Bezier曲线及其等距线的光顺样条逼近

Faired Spline Approximation of Bézier Curves and Offset Curves

给出了用低次 Bézier 样条曲线逼近高次 Bézier 曲线及其等距线的方法,并且可以对逼近曲线进行光顺处理.

Although Farouki in 1990[3,4] studied algebraic properties of curves and their offset curves,the opin-ion of the authors is that,as regards approximation algorithm,the study by Hoschek in 1988[2] remainsthe best.But the authors deem that,in order to facilitate design of geometric shape,it is necessary to gobeyond Hoschek's paper and study how to devise an algorithm that is needed to improve the fairing ofcurves and their offset curves.The authors use low order spline Bézier curves—3rd order for G^1 continuity and 5th order forG^2 continuity—to obtain good faired approximation of high order Bézier curves.The authors' method isused in adjusting curvatures of the curves so as to make the rate of variation of curvature fairly constant,thus making it possible to obtain good quality fairing needed in engineering application.In the design ofairplanes,automobiles and ships,the design and computation of offset curves is required for all geometricmodeling systems.Expressions for offset curves are in general much more complicated than those for thecurves themselves.The authors' new algorithm is also capable of obtaining faired approximation of offsetcurves.At present,improving the fairing of curve is achieved by adjustment of interpolating points in engi-neering design.The adjustment of interpolating points is troublesome because it causes in general a changein the geometric shape of curve.The authors' new algorithm solves nonlinear equations withQuasi-Newton's method and the least squares method is used.Thus,instead of adjusting interpolatingpoints of curves,it obtains optimun Bézier control vertices and then obtains curves and offset curves withoptimum faired approximation.