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SCIENTIA SINICA Informationis, Volume 47, Issue 12: 1694-1708(2017) https://doi.org/10.1360/N112016-00240

Construction of a class of quintic curves with rational offsets

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  • ReceivedOct 9, 2016
  • AcceptedDec 11, 2016
  • PublishedMay 18, 2017

Abstract

In this paper, a method for constructing a class of quintic curves with rational offsets is presented. Although this class of planar curves does not include PH curves, such curves are widely applied in CAD because they have rational offsets. A complex variate model is employed to deduce the proposed method. The quintic OR curves are first classified into two classes according to the different factorization of their hodographs, and are then discussed. With the given $C^1$ Hermite data, the curves are determined by specifying a real parameter. This real parameter can be used to adjust the shape of the constructed curves, and affects the parameter values of the cusps.


Funded by

国家自然科学基金(61272300,61100084)

浙江省一流学科A类 (浙江财经大学统计学)资助 和浙江省教育厅科研基金(Y201223321)


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  • Figure 1

    (Color online) A necessary and sufficient condition on the control polygon of a class I quintic OR curve: there are points $\boldsymbol{Q}_i$, $i=0,1,2$, such that $\Delta\boldsymbol{P}_0~:~4(\boldsymbol{Q}_0~-~\boldsymbol{P}_1)~=~2(\boldsymbol{P}_2~-~\boldsymbol{Q}_0)~:~3(\boldsymbol{Q}_1~-~\boldsymbol{P}_2)~=~3(\boldsymbol{P}_3~-~\boldsymbol{Q}_1)~:~2(\boldsymbol{Q}_2~-~\boldsymbol{P}_3)~=~4(\boldsymbol{P}_4~-~\boldsymbol{Q}_2)~:~\Delta\boldsymbol{P}_4$, where the ratio determines the parameter value of the cusp

  • Figure 2

    (Color online) $C^1$ Hermite interpolation using class I quintic curves with rational offsets. Given $\boldsymbol{P}_0~=~0,~\boldsymbol{P}_1~=~-1~+~2{\rm{i}},~\boldsymbol{P}_4~=~10~+~4{\rm{i}},~\boldsymbol{P}_5~=~8~+~{\rm{i}}$, and $a_0~=~2$, we get four class I quintic OR curves: (a) $\boldsymbol{P}_2\approx~1.187~+~5.650{\rm{i}}$, $\boldsymbol{P}_3~\approx~7.125~+~6.050{\rm{i}}$; (b) $\boldsymbol{P}_2~\approx~2.858~+~8.064{\rm{i}}$, $\boldsymbol{P}_3~\approx~8.079~+~7.429{\rm{i}}$; (c) $\boldsymbol{P}_2~\approx~10.193~-~3.358{\rm{i}}$, $\boldsymbol{P}_3~\approx~12.271~+~0.903{\rm{i}}$; (d) $\boldsymbol{P}_2~\approx~-~2.470~-~3.361{\rm{i}}$, $\boldsymbol{P}_3~\approx~5.035~+~0.901{\rm{i}}$

  • Figure 3

    (Color online) Selection of the parameter $a_0$ can be used to adjust the shape of the resultant class I quintic OR curve

  • Figure 4

    (Color online) $C^1$ Hermite interpolation using class II quintic OR curves. Given $\boldsymbol{P}_i$, $i=0,1,4,5$, and a real parameter $a_0$, there is a unique quintic OR curve meeting the condition

  •   

    Algorithm 1 Construct class I

    Require: 复数表示的两端控制顶点 $\boldsymbol{P}_i$, $i=0,1,4,5$; 一个实参数 $a_0$.

    Output: 复数表示的4组控制顶点$\boldsymbol{P}_2^{(i)}$, $\boldsymbol{P}_3^{(i)}$, $i=0,\ldots,3$.

    计算$\boldsymbol{A}~=~\frac{5}{a_0}\Delta\boldsymbol{P}_0$, $\boldsymbol{D}~=~5\Delta\boldsymbol{P}_4$;

    构造关于$\boldsymbol{B}$和$\boldsymbol{C}$的二元四次方程(10);

    构造关于$\boldsymbol{B}$和$\boldsymbol{C}$的二元一次方程(9);

    用数值求解方法求解(9)和(10)构成的非线性方程组, 求解得到$\boldsymbol{B}^{(i)}$和$\boldsymbol{C}^{(i)}$, $i=0,\ldots,3$;

    代入(8)求得$\boldsymbol{P}_2^{(i)}$, $\boldsymbol{P}_3^{(i)}$, $i=0,\ldots,3$.

  •   

    Algorithm 2 Construct class II

    Require: 复数表示的两端控制顶点 $\boldsymbol{P}_i$, $i=0,1,4,5$, 一个实参数 $a_0$.

    Output: 复数表示的控制顶点$\boldsymbol{P}_2$, $\boldsymbol{P}_3$.

    计算$\boldsymbol{z}_0~=~\frac{5\Delta\boldsymbol{P}_0}{a_0}$, $\boldsymbol{z}_1~=~5\Delta\boldsymbol{P}_4$;

    构造二元一次复方程(12);

    将方程(12)的实部和虚部分解, 得二元一次实方程组并求解$a_1$, $a_2$;

    代入(11)求得控制顶点$\boldsymbol{P}_2$, $\boldsymbol{P}_3$.

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