General Rational Parametric Curve To The Rational B¨¦zier Curve Into The Study

Because NURBS can express the complicated curves and surfaces composed with elementary and other curves and surfaces uniformly, and it has the advantages of local region adjusting and continuous rank adjusting, NURBS enhances the ability of surface modelling in the system of CAD/CAM, and is becoming the core of geometry formative technology. As the special form of the NURBS, rational Bézier curve has been used more widely.Adjusting the shape of the curve is needed in the surface modelling. Because a general rational parametric curve is short of weights and control points, its interaction is poor, while the rational Bézier curve can adjust its shape conveniently by adjusting its weights or control points. So how to convert a given general rational parametric curve to a rational Bézier curve is deserved studying.Firstly, the paper introduces the development of NURBS, the brief concise concepts of NURBS, Bézier curve, rational Bézier curve and general rational parametric curve. The essentiality and the feasibility of converting the given general rational parametric curve to a rational Bézier curve are presented.Then this paper discusses the question of converting a given general rational parametric curve to a rational Bézier curve, and presents a method-elevation method. In general, a rational parametric curve of degree n cannot be converted into a rational Bézier curve of degree n with nonzero weights. Using the method of degree elevation to the rational Bézier curve of degree n, we can get the corresponding rational Bézier curve, all control points of this rational Bézier curve can effect the shape of the curve, that is to say , the weights of all points are nonzero and the degree is the lowest.After discussing the general way of converting a general rational parametric curve to a rational Bézier curve, the paper presents the converting results of general rational parametric curves of degree n (n=2,3,4,5).The elevation method also can be used to produce the rational Bézier curve with positive weights, which widens the applicability.At last , the paper presents the future of studing, that is whether there exists a certain converting formula from a general rational parametric curve to a rational Bézier curve.