任意阶参数连续的形状可调过渡曲线

Shape adjustable transition curves with arbitrary parameter continuity

目的针对现有研究未能给出可以使过渡曲线在端点处与被过渡曲线之间达到C^k(k为任意自然数)连续的多项式势函数统一表达式的问题展开研究,以期用简单有效的方式解决这一问题。方法从过渡曲线的方程出发,借助莱布尼兹公式得出其k阶导矢表达式,根据预设的连续性目标,反推出可使过渡曲线在端点处达到C^k连续的势函数需满足的基本条件。由这些基本条件中所含条件的数量,以及对势函数、过渡曲线的其他期望所对应的条件个数的总量,确定多项式势函数的次数,将势函数表达成相同次数的Bernstein基函数的线性组合,组合系数待定。由势函数需满足的基本条件、其他预期条件,以及Bernstein基函数在端点处的函数值、导数值等信息,得出关于待定系数的方程组。解该方程组,得出满足所有预期目标,并含一个自由参数的多项式势函数的统一表达式。结果势函数中存在两个参数k和λ,k用于控制过渡曲线与被过渡曲线在端点处的连续阶,在k取定以后,λ可用于控制过渡曲线与被过渡曲线的贴近程度。势函数具有对称性、中点性、有界性,分析了当k固定时,势函数关于变量t和参数λ的单调性,分析了使势函数图形只存在唯一拐点时,自由参数的取值范围。由该势函数构造的过渡曲线,取一般参数时在端点处可达C^k连续,取特殊参数时可达C^(k+1)连续。分析了过渡曲线的形状特征,当k取定时,λ的值越大,过渡曲线越贴近被过渡曲线。结论实验数据验证了理论分析结果的正确性,同时直观显示了所给方法的有效性。

Objective The existing research has failed to provide a general expression of the polynomial potential function, which can enable the transition curve to reach Ck (where k is an arbitrary natural number) continuity at endpoints. This research aims to solve this problem in a simple and effective manner. Method First, by using the equation of the transition curve, the kth derivative of the transition curve is obtained with the help of the Leibniz formula. According to the predetermined continuity goal, the basic conditions, which should be met by the potential function to enable the transition curve to reach Ck continuity at endpoints, are deduced. Second, according to the total number of conditions contained in the basic conditions and those corresponding to other expectations of the potential function and transition curve, the degree of the polynomial potential function is determined. The potential function is expressed as a linear combination of the Bernstein basis functions with the same degree, and the combination coefficients are established. Finally, according to the basic and other expected conditions to be satisfied by the potential function, as well as the function and derivative values of the Bernstein basis functions at endpoints, an equation set is obtained for the undetermined coefficients. To solve the equation set, the unified expression of the polynomial potential function, which satisfies all expected goals and contains a free parameter, should be obtained. Result Two parameters exist in the potential function, namely, k and λ. Parameter k is used to control the continuity order between the transition and initial curves at the endpoints. After k is determined, parameter λ can be used to control the degree of proximity between the transition and initial curves. The potential function has symmetry, midpoint property, and boundedness. The monotonicity of potential function with respect to the variable t and the parameter λ is analyzed when k is fixed. The value range of the free parameter, which depicts the curve of the potential function and has a unique inflection point, is analyzed. For the general parameter values, the transition curve that is constructed by the potential function can reach Ck continuity at the endpoints. For the special parameter values, the transition curve can Ck+1 reach continuity. The shape characteristics of the transition curve are further analyzed. When the value of k is set, the greater the value of λ and the closer the transition to the initial curve. Conclusion The numerical examples verify the correctness of the theoretical analytical results and the effectiveness of the proposed method.