Approximation Of Bézier Polynomial Curves By Generalized Quasi-control Polygon

The representation and approximation of curves and surfaces is the key research content of Computer Aided Geometric Design(CAGD).CAGD attaches great importance to the mathematical theory and the structure of geometric bodies of CAD/CAM.It uses mathematical theories to describe the relationship between fits,inclusions,and constraints among geometric shapes such as curves,surfaces,parts,assem blies,etc.And using computer means analyze,adjust and optimize these geometric shapes to achieve the expected goals of product design.Bézier curve has excellent properties such as variation reduction,subdivision,and good shape control ability so it has been widely used in curve modeling and occupies a very important position in the CAGD field.When we use curves for practical applications such as rendering,cross-testing or design,a question arises,that is,how close the polyline polygon is to the exact curve geometry.Therefore,the use of polyline polygons to approximate the curve and give its approximation error bound is one of the research issues that CAGD has been paying attention to for a long time.The simplest way a broken line polygon approximates Bézier curve is using the Bézier control polygon which is produced by connecting the Bézier control points.In 1999,Nairn et al.obtained a sharp,quantitative bound on the distance between a Bézier polynomial curve and its Bézier control polygon.The bound depends on the maximal absolute difference of control point and the degree of Bézier polynomial curve.In 2005,Zhang and Wang defined a quasi-control polygon by connecting the points which are the kind of linear combination of Bézier control points.Zhang and Wang used it to approximate Bézier polynomial curve and the effect of using quasi-control polygon to approximate Bézier curve is better than that of using control polygon to approximate.In this paper,based on the work of Zhang and Wang,the definition of quasi-control points is generalized.The initial linear combination coefficients of each three adjacent control points are extended from the original 1/4,1/2,and 1/4 to more general coefficients l,1-2l,and l.Therefore,a new type of quasi-control points is defined,which is called generalized quasi-control points.Connecting these generalized quasi-control points,a new type of approximating broken line polygon is obtained,which is called generalized quasi-control polygons.In this paper,the generalized quasi-control polygon is used to approximate the polynomial curve.The selection of the generalized quasi-control polygon is related to the degree of the Bézier polynomial curve and it's more targeted.We give the approximation theorem and error bounds for approximating Bézier polynomial curves with generalized quasi-control polygons,and give a detailed proof of this theorem.Obviously,compared with the above two methods,the method proposed in this paper is equally simple and accessible to complete and the error of approximating the curve is better.Finally,this paper gives typical examples to compare and analyze the effect of approximating polynomial curve with using the control polygon,the quasi-control and the generalized quasi-control polygon to approximate Bézier polynomial curve.And the result of the examples shows that the effect of approximating Bézier polynomial curve with generalized quasi-control polygon is better.