摘要
为得到能使过渡曲线在端点处达到C^k(k为任意自然数)连续的多项式势函数的通用表达式,由连续条件反推的势函数需具备的条件,根据条件个数确定势函数的最低次数,将势函数表示成Bernstein基函数的线性组合,组合系数待定。根据Bernstein基函数的端点信息确定关于待定系数的方程组,解之得出满足连续性要求的势函数。考虑到由该势函数构造的过渡曲线形状由被过渡曲线唯一确定,又将势函数次数增加一次,得出能使过渡曲线在端点处达到任意C^k连续并且形状可调的多项式势函数的通用表达式。借助Bernstein基函数的升阶公式给出了两种势函数之间的关系,分析了势函数的性质以及相应过渡曲线的特征,给出了势函数以及过渡曲线的图例,验证了理论分析结果的正确性及所给方法的有效性。
In order to obtain the general expression of polynomial potential function which can make the corresponding transition curve reach C^k(k is an arbitrary natural number)continuity at the endpoints,based on the required conditions of the potential function deduced from the continuity condition,the minimal times of the polynomial potential function is determined according to the number of the required conditions,and the potential function is expressed as a linear combination of the classical Bernstein basis functions which are yet to be determined.According to the function values and derivative values at the endpoints of the Bernstein basis functions,the required conditions of the potential function are converted into an equation set about the undetermined combination coefficients.Solving the equation,we obtain the general expression of the potential function which satisfies the expected continuity condition.Considering that the shape of the transition curve is uniquely determined by the curves needed to be transferred,another new polynomial potential function with one higher degree is constructed.By similar derivation process,the general expression of the new potential function which can make the transition curve reach arbitrary C^k continuity and with shape adjustability is obtained.With the help of the degree elevation formula of the classical Bernstein basis functions,the relationship between the two kinds of potential functions is deduced.The properties of the potential functions and the characteristics of the corresponding transition curves are also analyzed.Legends of the potential functions and the transition curves are put forward and they can verify the correctness of the theoretical analysis results and the validity of the presented method.
作者
严兰兰
樊继秋
黄涛
YAN Lan-lan;FAN Ji-qiu;HUANG Tao(College of Science,East China University of Technology,Nanchang Jiangxi 330013,China)
出处
《图学学报》
CSCD
北大核心
2019年第1期59-69,共11页
Journal of Graphics
基金
国家自然科学基金项目(11261003
11761008)
江西省自然科学基金项目(20161BAB211028)
江西省教育厅科技项目(GJJ160558)
