发布时间: 2019-01-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.180231 2019 | Volume 24 | Number 1 计算机图形学

 收稿日期: 2018-04-18; 修回日期: 2018-07-21 基金项目: 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） 第一作者简介: 严兰兰, 1982年生, 女, 副教授, 博士, 主要研究方向为计算机辅助几何设计。E-mail:yxh821011@aliyun.com;樊继秋, 男, 讲师, 主要研究方向为代数几何。E-mail:jqfan@ecit.cn;周其华, 男, 副教授, 主要研究方向为几何分析。E-mail:qhzhou@ecit.cn. 中图法分类号: TP391.72 文献标识码: A 文章编号: 1006-8961(2019)01-0084-12

# 关键词

Shape adjustable transition curves with arbitrary parameter continuity
Yan Lanlan, Fan Jiqiu, Zhou Qihua
College of Science, East China University of Technology, Nanchang 330013, China
Supported by: National Natural Science Foundation of China (11261003, 11761008)

# Abstract

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 $\lambda$. 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 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 $\lambda$ 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.

# Key words

curve design; transition curve; shape adjustment; parameter continuity; potential function

# 1.1 基于Metaball的过渡曲线

 $\mathit{\boldsymbol{G}}\left( t \right) = \mathit{\boldsymbol{P}}\left( t \right)f\left( t \right) + \mathit{\boldsymbol{Q}}\left( t \right)\left( {1 - f\left( t \right)} \right)$ (1)

2) 端点性。即${f_k}\left( 0 \right) = 1, {f_k}\left( 1 \right) = 0$

3) 导数性。对于一般的参数$\lambda$，有

 $f_k^{\left( i \right)}\left( 0 \right) = f_k^{\left( i \right)}\left( 1 \right) = 0$

4) 对称性。即${f_k}\left( t \right) + {f_k}\left( {1 - t} \right) = 1$

5) 中点性。即${f_k}\left( {0.5} \right) = 0.5$

6) 单调性。对于固定的自然数$k$，当参数$\lambda \in \left[ { - \frac{k}{2}, 1} \right]$时，固定$\lambda$的值，${f_k}(t)$关于变量$t{\rm{ }}(t \in \left[ {0, 1} \right])$单调递减；对于固定的自然数$k$，当变量$t \in \left[ {0, \frac{1}{2}} \right]$时，固定$t$的值，${f_k}(t)$关于参数$\lambda$单调递增；当变量$t \in \left[ {\frac{1}{2}, 1} \right]$时，固定$t$的值，${f_k}(t)$关于参数$\lambda$单调递减。

7) 有界性。即$\forall t \in \left[ {0, 1} \right]$，有$0 \le {f_k}\left( t \right) \le 1$

8) 凹凸性。当$\lambda \in \left[ {\frac{2}{3} - \frac{k}{6}, 1} \right]$时，在参数区间$t \in \left[ {0, 1} \right]$上，势函数${f_k}(t)$的图形只有唯一拐点$\left( {\frac{1}{2}, \frac{1}{2}} \right)$，经过该点时，${f_k}(t)$的图形从凸曲线变为凹曲线。

 $\begin{array}{*{20}{c}} {{f_{k - 1}}\left( {t;1} \right) = \sum\limits_{i = 0}^{k - 1} {B_i^{2k + 1}\left( t \right)} + B_k^{2k + 1}\left( t \right) = }\\ {\sum\limits_{i = 0}^k {B_i^{2k + 1}\left( t \right)} } \end{array}$ (15)

 $\begin{array}{*{20}{c}} {{f_{k - 1}}\left( {t;1} \right) = }\\ {\sum\limits_{i = 0}^k {\left[ {\left( {1 - \frac{i}{{2k + 2}}} \right)B_i^{2k + 2}\left( t \right) + \frac{{i + 1}}{{2k + 2}}B_i^{2k + 2}\left( t \right)} \right]} = }\\ {\sum\limits_{i = 0}^k {\frac{{2k + 2 - i}}{{2k + 2}}B_i^{2k + 2}\left( t \right)} + \sum\limits_{i = 1}^{k + 1} {\frac{i}{{2k + 2}}B_i^{2k + 2}\left( t \right)} = }\\ {B_0^{2k + 2}\left( t \right) + \sum\limits_{i = 1}^k {B_i^{2k + 2}\left( t \right)} + \frac{1}{2}B_{k + 1}^{2k + 2}\left( t \right) = }\\ {\sum\limits_{i = 0}^k {B_i^{2k + 2}\left( t \right)} + \frac{1}{2}B_{k + 1}^{2k + 2}\left( t \right)} \end{array}$ (16)

 $\begin{array}{*{20}{c}} {{f_{k - 1}}\left( {t;1} \right) = }\\ {\sum\limits_{i = 0}^k {\left[ {\left( {1 - \frac{i}{{2k + 3}}} \right)B_i^{2k + 3}\left( t \right) + \frac{{i + 1}}{{2k + 3}}B_{i + 1}^{2k + 3}\left( t \right)} \right]} + }\\ {\frac{1}{2}\left[ {\left( {1 - \frac{{k + 1}}{{2k + 3}}} \right)B_{k + 1}^{2k + 3}\left( t \right) + \frac{{k + 2}}{{2k + 3}}B_{k + 2}^{2k + 3}\left( t \right)} \right] = }\\ {\sum\limits_{i = 0}^k {\frac{{2k + 3 - i}}{{2k + 3}}B_i^{2k + 3}\left( t \right)} + \sum\limits_{i = 1}^{k + 1} {\frac{i}{{2k + 3}}B_i^{2k + 3}\left( t \right)} + }\\ {\frac{{k + 2}}{{4k + 6}}B_{k + 1}^{2k + 3}\left( t \right) + \frac{{k + 2}}{{4k + 6}}B_{k + 2}^{2k + 3}\left( t \right) = }\\ {B_0^{2k + 3}\left( t \right) + \sum\limits_{i = 1}^k {B_i^{2k + 3}\left( t \right)} + \frac{{k + 1}}{{2k + 3}}B_{k + 1}^{2k + 3}\left( t \right) + }\\ {\frac{{k + 2}}{{4k + 6}}B_{k + 1}^{2k + 3}\left( t \right) + \frac{{k + 2}}{{4k + 6}}B_{k + 2}^{2k + 3}\left( t \right) = }\\ {\sum\limits_{i = 0}^k {\frac{i}{{2k + 3}}B_i^{2k + 3}\left( t \right)} + \frac{{3k + 4}}{{4k + 6}}B_{k + 1}^{2k + 3}\left( t \right) + }\\ {\frac{{k + 2}}{{4k + 6}}B_{k + 2}^{2k + 3}\left( t \right)} \end{array}$ (17)

 ${f_k}\left( t \right) = \sum\limits_{i = 0}^{k + 1} {B_i^{2k + 3}\left( t \right)}$

 $\frac{{{{\rm{d}}^{k + 1}}{f_k}\left( t \right)}}{{{\rm{d}}{t^{k + 1}}}} = \sum\limits_{i = 0}^{k + 1} {\frac{{{{\rm{d}}^{k + 1}}B_i^{2k + 3}\left( t \right)}}{{{\rm{d}}{t^{k + 1}}}}}$ (18)

 $\begin{array}{*{20}{c}} {\frac{{{{\rm{d}}^{k + 1}}{f_k}\left( t \right)}}{{{\rm{d}}{t^{k + 1}}}}\left| {_{t = 0}} \right. = \frac{{\left( {2k + 3} \right)!}}{{\left( {k + 2} \right)!}}\sum\limits_{i = 0}^{k + 1} {{{\left( { - 1} \right)}^{k + 1 - i}}C_{k + 1}^i} = }\\ {\frac{{\left( {2k + 3} \right)!}}{{\left( {k + 2} \right)!}}\sum\limits_{i = 0}^{k + 1} {C_{k + 1}^i \cdot {1^i} \cdot {{\left( { - 1} \right)}^{k + 1 - i}}} = }\\ {\frac{{\left( {2k + 3} \right)!}}{{\left( {k + 2} \right)!}}{{\left[ {1 + \left( { - 1} \right)} \right]}^{k + 1}} = 0} \end{array}$

 $\frac{{{{\rm{d}}^{k + 1}}{f_k}\left( t \right)}}{{{\rm{d}}{t^{k + 1}}}}\left| {_{t = 1}} \right. = \sum\limits_{i = 0}^{k + 1} {\frac{{{{\rm{d}}^{k + 1}}B_i^{2k + 3}\left( t \right)}}{{{\rm{d}}{t^{k + 1}}}}\left| {_{t = 1}} \right.} = \sum\limits_{i = 0}^{k + 1} 0 = 0$

 ${f_k}\left( {0.5} \right) + {f_k}\left( {0.5} \right) = 1 \Rightarrow {f_k}\left( {0.5} \right) = 0.5$

 ${{f'}_k}\left( t \right) = C_{2k + 3}^{k + 1}{t^k}{\left( {1 - t} \right)^k}g\left( t \right)$ (19)

 $\begin{array}{*{20}{c}} {g\left( t \right) = \left( {4k\lambda + 6\lambda - 3k - 4} \right) \times }\\ {\left( {{t^2} - t} \right) + \left( {\lambda - 1} \right)\left( {k + 1} \right)} \end{array}$

 $\left\{ \begin{array}{l} g\left( 0 \right) = g\left( 1 \right) = \left( {\lambda - 1} \right)\left( {\lambda + 1} \right) \le 0\\ g\left( {\frac{1}{2}} \right) = - \frac{1}{4}\left( {k + 2\lambda } \right) \le 0 \end{array} \right.$

 $\begin{array}{*{20}{c}} {\frac{{{\rm{d}}{f_k}\left( t \right)}}{{{\rm{d}}\lambda }} = B_{k + 1}^{2k + 3}\left( t \right) - B_{k + 2}^{2k + 3}\left( t \right) = }\\ {C_{2j + 3}^{k + 1}{{\left[ {t\left( {1 - t} \right)} \right]}^{k + 1}}\left( {1 - 2t} \right)} \end{array}$

$t \in \left[ {0, \frac{1}{2}} \right]$时，$\frac{{{\rm{d}}{f_k}(t)}}{{{\rm{d}}\lambda }} \ge 0$，表明${f_k}(t)$关于$\lambda$单调递增，而当$t \in \left[ {\frac{1}{2}, 1} \right]$时，$\frac{{{\rm{d}}{f_k}(t)}}{{{\rm{d}}\lambda }} \le 0$，表明${f_k}(t)$关于$\lambda$单调递减。

 $\begin{array}{*{20}{c}} {{{f''}_k}\left( t \right) = C_{2k + 3}^{k + 1}{t^{k - 1}}{{\left( {1 - t} \right)}^{k + 1}} \times }\\ {\left( {k + 1} \right)\left( {1 - 2t} \right)h\left( t \right)} \end{array}$ (20)

 $h\left( t \right) = \left( {4k\lambda + 6\lambda - 3k - 4} \right)\left( {{t^2} - t} \right) + \left( {\lambda - 1} \right)k$

 $\left\{ \begin{array}{l} h\left( 0 \right) = h\left( 1 \right) = \left( {\lambda - 1} \right)k \le 0\\ h\left( {\frac{1}{2}} \right) = 1 - \frac{1}{4}k - \frac{3}{2}\lambda \le 0 \end{array} \right.$

# 3.1 过渡曲线的特征

2) 由${f_k}(t)$的导数性可知

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{G}}^{\left( i \right)}}\left( 0 \right) = {\mathit{\boldsymbol{P}}^{\left( i \right)}}\left( 0 \right)\\ {\mathit{\boldsymbol{G}}^{\left( i \right)}}\left( 1 \right) = {\mathit{\boldsymbol{P}}^{\left( i \right)}}\left( 1 \right) \end{array} \right.$ (22)

4) 由${f_k}(t)$的单调性可知，对于指定的$k$值，当$\lambda \in \left[ { - \frac{k}{2}, 1} \right]$时，对于固定的$\lambda$值，随着变量$t$的增大($t$从0变化到1)，曲线$\mathit{\boldsymbol{P}}\left( t \right)$的权重逐渐减小(从1降为0)，而曲线$\mathit{\boldsymbol{Q}}\left( t \right)$的权重逐渐增加(从0增为1)，因此曲线$\mathit{\boldsymbol{P}}\left( t \right)$对过渡曲线的影响逐渐减弱，曲线$\mathit{\boldsymbol{Q}}\left( t \right)$对过渡曲线的影响逐渐增强。而在$t = \frac{1}{2}$处，曲线$\mathit{\boldsymbol{P}}\left( t \right) $$\mathit{\boldsymbol{Q}}\left( t \right) 对过渡曲线的影响相当(权重均为 \frac{1}{2})。因此过渡曲线\mathit{\boldsymbol{G}}\left( t \right) 的形状在前半段相似于曲线\mathit{\boldsymbol{P}}\left( t \right) ，在后半段相似于曲线\mathit{\boldsymbol{Q}}\left( t \right) 另外，对于指定的k 值和 t值，当 t \in \left[ {0, \frac{1}{2}} \right]时，随着参数 \lambda 的增加，曲线\mathit{\boldsymbol{P}}\left( t \right) 的权重逐渐增加，当t \in \left( {\frac{1}{2}, 1} \right) 时，随着参数 \lambda 的增加，曲线\mathit{\boldsymbol{Q}}\left( t \right) 的权重逐渐增加。因此 \lambda 越大，过渡曲线\mathit{\boldsymbol{G}}\left( t \right) 的形状在前半段越相似于曲线\mathit{\boldsymbol{P}}\left( t \right) ，在后半段越相似于曲线\mathit{\boldsymbol{Q}}\left( t \right) 5) 由{f_k}(t) 的有界性可知，过渡曲线\mathit{\boldsymbol{G}}\left( t \right) 为被过渡曲线\mathit{\boldsymbol{P}}\left( t \right)$$\mathit{\boldsymbol{Q}}\left( t \right)$的凸组合。

# 3.2 过渡曲线的图例

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{P}}\left( t \right):x = - \cos \left( {\frac{{\rm{ \mathsf{ π} }}}{2}t} \right),y = 1 - \sin \left( {\frac{{\rm{ \mathsf{ π} }}}{2}t} \right)\\ \mathit{\boldsymbol{Q}}\left( t \right):x = 1 - \cos \left( {\frac{{\rm{ \mathsf{ π} }}}{3}t} \right),y = \sin \left( {\frac{{\rm{ \mathsf{ π} }}}{3}t} \right) \end{array} \right.$

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{P}}\left( t \right):x = - \sin \left( {\frac{{\rm{ \mathsf{ π} }}}{2}t} \right),y = 1 - \cos \left( {\frac{{\rm{ \mathsf{ π} }}}{2}t} \right)\\ \mathit{\boldsymbol{Q}}\left( t \right):x = 1 - \cos \left( {\frac{{{\rm{2 \mathsf{ π} }}}}{3}t} \right),y = \sin \left( {\frac{{{\rm{2 \mathsf{ π} }}}}{3}t} \right) \end{array} \right.$

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