NURBS Representation Of The Ellipse And Its Application In Surface Modeling

Because of the huge function of NURBS in the field of CAD/CAM today,it has been used to express the curves and curved surfaces by a variety of popular geometric modeling system as the only way.Not only,it retains the advantages of B-spline methods,also it can accurately represent common quadratic curve and secondary surface in the engineering.Conic section and quadric surfaces are widely used in engineering,tertiary NURBS curves and surfaces are most common in engineering applications,but conic curve is often used to construct a common curve surface,especially in quadric surface.The standard rational cubic Bezier of secondary conic curve represents is not unique,a same conic curve has infinite variety of standard rational cubic Bezier representation,but the inside weight factor ? must satisfy ?>-2/1.This paper focuses on the whole ellipse represents an infinite number of cubic NURBS and Its Application in common surface modeling discuss as the following aspects:Firstly,we use the Prautzsch algorithm to degree elevation a kind of secondary NURBS representation of the whole ellipse programs to get its times NURBS representation.Based on it,we point that actually the same cubic NURBS representation of different ellipse is corresponding to the rational linear transformation parameters,and discussing the whole ellipse with unlimited variety cubic NURBS representation problem,in turn,It reached the same cubic NURBS representation of different ellipse.Secondly,it discusses the common curved surfaces NURBS representation issues such as cylindrical surface,conical surface,rotating surface and ellipsoidal surface,and then giving the representing method of the same surface with parameters of the infinite variety of NURBS with time.For different values of parameters we will get different control points and weight factor,but they are represented by the same surface.Finally,further noted,the same ellipse with different representation of times NURBS is corresponding to the rational linear transformation parameters.Not only we can change the walue of the control vertex position and weight factors by adjusting the values of the parameters,also change the correspondence between the point of the curve and the point within the parameter,which is having the different effects of the parameters.The essential difference between the same surface with different NURBS representation is that the parameters of the curved surface condition on different points.It provides an effective control means for us to seek appropriate parameterization when constructing NURBS curved surfaces.