Research On Combined Subdivision Curves Modeling

This thesis mainly discusses the general framework of combined approximating and interpolating subdivision scheme,including the construction of four-point ternary general framework,the analysis of the important properties of combined subdivision scheme and a kind of combined subdivision scheme are constructed from the perspective of Laurent polynomial.In the second chapter,some properties of combined approximating and interpolating ternary subdivision scheme,such as support,continuity and fractal property are analyzed by means of the Laurent polynomials.The sufficient condition of C~3 continuity properties is proved.It is pointed out that the parameter can be adjusted to control the shape of the limit curve,which generates brand-new ternary schemes,and some comparisons with other methods are given.In the third chapter,one new combined ternary 4-point subdivision scheme with multiple parameters is proposed.A set of nice properties,such as support,continuities and polynomial generation property,have been briefly discussed.The new combined scheme not only contains a lot of classical ternary schemes as specified cases,but also generates brand-new ternary schemes.Compare to other approximating subdivision schemes,limit curves generated by the given scheme are more consistent with the corresponding control polygons and keep detailed features.Given examples show the effectiveness of the scheme.Furthermore,fractal property of the scheme has been analyzed and nice fractal images are given.In the fourth chapter,based on B-spline with best smoothness,a Laurent polynomial is constructed by means of its relationship with the generated polynomial of the subdivision scheme.This Laurent polynomial not only generalizes a family of classical subdivision schemes,but also generate C~3-continuous scheme with detailed feature.The support size and the necessary and sufficient condition for C~3 continuity of the limit curve are analyzed for a combined scheme.Finally,numerical examples are given demonstrate the influence of parameters on the limit curves.