【作者】 郝茹；

【导师】 刘润涛；

【作者基本信息】 哈尔滨理工大学 ， 应用数学， 2009， 硕士

【摘要】 计算机辅助几何设计CAGD(Computer Aided Geometric Design)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。Bézier曲线和曲面广泛应用于计算机图形学,并且在形状设计方面有很好的性质。近年来Bézier曲线、曲面在工程设计应用中,有着极其重要的作用。如飞机、船体等外形曲面设计,首先是通过设计出经过一定型值点的曲线作为线型,再在这些线型上插值出相应的外形曲面。因此Bézier曲线曲面在实践中表现出强大的生命力,受到学术界的广泛重视。本文首先从有理Bézier曲线的定义出发,阐述它的性质,通过权因子而不是控制顶点来修改有理四次Bézier曲线的形状,实现了相邻曲线段的G 2的连续拼接;进而实现了相邻三段曲线的G 2的连续拼接。同时讨论了有理四次Bézier曲线参数化,应用重心坐标推导出有理四次Bézier曲线的表达式,通过给定5个控制顶点和位于这些顶点凸包内四次有理曲线上一点,反算出了该点的参数和内权因子,研究了在曲线形状不变的情况下,空间有理四次Bézier曲线权因子变换和参数变换的等效性。通过改变有理四次贝齐尔曲线的控制顶点或权因子是实现形状修改的途径之一。实际应用中人们更希望能实现有预定目标的修改,譬如说使修改过的曲线经过某个点,本文也给出了既可以修改控制顶点,也可以修改权因子来实现这个目标的方法。采用在公共边界处曲线连续和切平面光滑连续的性质,研究了有理Bézier曲面的拼接问题,给出了具有公共边界曲线的两张双四次有理Bézier曲面G1光滑拼接条件,从而得到更多的可调形状参数。更多还原

【Abstract】 Computer aided geometric design(CAGD) mainly researches curves and surfaces which vary in the free and complex forms, that is to say, curves and surfaces of freedom forms. Bézier curves and surfaces are widely applied in computer graphics and have many good properties in shape design of industrial products.In recent years, Bézier curves and surfaces have extremely important function in the applications of engineering design, such as air planes, ships and so on. Firstly a curve shape is obtained by designing a curve that crosses some data points. Then the corresponding shape surface is formed by interpolation on these curve shapes. So Bézier curves and surfaces show their strong vitality in practice and receive wide attentions.Firstly, the definition of rational quartic Bézier curve is introduced and its properties are described. The shape of rational quartic Bézier curve is modified by changing the weights corresponding to control points instead of control points themselves. The G2 continuity for the joining between two adjacent rational quartic Bézier curves is realized. Based on this the G2continuity for the joining among three adjacent quartic rational Bézier curves are achieved further.Secondly, the parameterization of rational quartic Bézier curve is discussed. The expression formula of rational quartic Bézier curve is deduced by barycentric coordinates. The parameter and inner weight factors are calculated inversely by given five control points and a point on the rational quartic Bézier curve within the convex hull formed by all the control points of the curve. The equivalent property between the weight transformation for spatial rational quartic Bézier curve and parametric transformation of the curve under the condition of keeping the shape unchanged is investigated.Altering the control points or weights of rational quartic Bézier curve is one of important approaches in achieving the shape modification. In practical applications, the modification with predestinate target is more hoped to be realized, for example, to make the modified curve pass a given point. The method to make the modification is given both by modifying the control points and by modifying weight factors.According to the theory of Bézier surface, by taking the property that the curve and its tangent plane is continuous on the common boundary, the joining between adjacent rational Bezier surface is studied and the G1 continuity conditions of two adjacent rational Bézier surfaces of double four degrees are given, therefore more adjustable shape parameters can be obtained.更多还原