【作者】 王燕；

【导师】 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算数学， 2009， 硕士

【摘要】 为了克服代数多项式空间曲线曲面造型的不足,很多学者提出了基于非代数多项式空间的其它形式的曲线曲面造型方法。本文在他们研究的基础上,主要做了以下工作:第一,在研究四次C-曲线性质的基础上,讨论了与给定多边形相切的分段四次C-Bézier曲线和四次C-B样条闭曲线和开曲线。所构造的C-Bézier曲线是C 1连续的,且对切线多边形是保形的。四次C-B样条闭曲线和开曲线是C 3连续的,且对切线多边形也是保形的。所构造曲线段的控制顶点由切线多边形的顶点直接计算产生。最后以实例表明,本文的方法是有效的。第二,在讨论三次H-Bézier曲线性质的基础上,提出了三次H-Bézier曲线的任意分割算法,即对三次H-Bézier曲线上任意一点,求该点把曲线分成的两个子曲线段的控制参数和控制顶点;给出了三次H-Bézier曲线与三次Bézier曲线的拼接条件,以及三次H-Bézier曲线在曲面造型中应用的例子。采用本文方法所得结果简单、直观,有效地增强了三次H-Bézier方法控制及表达曲线形状的能力。第三,给出了五次H-Bézier曲线的细分公式,并且证明了细分过程产生的控制多边形序列收敛于原曲线。在收敛性的基础上,证明了五次H-Bézier曲线的两个重要性质:保凸性和变差缩减性。更多还原

【Abstract】 In order to overcome the shortcomings of curves and surfaces modeling in algebraic polynomial space, many scholars propose other forms of curves and surfaces in the non-algebraic polynomial space. Based on the study of the scholars, this thesis does some study as follows:Firstly, through analysis of the properties of quartic C-curves, we present an approach of constructing planar piecewise quartic C-Bézier curves and quartic C-B spline curves with all edges tangent to a given control polygon. The C-Bézier curve segments are joined together with C 1continuity and the Quartic C-B spline closed curves and open curves are C 3 continuous. All curves are shape preserving to their tangent polygons. All control points of the curve segments can be calculated simply by the vertices of the given tangent polygon. Finally some numerical examples illustrate that the method given in this paper is effective.Secondly, based on the analysis of the properties of cubic H-Bézier curves, a subdivision algorithm is proposed, to compute the control parameters and control points of the two subcurves subdivided by any point of cubic H-Bézier curves. The connection conditions between cubic H-Bézier curves and cubic Bézier curves are derived and the applications of cubic H-Bézier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bézier curves.Thirdly, an effective subdivision formula for H-Bézier curves of degree five is presented. Furthermore, it is proved that the control polygons generated by the subdivision converge to the original H-Bézier curves of degree five. Two important properties, the variation diminishing (V-D) property and convexity preserving property, are proved for H-Bézier curves of degree five.更多还原