated, AIM) proposed by Suand Liu, Stone and DeRose’s Geometric Characterization Method (abbreviated, GCM) basedon the position of the control points, and Ye’s method based on the theory of envelopes andtopological mappings, which will be referred to as Ye Method in what follows. Althoughmany classic results have been obtained in the study of shape features concerning planarparametric cubic curves and their rational forms, very few practical results have beenpresented with regard to algebraic polynomial parametric curves with higher degree andnonalgebraic parametric curves.The purpose of this dissertation is to study the geometric characteristics of parametriccurves with differential shapes, criteria for distinguishing local convexity from globalconvexity, and shape analysis of some planar quartic polynomial curves as well astrigonometric parameter curves. The main contribution lies in the following aspects:1. The shape diagrams of Ye Method are classified into three kinds of Bézier types andthree kinds of B-spline types, and the relationship of shape diagrams of AIM, GCM and YeMethod is ascertained. Moreover, the distribution regions corresponding to local convexityand global convexity of all these shape diagrams are partitioned.2. To overcome the shortcomings of the aforementioned methods, we propose a novelmethod, called Characteristic Space Method (abbreviated, CSM), to determine the shapefeatures of parametric curves. The main idea of our method is to construct characteristic cones(including cusp cones and two end loop cones) in terms of three side vectors in the geometriccharacteristic equations of parametric curves. The characteristic cones and their tangentplanes partition the3-dimension space into different characteristic regions according to shapefeatures, which is called the characteristic space of parametric curves. By CSM, we obtainsome useful theorems on characteristic spaces of cubic Bézier curves and cubic B-spline curves. It is shown that the shape diagrams of GCM and Ye Method can be derived fromcharacteristic spaces by virtue of planar slices, which are vertical to one of the axes; while theshape diagrams of AIM can be obtained by nonplanar section through the characteristic space.The advantages of our method are three-fold: Firstly, CSM does not rely on the parallelassumption of control side vectors. Secondly, the shape features of parametric curves can becompletely determined by the position of the characteristic point in a characteristic space,except for the trivial case in which the four control points are collinear. Last but not least, thecomputation of the characteristic point in CSM needs only three determinants about thecontrol side vectors, and the region determination of the characteristic point involves onlyplanes and three cones. Therefore, this method is more suitable to the automatic determinationof shape features by computer programs.3. Using Ye Method, we obtain the shape conditions of planar rational cubic Béziercurves and four classes of parametric curves with shape parameters, including planar quarticpolynomial curves, spline curves, planar trigonometric Bézier curves and C-B-spline curves.We find that the structures of these shape diagrams are very similar to those of cubicparametric curves.Furthermore, we also discuss the influences of shape parameters on the associated shapediagrams. The obtained results can enable the user to place control points or choose shapeparameters such that the resulting curve is globally or locally convex, possessing wantedsingularities or inflection points, or with the desired shape. In addition, some special shapediagrams are also discussed when the shape parameters beyond the restricted interval.4. For the above mentioned parametric curves, we construct their cusp cones and endloop cones. Finally, we investigate the impact of shape parameters on characteristic space.更多还原