Research On Hermite Interpolation Of Quadric Parabolic-PH Curve

PH(Pythagorean Hodographs)curve is a kind of special polynomial parametric curve with precise rational offset curve,which has unique advantages and keeps complete consistency with standard B-spline representation and Bézier representation compared with classical polynomial parametric curve.Therefore,the study of PH curve has important theoretical significance and application value,and it is one of the focuses in Computer Aided Geometric Design(CAGD)research in recent years.Along with the continuous deepening of scholars'research,many important research results have emerged,but the analysis shows that there are many problems to be studied about PH curve,and there is still much work to be improved and supplemented.Among them,the interpolation problem of PH curve has been widely used in many modern industrial fields such as robot design,machinery industry,aerospace industry,etc,and Hermite interpolation with known endpoint data is a commonly method to constructe curve in CAGD.This dissertation focuses on an even-order rational offset curve with a low order,that is,the Hermite interpolation problem of the quadratic parabolic-PH curve.In this dissertation,the parameters are introduced based on M?bius transformation,and the C~2 Hermite interpolation of quadratic rational parabolic-PH curves is constructed by the complex analysis method.The numerical examples show the effectiveness of the algorithm,combined with the minimum absolute rotation number and the elastic bending energy minimization,giving a selection method for judging the optimal curve satisfying the interpolation condition,and the comparison analysis results with other interpolation methods are illustrated by specific examples.The C~2 Hermite interpolation of quadratic rational parabolic-PH curves under the M?bius transformation constructed in this dissertation achieves a higher continuity interpolation condition with lower order of curves,simpler calculation,and more obvious interpolation effect,having more natural geometry than the construction of traditional PH curve.In addition,the curve construction of quadratic parabolic-PH curves under other continuous interpolation conditions is derived.The GC~2 Hermite interpolation of quadratic parabolic-PH curves is constructed by complex analysis method.The specific interpolation algorithm and the corresponding Bézier curve representation and control poingts expression are given.Numerical examples verify the feasibility of the algorithm.The construction of GC~2 Hermite interpolation of the quadratic parabolic-PH curve enriches the construction method of the quadratic parabolic-PH curve under the other types of continuous interpolation conditions,which has certain significance for the related research and development of even-order rational equidistant curves.