发布时间: 2019-01-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.180318 2019 | Volume 24 | Number 1 计算机图形学

Möbius变换下四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}}$ Hermite插值

 收稿日期: 2018-05-23; 修回日期: 2018-08-13 基金项目: 国家自然科学基金项目（11471093） 第一作者简介: 郭宇, 1992年生, 男, 助教, 硕士研究生, 主要研究方向为计算机辅助几何设计、应用数值逼近。E-mail:296997289@qq.com;江平, 女, 博士, 教授, 硕士生导师, 主要研究方向为应用数值逼近、计算机辅助几何设计。E-mail:jiangping_72@sina.com;王剑敏, 男, 硕士研究生, 主要研究方向为计算机辅助几何设计。E-mail:1277241041@qq.com. 中图法分类号: TP391 文献标识码: A 文章编号: 1006-8961(2019)01-0096-07

# 关键词

Möbius变换; 有理等距曲线; ${{\rm{C}}^{\rm{2}}}$Hermite插值; 绝对旋转数; 弹性弯曲能量

${{\rm{C}}^{\rm{2}}}$ Hermite interpolation based on quartic rational parabolic-PH curves by using Möbius transformation
Guo Yu, Jiang Ping, Wang Jianmin, Liu Zhi
School of Mathematics, Hefei University of Technology, Hefei 230000, China
Supported by: National Natural Science Foundation of China (11471093)

# Abstract

Objective The offset curve, also known as the parallel curve, refers to the locus of points along the normal vector direction with distance d. In recent years, the offset curve has played an important role in many fields and is widely applied in computer-aided geometric design (CAGD). In general, the arc length and offset curve of the polynomial curve have no rational form, and the offset-rational (OR) curve is a special polynomial parameter curve with exactly rational offset curves. The special properties of the curve have attracted the attention of many researchers. In recent years, the interpolation problem of OR curves has been widely studied. The problem of curve interpolation is widely used in many modern industrial fields, such as robot design, machinery industry, and space industry. The Hermite interpolation of given endpoint is a common method to construct a curve in CAGD. The ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation problem of the quartic parabolic-PH curve, which is an even order of offset rational curve is discussed in this paper. Method Based on the parameters introduced by M bius transformation, a bijective linear fractional transformation, the ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation of quartic rational parabolic-PH curve is constructed through complex analysis. The data $H_C^2 = \{ {R_0},{R_1},{T_0},{T_1},{V_0},{V_1}\}$ to be interpolated is given with R0 and R1 referring to the two end points, T0 and T1 for the tangent vectors at R0 and R1, and V0 and V1 for the second tangent vectors at R0 and R1. By appropriate transformation, rotation, and scaling, making R0=0 and R1=1, we can further obtain the interpolation conditions for ${{\rm{C}}^{\rm{2}}}$ curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation, whose tangents have three orders. By supposing the expression of $r\left( t \right)$, $F\left( t \right)$, $G\left( t \right)$, the first-and second-order derivative of the curve can be obtained. The corresponding expression of the control points and the Bézier curve can be obtained by using the integral relation formula. The exact value of the parameter are calculated by the ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation condition of the curve. Then, the quartic rational parabolic-PH curves formed by the M bius transformation are finally constructed. Result By providing a set of "reasonable" endpoints to be interpolated, we can obtain 12 ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation curves from the transformed quartic polynomial parabolic-PH curve under the initial interpolation condition and further obtain the ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation curves of the 12 quartic rational parabolic-PH curves under the initial interpolation condition. Numerical examples show the effectiveness of the algorithm. It is not clear and convenient to choose the appropriate interpolation curve from the 12 interpolation curves. We need to select the curves that satisfy the interpolated condition and can elastically handle the inflection points. Other interpolation curves may have cusp points, node points, closed loops, or obviously inconsistent with geometric design requirements. By combining the minimum absolute rotation number and the elastic bending energy minimization, the selection method for determining the optimal curve satisfying the interpolation condition is put forward. When the absolute rotation number and the elastic bending energy of the interpolated curve are minimized, the optimal curve is often obtained, which has better smoothness and natural shape that meet the needs of geometric design. The examples illustrate that the traditional quartic parabolic-PH curve can construct ${{\rm{C}}^1}$ Hermite interpolation curves. However, the constraint of interpolation condition does not allow the direct construction of a curve with higher continuity. For traditional quintic PH curve, we cannot directly construct a curve with a continuity higher than ${{\rm{C}}^1}$, whereas through M bius transformation, we can achieve ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation, which has a higher continuity than the traditional method. For the same set of given data, we construct the ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation curve from quintic rational PH curve and quartic rational parabolic-PH curve. Compared with the 18 quintic rational PH curves, we can achieve the optimal curve from the 12 quartic rational parabolic-PH curves with lower elastic bending energy. Hence, the quartic rational parabolic-PH curves constructed by our method have more natural geometry than the traditional quintic rational PH curves. Although parabolic-PH curves with eight degree can be used to construct ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation curves, the solution is complex, and the computation is large. Hence, through analysis and comparison, the quartic rational parabolic-PH curve presented in this paper has a simpler computation than quintic PH curves and parabolic-PH curves with eight degree. The interpolation results of the quartic rational parabolic-PH curve is more obvious, and the optimal curve best meets the requirement for the geometric design. Conclusion The use of ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation of quartic rational parabolic-PH curves constructed by the introduction of M bius transformation not only ensures low degree of interpolation curve but also ensures a higher continuity of interpolation conditions. It makes the calculation simpler and the interpolation effect more obvious compared with the traditional PH curve with odd number of order. Related research on the sub PH curve is of certain significance. This report is significant for the study of PH curves with even number of degree.

# Key words

Möbius transformation; offset-rational curves; ${{\rm{C}}^{\rm{2}}}$ Hermite interpolation; absolute rotation number; elastic bending energy

# 1 预备知识

$\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}:P\left( t \right) = \left( {x\left( t \right), y\left( t \right)} \right) = \sum\limits_{i = 0}^n {} {P_i}B_i^n\left( t \right)$$n次已适当参数化的平面多项式曲线，其在复平面上的相应表示为 Z\left( t \right) = \sum\limits_{i = 0}^n {} {Z_i}B_i^n\left( t \right), {Z_i} = {\rm{ }}{x_i} + {\rm{i}}{y_i}是控制顶点{\mathit{\boldsymbol{P}}_i} = \left( {{x_i}, {y_i}} \right) 对应的复数表示， \left\{ {B_i^n\left( t \right)} \right\}_{i = 0}^n$$ n$次Bernstein多项式基函数。

 $Z'\left( t \right) = r\left( t \right)F\left( t \right){G^2}\left( t \right)$ (1)

 $\begin{array}{*{20}{c}} {s'\left( t \right) = \mathit{\Phi '}\left( {Z\left( t \right)} \right)Z'\left( t \right) = }\\ {\frac{{ad - bc}}{{{{\left( {cz + d} \right)}^2}}}{r_1}\left( t \right){F_1}\left( t \right)G_1^2\left( t \right)} \end{array}$

$G_2^2\left( t \right){\rm{ }} =\frac{{ad - bc}}{{{{\left( {cz + d} \right)}^2}}}G_1^2\left( t \right)$，即$s\prime \left( t \right) = {r_1}\left( t \right){F_1}\left( t \right)G_2^2\left( t \right), {G_2}\left( t \right)$为复系数多项式，故由引理1可得，$s(t)$ 为抛物-PH曲线。

 $\begin{array}{*{20}{c}} {\left| {s'\left( t \right)} \right| = \frac{{\left| {ad - bc} \right|}}{{{{\left| {cZ\left( t \right) + d} \right|}^2}}}\left| {Z'\left( t \right)} \right| = }\\ {\frac{{\left| {ad - bc} \right|}}{{{\mathop{\rm Re}\nolimits} {{\left( {cZ\left( t \right) + d} \right)}^2} + {\mathop{\rm Im}\nolimits} {{\left( {cZ\left( t \right) + d} \right)}^2}}}\left| {Z'\left( t \right)} \right|} \end{array}$

$s(t)$为有理曲线。

 $\kappa \left( t \right) = \frac{{\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right) \times {{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }^\prime \left( t \right)} \right|}}{{{{\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right)} \right|}^3}}}$

Table 1 Absolute rotation number for 12 curves $\mathit{\Phi}(Z(t))$ in Fig. 2

 1 2 3 4 5 6 7 8 9 10 11 12 R 0.988 9 0.872 5 1.286 6 0.950 1 1.316 3 0.858 1 0.473 7 0.387 5 0.400 7 0.355 7 0.449 8 0.386 8

Table 2 Elastic bending energy for 12 curves $\mathit{\Phi}(Z(t))$ in Fig. 2

 1 2 3 4 5 6 7 8 9 10 11 12 E 822.36 80.212 154.12 209.99 89.828 130.43 6.313 9 4.968 8 4.934 1 4.416 8 5.970 2 4.930 6

Table 3 Comparison of elastic bending energy corresponding to the optimal curve

 最优曲线对应的E值 五次有理PH曲线 1.33 四次有理抛物-PH曲线 0.839 9

 $r\left( t \right) = 1,F\left( t \right) = a\left( {1 - t} \right) + bt$

 $G\left( t \right) = {\left( {1 - t} \right)^3} + c{\left( {1 - t} \right)^2}t + d\left( {1 - t} \right){t^2} + e{t^3}$

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