【作者】 段小娟；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 应用数学， 2009， 硕士

【摘要】 PH曲线及样条和混合多项式曲线是近年来CAGD研究中的热点问题之一。本文围绕这两方面作了一些探讨研究。一、基于控制三角形周长最短设计平面三次PH样条;二、给出了平面三次F-曲线上拐点和奇异点存在的充要条件,并给出了在张纪文第一种三次F-Bezier定义下F-Bezier曲线的形状分类图。第二章给出了平面三次PH样条基于控制三角形周长最短下的设计方法。主要采用以控制三角形周长最短为评价标准,PH样条在型值点以向量长度和转角方式给出的情况下,对各型值点处的切线斜率进行设计。并给出了控制多边形的周长表示和PH样条的弧长表示。第三章给出了平面上广义F-曲线,三次F-曲线和张纪文第一种三次F-Bezier定义下F-Bezier曲线上的拐点和奇异点(包括尖点和重点)存在的充分必要条件。采用的主要方法是根据曲线的相对曲率定义了曲线的特征函数,通过适当的转化,使其成为实数域上的一个二次多项式函数,其实根(零点)对应于曲线上的拐点和奇异点。根据特征函数的零点分布规律容易得到对应曲线上的拐点和奇异点的分布规律。最后给出了F-Bezier曲线的形状分类图。由于F-曲线和C-曲线、H-曲线关系,实际上将C-曲线、H-曲线上拐点和奇异点分类用F-曲线统一起来。这些结果可用于检测F-曲线的拐点和奇异点,也可用在曲线造型设计和光顺插值中避免或排除多余拐点和奇异点。更多还原

【Abstract】 This paper focuses on two problems,PH spline and hybrid polynomial curves, which are regarded as hot spots in recent CAGD research.First,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Second,we investigate the distribution of inflection points and singularities of planar cubic F-curve.And also we give the shape classification map of cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.In Chapter two,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Using the minimum perimeter of control triangle as the evaluation of PH-spline,we design the slope of the tangent point.The data points is given by length and angle.Also we represent the perimeter of the control polygon and the arc length of the PH spline.In Chapter three,we give the necessary and sufficient condition leading to inflection points and singularities(including cusps and loops) on planar generalized F-curve,planar cubic F-curve and cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.An eigenfunction of cubic F-curves according to the relative curvature of curves is introduced.Through the appropriate conversion,the eigenfunction can be made a quadratic polynomial function in real domain,whose zeros correspond to the points of inflexion and singularities of the curves above.Finally,the shape classification map of cubic F-Bezier curve is given.Thus the classification of inflection points and singularities on C-curves and H-curves are united by F-curve.These results can be used to detect the interior inflection point and singular points on F-curves and can also be used to avoid or exclude redundant inflection points and singularities.更多还原