Research And Application On Several Categories Of Blending Subdivision Modeling Methods

Subdivision scheme is an efficient method for generating smooth curves and surfaces in Computer Aided Geometric Design.In this dissertation,the blending subdivision scheme that integrates interpolating and approximating is investigated from binary to ternary on static and non-static,curves and surfaces.The main contributions are as follows.In the first chapter,the basic concept,research background and significance of subdivision scheme are introduced firstly.Secondly,the status and main problems of this research at home and abroad are analyzed.Finally,the content and chapter arrangement of this dissertation are given.In the second chapter,from the perspective of geometry,based on the geometric interpretation of the four-point binary interpolating subdivision scheme,we analyze the geometric meaning of the four-point ternary interpolating subdivision scheme,and modify the scheme to combine approximating subdivision,then a blending ternary subdivision scheme with parameters is obtained.Many existing interpolating subdivision schemes and approximating subdivision schemes can be seen as special cases of this scheme,we also use generating polynomial method to analyze the C~k continuity of limit curve produced by the blending subdivision.A new C~4 continuous five-point ternary curve subdivision scheme is obtained.Numerical examples show that the proposed blending subdivision scheme can be used to control the shape of limit curves by selecting appropriate parameters.In the third chapter,the Laurent polynomials of B-spline subdivision scheme and the generalized n-th B-spline ternary subdivision scheme are proposed.Moreover,we derive three distinguishing subdivision schemes,an approximating subdivision based on interpolating subdivision scheme,an interpolating subdivision based on approximating subdivision scheme and a blending subdivision scheme that integrates interpolating and approximating,which all utilize the interpolating four-point ternary subdivision scheme proposed by Hassan and generalized cubic B-spline ternary subdivision scheme.Each subdivision scheme can unify the ternary interpolating and approximating subdivision.We give the geometric interpretation and analyze the continuity of the scheme.Numerical examples are also given to demonstrate the influence of parameter on the limit curves.In the fourth chapter,we propose four-point ternary interpolating subdivision schemes for regular and irregular quadrilateral meshes,and a tensor product B-spline surface subdivision scheme on the quadrilateral mesh.The proposed 1-9 interpolation surface subdivision scheme and tensor product B-spline surface subdivision scheme are used to obtain a kind of ternary surface subdivision scheme that blends interpolation and approximation.Here,the continuity of this scheme is analyzed.The numerical example shows that the method is reasonable and effective.In the fifth chapter,three different non-stationary four-point binary blending subdivision schemes are constructed according to the relationship between the non-stationary interpolating four-point subdivision scheme and cubic exponential B-spline subdivision scheme.Among them are a non-stationary approximating subdivision scheme based on non-stationary interpolating subdivision,a non-stationary interpolating subdivision scheme based on non-stationary approximating subdivision,and a non-stationary blending subdivision scheme that integrates interpolating and approximating.For many existing interpolating subdivision schemes and approximating subdivision schemes,we find they are special cases of the proposed blending subdivision schemes.Geometric interpretation of these schemes are given,and some other properties of these schemes are analyzed such as C~k continuity,the exponential polynomial generation and reproduction.Numerical examples show that the proposed blending subdivision scheme can be used to control the shape of limit curves by selecting appropriate parameters.Simultaneously,these non-stationary subdivision schemes can reproduce conic curves.In the sixth chapter,in order to obtain the non-stationary ternary subdivision scheme unifying interpolation and approximation,a non-stationary four-point binary blending subdivision scheme is constructed.What's more,existing interpolating subdivision schemes and approximating subdivision schemes may be special cases of the blending subdivision scheme which we proposed here.The scheme are explained geometrically,and some properties of the scheme is analyzed such as continuity,reproduction.Numerical examples show that the proposed blending subdivision scheme can be used to control the shape of limit curves by selecting appropriate parameters.When the initial control polygon is a square,through selecting the appropriate parameters,the subdivision schemes can reproduce the circle.In the seventh chapter,we summarize the work of this dissertation and look forward to the future work.