Mbius变换下四次有理抛物-PH曲线的C~2 Hermite插值
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摘要
目的曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用,而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法,本文讨论了一种偶数次有理等距曲线,即四次抛物-PH曲线的C2Hermite插值问题。方法基于Mbius变换引入参数,利用复分析的方法构造了四次有理抛物-PH曲线的C~2 Hermite插值,给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果通过给出"合理"的端点插值数据,以数值实例表明了该算法的有效性,所得12条插值曲线中,结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法,并以具体实例说明了与其他插值方法的对比分析结果。结论本文构造了Mbius变换下的四次有理抛物-PH曲线的C~2 Hermite插值,在保证曲线次数较低的情况下,达到了连续性更高的插值条件,计算更为简单,插值效果明显,较之传统奇数次PH曲线具有更加自然的几何形状,对偶数次PH曲线的相关研究具有一定意义。
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 C~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 C~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 isgiven with R_0 and R_1 referring to the two end points,T_0 and T_1 for the tangent vectors at R_0 and R_1,and V_0 and V_1 for the second tangent vectors at R_0 and R_1. By appropriate transformation,rotation,and scaling,making R_0= 0 and R_1= 1,we can further obtain the interpolation conditions for C~2 curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for C~2 Hermite interpolation,whose tangents have three orders. By supposing the expression of r( t),F( t),G( t),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 C~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 C~2 Hermite interpolation curves from the transformed quartic polynomial parabolic-PH curve under the initial interpolation condition and further obtain the C~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 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 C~1,whereas through M bius transformation,we can achieve C~2 Hermite interpolation,which has a higher continuity than the traditional method. For the same set of given data,we construct the C~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 C~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 C~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.
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 C~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 C~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 isgiven with R_0 and R_1 referring to the two end points,T_0 and T_1 for the tangent vectors at R_0 and R_1,and V_0 and V_1 for the second tangent vectors at R_0 and R_1. By appropriate transformation,rotation,and scaling,making R_0= 0 and R_1= 1,we can further obtain the interpolation conditions for C~2 curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for C~2 Hermite interpolation,whose tangents have three orders. By supposing the expression of r( t),F( t),G( t),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 C~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 C~2 Hermite interpolation curves from the transformed quartic polynomial parabolic-PH curve under the initial interpolation condition and further obtain the C~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 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 C~1,whereas through M bius transformation,we can achieve C~2 Hermite interpolation,which has a higher continuity than the traditional method. For the same set of given data,we construct the C~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 C~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 C~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.
引文
[1] Wang G J,Wang G Z,Zheng J M. Computer Aided Geometric Design[M]. Beijing:Higher Education Press,2001.[王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育出版社,2001.]
[2] Farouki R T,Sakkalis T. Pythagorean Hodographs[M]. New York:IBM Corp,1990.
[3] Farouki R T,Neff C A. Hermite Interpolation by Pythagorean Hodograph Quintics[M]. Washington:American Mathematical Society,1995.
[4] Farouki R T. The elastic bending energy of Pythagorean-hodograph curves[J]. Computer Aided Geometric Design,1996,13(3):227-241.[DOI:10. 1016/0167-8396(95)00024-0.]
[5] Zheng J M. A rational parametrization of the offset curve[J].Chinese Science Bulletin,1994,(20):1915.[郑建民.关于等距曲线的有理参数化[J].科学通报,1994,(20):1915.]
[6] Lv W. Offset-rational parametric plane curves[J]. Computer Aided Geometric Design,1995,12(6):601-616.[DOI:10.1016/0167-8396(94)00036-R.]
[7] Chen G D,Wang G J. C2hermite interpolation by offset-rational curves[J]. Journal of Engineering Graphics,2000,21(3):64-69.[陈国栋,王国瑾.有理等距曲线的C2Hermite插值[J].工程图学学报,2000,21(3):64-69.][DOI:10. 3969/j.issn. 1003-0158. 2000. 03. 012.]
[8] BartoňM,Jüttler B,Wang W P. Construction of rational curves with rational rotation-minimizing frames via M9bius transformations[C]//Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces. Berlin:SpringerVerlag,2010,5862:15-25.
[9] Lee S,Lee H C,Lee M R,et al. Hermite interpolation using M9bius transformations of planar Pythagorean-Hodograph cubics[J]. Abstract and Applied Analysis,2012,2012:560246.
[10] Gui X S. The construction of quartic pythagorean hodograph curve[D]. Hefei:Hefei University of Technology,2010.[桂校生.四次Pythagorean Hodograph速端曲线的构造[D].合肥:合肥工业大学,2010.]
[11] Zhang W,Wang G J. Involutes of quartic PH curves and their geometric Hermite interpolation[J]. Journal of Computer-Aided Design&Computer Graphics,2011,23(2):216-222.[张威,王国瑾.四次PH曲线的渐开线及其几何Hermite螺线插值[J].计算机辅助设计与图形学学报,2011,23(2):216-222.]
[12] Ding C. Research on the issues and application of the PH curves[D]. Hefei:Hefei University of Technology,2015.[丁晨. PH曲线的若干问题及应用研究[D].合肥:合肥工业大学,2015.]
[2] Farouki R T,Sakkalis T. Pythagorean Hodographs[M]. New York:IBM Corp,1990.
[3] Farouki R T,Neff C A. Hermite Interpolation by Pythagorean Hodograph Quintics[M]. Washington:American Mathematical Society,1995.
[4] Farouki R T. The elastic bending energy of Pythagorean-hodograph curves[J]. Computer Aided Geometric Design,1996,13(3):227-241.[DOI:10. 1016/0167-8396(95)00024-0.]
[5] Zheng J M. A rational parametrization of the offset curve[J].Chinese Science Bulletin,1994,(20):1915.[郑建民.关于等距曲线的有理参数化[J].科学通报,1994,(20):1915.]
[6] Lv W. Offset-rational parametric plane curves[J]. Computer Aided Geometric Design,1995,12(6):601-616.[DOI:10.1016/0167-8396(94)00036-R.]
[7] Chen G D,Wang G J. C2hermite interpolation by offset-rational curves[J]. Journal of Engineering Graphics,2000,21(3):64-69.[陈国栋,王国瑾.有理等距曲线的C2Hermite插值[J].工程图学学报,2000,21(3):64-69.][DOI:10. 3969/j.issn. 1003-0158. 2000. 03. 012.]
[8] BartoňM,Jüttler B,Wang W P. Construction of rational curves with rational rotation-minimizing frames via M9bius transformations[C]//Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces. Berlin:SpringerVerlag,2010,5862:15-25.
[9] Lee S,Lee H C,Lee M R,et al. Hermite interpolation using M9bius transformations of planar Pythagorean-Hodograph cubics[J]. Abstract and Applied Analysis,2012,2012:560246.
[10] Gui X S. The construction of quartic pythagorean hodograph curve[D]. Hefei:Hefei University of Technology,2010.[桂校生.四次Pythagorean Hodograph速端曲线的构造[D].合肥:合肥工业大学,2010.]
[11] Zhang W,Wang G J. Involutes of quartic PH curves and their geometric Hermite interpolation[J]. Journal of Computer-Aided Design&Computer Graphics,2011,23(2):216-222.[张威,王国瑾.四次PH曲线的渐开线及其几何Hermite螺线插值[J].计算机辅助设计与图形学学报,2011,23(2):216-222.]
[12] Ding C. Research on the issues and application of the PH curves[D]. Hefei:Hefei University of Technology,2015.[丁晨. PH曲线的若干问题及应用研究[D].合肥:合肥工业大学,2015.]