Approximate Degree Reduction Of C-Bézier Curves

The algorithmic study for the degree reduction of parameter curves, especiallyfor the degree reduction of curves with constraints of endpoints continuity inL² -norm, is one of the popular projects in the field of Computer Aided GeometricDesign these days. C-Bézier curves are some parameter curves defined overF_n={1, t,…, t^n-2, sin t, cos t}. They are similar to Bézier curves in many properties.Besides, they can better figure and deal with conic sections. In this paper, based onthe definitions as well as properties of both C-Bézier basis and C-Bézier curves,first it analyses the explicit expression of the C-Bézier basis which is the linearcombination of the basis 1, t,…,t^n-2, sint, cost. Then it respectively studies theapproximation of one-degree reduction of C-Bézier curves in L²-norm inunconstraint conditions and in constraint of endpoint with parameter continuity orgeometric continuity.In chapter one of the paper,the development of the parameter curves andsurfaces theorems is introduced and the development of the degree reduction ofparameter curves theorems is summarized. In chapter two, after reviewing therecursive construcion as well as the properties of the C-Bézier basis and thedefinition as well as the properties of the C-Bézier curve, this paper gives adirect way by using the properties of C-Bézier basis to show the explicitexpression of the n degree C-Bézier basis. It shows the explicit expression of n+1degree C-Bézier basis by using the recursive construcion of the C-Bézier basis.Finally, it studies one-degree reduction of C-Bézier curves in L² -norm withunconstraint condition and the error forecast. In chapter three, it studies theapproximation of one-degree reduction of C-Bézier curves in L² -norm withconstraint of endpoint parameter continuity or geometric continuity.C¹,C²,G¹,G² continuity is discussed in detail and some examples as well as comparisons are given. In the last chapter, the paper draws the conclusion and makessome prospects of the further work.