Research On Generalized De Boor-Cox Formulas For Changeable Degree B-Splines In CAD

In the field of computer-aided geometric design,B-spline is a mature tool for free-form curve and surface modeling.Its basis functions are piecewise polynomials defined on parameter intervals,which form the unique normalized B-basis within the corresponding piecewise polynomial space.And the basis of B-splines can be calculated by the famous de Boor Cox formula,which is the key reason why B-splines are widely used.However,B-splines require the same degree on each segment,and in practical curve design,it is often necessary to use high-degree polynomials to represent low-degree polynomials,resulting in inefficient data usage.Changeable-degree B-splines allow basis functions to have different degrees on different segments,and like B-spline bases,they are the unique normalized B-basis in their space,making them a direct extension of B-splines.However,their basis functions are defined by integral recursion,which results in inefficient computation.Moreover,in most cases,there is no de Boor-Cox formula for changeable-degree B-spline functions similar to that of B-splines.Therefore,a more efficient method is needed to compute changeable-degree B-spline basis functions.This thesis presents the following contributions to this issue:(1)This thesis introduces an algorithm for generating changeable-degree B-spline basis functions,based on the Bernstein basis representation.In this context,the Bernstein basis representation refers to the coordinates of the changeable-degree B-spline functions under the Bernstein basis,which can be stored as vectors or matrices.Any Bernstein polynomials can be directly obtained based on the expression,so it is only necessary to calculate the representation matrix of the changeable-degree B-spline basis under the Bernstein basis to obtain the desired result.Unlike existing global algorithms,the algorithm proposed in this thesis outputs the basis representation matrix through a recursive approach,which enables the computation of partial or whole sets of changeable-degree B-spline basis functions.The idea behind computing the basis function is consistent with the traditional spline method,but instead of polynomials,vectors are used in the recursive process.This substitution allows for the fast calculation of the basis function values at certain points,utilizing the local support property of the changeabledegree B-spline basis functions.Additionally,numerical experiments have demonstrated that the algorithm has high computational efficiency and excellent numerical accuracy.(2)This thesis presents two generalized de Boor-Cox formulas for changeable-degree Bsplines.The first formula is a structured multilevel generalization of the de Boor-Cox algorithm,which preserves the structure of the product between the changeable-degree B-spline function and the coefficient polynomial.The multilevel aspect involves changeable-degree B-spline functions at non-adjacent levels in the recursive construction,corresponding to the result of applying the de Boor-Cox algorithm multiple times to B-spline functions.This formula focuses on describing the relationship between changeable-degree B-spline functions at different levels in the recursive construction.The intermediate results generated by the algorithm for generating the basis representation matrix correspond to changeable-degree B-spline functions in the recursive construction and provide a useful tool for computing the coefficient polynomial.The second formula is a single-level generalization of the de Boor-Cox formula that preserves the coefficient polynomial’s structure as a simple first-degree polynomial.Similar to the de BoorCox formula,this formula consists of the sum of two polynomial products.To achieve this relationship,a transition function is constructed to replace the changeable-degree B-spline function,and there are multiple methods for constructing this transition function.This paper proposes two methods based on the Taylor expansion property of polynomials and the Bernstein basis representation theory.Apart from the construction of the transition function,this formula has a simple form and has the potential to be computationally efficient.Similar to the pyramid algorithm for B-splines,the entire recursive process of computing the changeable-degree Bspline basis using the second generalized de Boor-Cox formula can be described using a pyramid cluster of different heights,with additional transition function constructions.