Research On The Applications Of Dual Basis And Geometric Approximation In CAGD

Computer Aided Geometric Design(CAGD) is an emerging discipline along with the appearance of computers and the flourish of modern industrial such as aviation,shipbuilding,design and manufacturing of machine.Approximation and representation of curves and surfaces are the two fundamental theoretical themes in CAGD.In recent years,new curves have constantly been raised such as Wang-Ball curve,Said-Ball curve,SBGB curve,WSGB curve and WBGB curve.Compared with the most widely used Bezier curves,these curves also inherit the properties such as endpoints' property,convex hull property and shape preservation.In order to realize data exchange among different modeling systems and graphics systems,also in order to achieve mixed use of different curves in the same system and the consistency of different kinds of geometric operations such as splicing,derivative and drawing,we need to study conversions between different curves.The dual bases are widely applied to mutual transformation between different bases for the same curve. Therein degree reduction approximation,offset approximation and polynomial approximation of rational curves/surfaces have become the hotspots of investigation,since they directly relate to the efficiency,precision,quality and function of geometric design systems.In view of these facts,the dissertation gives a more in-depth discussion about the theories and applications of dual bases and geometric approximations.The main results in this dissertation are outlined as follows:●In respect of theories and applications of dual basis(1) Dual bases of NS-power basis and WBGB basis are constructed.By means of dual bases of NS-power basis and WBGB basis,we also obtained the Marsden identity and realized the conversions from Bezier curve to NS power curve and WBGB curve.These results are very useful for the applications of NS curve and WBGB curve and their popularizations in Computer Aided Geometric Design.(2) By introducing a set of parameters K,L,we further investigated the unifying representation of generalized Ball basis and presented Bezier-Said-Wang type generalized Ball curve (BSWGB curve).BSWGB basis unifies Said-Ball basis,Wang-Ball basis,SBGB basis,WSGB basis and WBGB basis and each of the latter five bases is the special case of the new one.We also used dual basis as a tool to further investigate BSWGB basis and deduce the dual basis of it. Accordingly,we discussed the dual bases of some paper as special cases.We also solve the two problems in practice:1) deducing the BSWGB basis representation of the power basis,namely, Marsden identity,2) deducing the conversion matrices from Bernstein basis to BSWGB basis.(3) By introducing the definition and operations of the inner-product matrix of two vector functions,explicit formulas for the dual basis functions of Said-Bezier type generalized Ball basis (SBGB) with respect to the Jacobi weight function are given.The dual basis functions satisfying boundary constraints are also considered.We also gave integral dual functional of SBGB basis and presented a direct solution of the least squares approximation polynomials satisfying interpolating conditions of functions by means of SBGB basis.As a result,the dissertation includes the weighted dual basis functions of Bernstein basis,Said-Ball basis and some intermediate bases and discusses the dual bases of paper([J(u|¨)t98],[RA07],[RA08]) as special cases.These results are very useful for the study of SBGB basis and wide range of application of SBGB basis.Using the above results, we can discuss the weighted dual basis functions of WBGB,WSGB,and BSWGB basis similarly. As its applications,the dissertation also gave the approximation algorithms of Bezier curves' offsets.●In respect of the applications of S-power basesS-power basis possesses good numerical condition,the conversion matrices between the Bernstein basis and the S-power basis are not ill-conditioned.S-power basis preserves the advantages of the power form such as simple form,easy to calculate(admits a Horner's evaluation scheme) and the coefficients of it obviously admit geometrical interpretation and can be used as shape control tools.The most important thing is S-power curve can keep high continuities on two endpoints automatically and especially suited for the piecewise representations of curves and surfaces.Sanchez-Reyes gave detailed studies to unitary S-power basis and gave a brief introduction to bivariate one.Based on these,we gave detailed discussions of division operations and extraction function of bivariate S-power basis.As the applications to bivariate S-power basis,we studied degree reduction of tensor product Bezier surfaces,polynomial approximation of rational Bezier surfaces and offset approximation of Bezier surfaces.The algorithms show that the complexity is low by using bivariate S-power basis,which only involves in additions,subtractions and multiplications,and the approximation surface automatically interpolates to the four corner points of the original one.●In respect of the applications of S-power bases(1) We obtained rational and polynomial approximation algorithms of offset curves by using "two points" Newton interpolation with confluents.The algorithms could keep high continuities on two endpoints and especially suited for the piecewise representation of curves.Increasing the orders of the polynomials and combining with subdivision algorithm,we can improve approximation precision easily and the curve could keep high continuities on subdivision points.(2) We obtained rational approximation algorithms of offset curves by using modified Thiele type interpolation.We also further considered rational approximation algorithms satisfying endpoints constraints.Our algorithms only involved multiplication,division and their complexity was O(n~2).It had a better improvement,compared with Li's algorithm(its complexity:O(n~3)). We also obtained rational approximation algorithms of offset surfaces by using modified bivariate Newton-Thiele blending interpolation algorithms and the algorithms only involved multiplication and division.