Research On The Theory And The Application Of Basic Spline Class Curves & Surfaces

Based on the theory of Basic Spline and Bernstein basis functions, the concept of basis functions, which are BSC (Basic Spline Class) , BBC (Bernstein-Bezier Class), TBSC(Trigonometric Basic-Spline Class) and TBBC(Trigonometric Bernstein-Bezier Class) functions are deduced firstly by means of rational linear parameter transformation and Trigonometric parameter transformation. Then, the systemic theory of BSC, BBC, TBSC and TBBC curves and surfaces are established. Some important problems will be mainly discussed in this paper, including: the formation and representation of BSC, BBC, TBSC, TBBC curves and surfaces, some significant properties and corresponding algorithm, the expression of normal curves and surfaces in the form of BSC, BBC, TBSC, TBBC, the methods of the reparametrization of curves and surfaces based on the theory of BSC, BBC, TBSC, TBBC curves and surfaces, the application of the theory of BSC, BBC, TBSC, TBBC curves and surfaces in the reparametrization of curves and surfaces and practical modeling.The whole paper is divided into six parts.The first one focuses on the notations and properties of Basic Spline and Bernstein basis functions. At the same time, the theory of Basic Spline and Bezier curves and surfaces are reviewed as the premise of further research.The second one emphasizes in the theory of BSC curves and surfaces. The formation method and parametric equation of these curves and surfaces are worked out. Their properties are discussed in details. Furthermore, the relation between Basic Splines and BSC curves and surfaces is given.The third part is focused on BBC curves and surfaces, which includes the equation and properties of the curves and surfaces, and the relationship between Be zier and BBC curves and surfaces. Some useful corresponding algorithms are also obtained.The fourth part is the theory of TBSC and TBBC curves and surfaces. Based on Trigonometric parametric transformation, TBSC and TBBC functions are obtained first. Then the theory of TBSC and TBBC curves and surfaces are discussed in details. It is not hard to conclude that the theory can also be used to solve the problem of reparametrization of curves and surfaces.The fifth part is the expression of normal curves and surfaces in the form of BSC and BBC.The last part involves the applications of the theory of BSC and BBC curves and surfaces in the field of reparametrization of curves and surfaces and entitative modeling.The details are as follows.The correlative knowledge of Basic Spline and Bernstein basis functions, thetheory of Basic Spline and Bezier Curves & Surface and their rational forms, all these are regarded as the academic background of further research.The theory of BSC curves & surfaces can be considered as the first innovation in this thesis. Firstly, by means of the reparametrization of Basic Spline basis functions, the Basic Spline Class is established and its properties are discussed. Secondly, the BSC form of curves and surfaces are derived and the generating algorithms are investigated. Thirdly, the relation between Basic Spline curves and surfaces and their BSC forms is obtained. We can conclude that the BSC representation of curves and surfaces is the generalized form of Basic Spline curves and surfaces. The theory of BSC curves and surfaces can be applied to solve the reparametrization problem of Basic Spline curves and surfaces.The theory of BBC curves & surfaces can be regarded as the second innovation in this dissertation. Firstly, the concept of Bernstein- Bezier Class functions are defined and their properties are demonstrated in details. Secondly, the BBC form of curves and surfaces are reasoned and the related algorithms are worded out. Thirdly, the connection between Bezier curves and surfaces and their BBC forms is gained. It is not hard to conclude that the BBC representation of curves and surfaces is the extensive form of Bezier curves and surfaces. The applied value of the theory of BBC curves and surfaces lies in that it can help realize the repa...