形状调配中带参数的过渡曲线设计
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摘要
为提高过渡曲线在端点处的连续阶,并赋予过渡曲线相对于固定基曲线的形状调整能力,从过渡曲线的方程出发,根据预设的连续性目标反推调配函数需满足的基本条件,将调配函数表达成Bernstein基函数的线性组合,组合系数待定,由基本条件和Bernstein基函数的端点性质得出关于待定系数的方程组,解该方程组得出调配函数初步表达式,再借助Bernstein基函数的升阶公式将初步表达式的次数提高两次,进而在表达式中引入自由参数。调配函数具有对称性、中点性、单调性、有界性,分析了保证调配函数图形只存在唯一拐点的自由参数取值范围。取一般参数时,过渡曲线在端点处可达拟C~3连续,取特殊参数时可达拟C~4连续,分析了过渡曲线的形状特征,数值实例验证了方法的正确性和有效性。
The purpose of the present study is to enhance the continuity of the transition curve at the endpoints, and give it the shape adjustment ability relative to the fixed base curves. Starting from the equation of the transition curve, according to the predetermined goal about the continuous order, the basic conditions that need to be satisfied by the blending functions are deduced. The blending function is expressed as a linear combination of Bernstein basis functions, and the combination coefficients are to be determined. Based on the basic conditions and the endpoint properties of Bernstein basis functions, an equation set about the undetermined coefficients is obtained. The initial expression of the blending function is obtained by solving the equation set. By using the degree elevation formula, the degree of the initial expression is raised twice. Then, the free parameter is introduced into the expression of the blending function. The blending function features in symmetry,midpoint property, monotonicity and boundedness. Additionally, the paper analyzes that value range of free parameter that can ensure the curve of the blending function has a unique inflection point. For general parameter values, the transition curve can reach C3 continuity. For special parameter values,the transition curve can reach C4 continuity. Furthermore, the shape characteristics of the transition curve are analyzed, and the numerical examples have also verified the correctness and effectiveness of the proposed method.
The purpose of the present study is to enhance the continuity of the transition curve at the endpoints, and give it the shape adjustment ability relative to the fixed base curves. Starting from the equation of the transition curve, according to the predetermined goal about the continuous order, the basic conditions that need to be satisfied by the blending functions are deduced. The blending function is expressed as a linear combination of Bernstein basis functions, and the combination coefficients are to be determined. Based on the basic conditions and the endpoint properties of Bernstein basis functions, an equation set about the undetermined coefficients is obtained. The initial expression of the blending function is obtained by solving the equation set. By using the degree elevation formula, the degree of the initial expression is raised twice. Then, the free parameter is introduced into the expression of the blending function. The blending function features in symmetry,midpoint property, monotonicity and boundedness. Additionally, the paper analyzes that value range of free parameter that can ensure the curve of the blending function has a unique inflection point. For general parameter values, the transition curve can reach C3 continuity. For special parameter values,the transition curve can reach C4 continuity. Furthermore, the shape characteristics of the transition curve are analyzed, and the numerical examples have also verified the correctness and effectiveness of the proposed method.
引文
[1]李重,金小刚,马利庄,等.多项式混合曲线曲面方法构造[J].计算机辅助设计与图形学学报,2009,21(5):579-583.
[2]SEDERBERG T W,GREENWOOD E.A physically based approach to 2-D shape blending[J].ACMSIGGRAPH Computer Graphics,1992,26(2):25-34.
[3]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.
[4]SONG Q Z,WANG J Z.Generating Gn parametric blending surfaces based on partial reparameterization of base surfaces[J].Computer Aided Design,2007,39(11):953-963.
[5]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.
[6]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.
[7]刘华勇,张大明,李璐.基于参数连续HC Bézier-like曲线的过渡曲线的构造[J].纯粹数学与应用数学,2011,27(1):69-74.
[8]刘华勇,李璐,王焕宝.Bézier-like曲线的形状调配[J].扬州大学学报:自然科学版,2012,15(1):65-68.
[9]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.
[10]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.
[11]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.
[12]李军成,宋来忠,刘成志.带参数的多项式调配函数与构造基于Metaball的过渡曲线[J].中国图象图形学报,2016,21(7):893-900.
[13]FARIN G.Curves and surfaces for CAGD[M].5nd ed.San Francisco:Morgan Kaufmann Publishers,2002:66.
[2]SEDERBERG T W,GREENWOOD E.A physically based approach to 2-D shape blending[J].ACMSIGGRAPH Computer Graphics,1992,26(2):25-34.
[3]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.
[4]SONG Q Z,WANG J Z.Generating Gn parametric blending surfaces based on partial reparameterization of base surfaces[J].Computer Aided Design,2007,39(11):953-963.
[5]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.
[6]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.
[7]刘华勇,张大明,李璐.基于参数连续HC Bézier-like曲线的过渡曲线的构造[J].纯粹数学与应用数学,2011,27(1):69-74.
[8]刘华勇,李璐,王焕宝.Bézier-like曲线的形状调配[J].扬州大学学报:自然科学版,2012,15(1):65-68.
[9]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.
[10]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.
[11]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.
[12]李军成,宋来忠,刘成志.带参数的多项式调配函数与构造基于Metaball的过渡曲线[J].中国图象图形学报,2016,21(7):893-900.
[13]FARIN G.Curves and surfaces for CAGD[M].5nd ed.San Francisco:Morgan Kaufmann Publishers,2002:66.
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