Research On The Issues And Application Of The PH Curves

Pythagrean-hodograph (PH) curve is a special kind of polynomial curve.PH curve not only retains full compatibility with standard Bezier/B-spline representation,but also offers unique computational advantages over "ordinary" polynomial parametric curves.For example,the arc length of PH curve and equidistance line can be computed precisely, when PH curve is polynomial or rational parametric curves, the corresponding arc function and the equidistance line are also a polynomial or rational parametric curves;In addition,PH curve interpolation of discrete points can get a relatively uniform distribution of curvature, and the curves are often more smooth than the classical polynomial curve.However,the research of PH curves are extensive and exist a lot of work to do. The several theory and application of PH curves are researched in this dissertation,major work is as follows;Firstly, the parameter is introduced by MObius transformation in the extended complex plane C∞=C∪{∞} Rational Pythagorean-Hodograph quintic of C2 Hermite interpolation is constructed and interpolation algorithm is given. The optimal curve satisfing interpolation conditions is given based on the "Elastic Bending Energy", the numerical examples show the effectiveness of the algorithm.Secondly,the sterographic projection preserves the invariance propetrty of PH. The spatial C1 Hermite date are projected to a xy-plane by sterographic projection. The xy-plane PH curves with undetermined junction points and default data are constructed. Then, by the inverse general sterographic projection, spherical rational PH curves are generated. The numerical examples show the effectiveness of the algorithm.