基于Metaball的C~k连续过渡曲线的构造
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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 andthey can verify the correctness of the theoretical analysis results and the validity of the presented method.
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 andthey can verify the correctness of the theoretical analysis results and the validity of the presented method.
引文
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[19]FARIN G.Curves and surfaces for CAGD[M].5th ed.San Francisco:Morgan Kaufmann Publishers,2002:66.
[2]董新华,马勇,柳强.渐开线齿轮齿根过渡曲线最佳线型实现方法[J].机械传动,2013,37(5):47-49.
[3]宋立权,赵学科,李智成,等.任意缸体旋叶式压缩机的叶片型线设计理论研究及应用[J].机械工程学报,2011,47(15):143-148.
[4]蔡华辉,王国瑾.三次C-Bézier螺线构造及其在道路设计中的应用[J].浙江大学学报:工学版,2010,44(1):68-74.
[5]禹鑫燚,邢双,欧林林,等.工业机器人CP运动指令的设计与实现[J].浙江工业大学学报,2017,45(5):568-573.
[6]张宏鑫,王国瑾.保持几何连续性的曲线形状调配[J].高校应用数学学报:A辑,2001,16(2):187-194.
[7]高晖,寿华好,缪永伟,等.3个控制顶点的类三次Bézier螺线[J].中国图象图形学报,2014,19(11):1677-1683.
[8]刘华勇,王焕宝,李璐,等.几何连续的Bézier-like曲线的形状调配[J],山东大学学报:理学版,2012,47(3):51-55.
[9]刘华勇,段小娟,张大明,等.基于三角Bézier-like的过渡曲线构造[J],浙江大学学报:理学版,2013,40(1):42-46.
[10]李重,马利庄,MEEK D.平面两圆弧相离情况下G2连续过渡曲线构造[J].计算机辅助设计与图形学学报,2006,18(2):265-269.
[11]郑志浩,汪国昭.三次PH曲线曲率单调性及过渡曲线构造[J].计算机辅助设计与图形学学报,2014,26(8):1219-1224.
[12]李凌丰,谭建荣,赵海霞.基于Metaball的过渡曲线[J].中国机械工程,2005,16(6):483-486.
[13]李军成,宋来忠,刘成志.带参数的多项式势函数与构造基于Metaball的过渡曲线[J].中国图象图形学报,2016,21(7):893-900.
[14]李军成,宋来忠,刘成志.形状调配中带参数的过渡曲线与曲面构造[J].计算机辅助设计与图形学学报,2016,28(12):2088-2096.
[15]严兰兰,梁炯丰,黄涛.带形状参数的Bézier曲线[J].合肥工业大学学报:自然科学版,2009,32(11):1783-1788.
[16]高晖,寿华好.势函数的构造及基于Metaball的过渡曲线[J].计算机辅助设计与图形学学报,2015,27(5):900-906.
[17]李军成,宋来忠.利用带形状参数的有理势函数构造基于Metaball的过渡曲线[J].浙江大学学报:理学版,2017,44(3):307-313.
[18]严兰兰,韩旭里.形状及光滑度可调的自动连续组合曲线曲面[J].计算机辅助设计与图形学学报,2014,26(10):1654-1662.
[19]FARIN G.Curves and surfaces for CAGD[M].5th ed.San Francisco:Morgan Kaufmann Publishers,2002:66.