Dual generalized B-spline functions and their applications in several approximation problems

A new construction method of dual generalized B-spline functions is presented in this paper. Due to the importance of generalized B-splines, several interesting applications of these dual functions in Computer Aided Manufacturing are also given. The purposes of the paper are as follows: 1) As is known to all, B-spline curves, surfaces and related techniques play important roles in the fields of Computer Aided Geometric Design, Computer Aided Manufacturing and Computer Aided Design for their perfect properties. 2) Recently, on the basis of recursive formula of classical B-spline bases, Juhász and Róth (2013) put forward the generalized B-spline bases, which are generated by monotone increasing and continuous “core” functions. 3) With these flexible “core” functions, generalized B-spline curves and surfaces not only hold almost the same perfect properties which classical B-splines hold but also show more flexibility in practical applications. It is necessary to further study dual generalized B-spline bases. The whole arrangement of the paper is divided into four parts. Firstly, a review of dual bases concluding the construction method is given. Secondly, one fresh method of constructing dual generalized B-spline functions is presented in order to extend their applications. With the help of the dual bases functions, one can easily find the best approximation of any function with expressions of generalized B-spline bases; furthermore, one has no need to consider the orthogonal bases of the corresponding space. After that, several applications of dual generalized B-spline functions in Computer Aided Manufacturing such as least square approximation, curve offsetting and degree reduction are illustrated. Numerical examples show that dual functions greatly simplify these approximation problems. Furthermore, better approximation effects can be achieved by changing core functions which exhibits the effects of "core". The paper not only focuses on the construction of generalized B-spline bases which has never been built before, but also applies the dual bases in the practical applications of Computer Aided Manufacturing. We hope it could be a useful bridge to connect theories and practical applications.