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Received：18 April 2018，
Revised：2018-7-21，
Published：16 January 2019
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目的
2
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
2
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
2
势函数中存在两个参数
k
和
$$ \lambda $$
，
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，
$$ \lambda $$
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
$$ \lambda $$
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，
$$ \lambda $$
的值越大，过渡曲线越贴近被过渡曲线。
结论
2
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
2
The existing research has failed to provide a general expression of the polynomial potential function
which can enable the transition curve to reach C
k
(where
k
is an arbitrary natural number) continuity at endpoints. This research aims to solve this problem in a simple and effective manner.
Method
2
First
by using the equation of the transition curve
the
k
th 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 C
k
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
2
Two parameters exist in the potential function
namely
k
and
$$ \lambda $$
. Parameter
k
is used to control the continuity order between the transition and initial curves at the endpoints. After
k
is determined
parameter
$$ \lambda $$
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
$$ \lambda $$
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 C
k
continuity at the endpoints. For the special parameter values
the transition curve can C
k
+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
$$ \lambda $$
and the closer the transition to the initial curve.
Conclusion
2
The numerical examples verify the correctness of the theoretical analytical results and the effectiveness of the proposed method.
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