The Research And Application On The Pyramid Algorithm Of Bivariate B-spline Basis Function

The pyramid algorithm is proposed by American mathematician Ron Goldman.It is a kind of dynamic programming algorithm.Because of its pyramid shape,the structure is clear and simple,widely used in polynomial interpolation and approximation.In complex free-form surface modeling,it is inevitable to calculate the tangent,curvature variables,which can be transformed into the basis function and derivative function.Therefore,it is an important task to study the fast and general solution of spline basis functions.Based on the traditional spline interpolation approximation theory,the application of pyramid algorithm in the construction of basis functions is studied systematically.Firstly,we construct the algorithm pyramid by linear interpolation based on Lagrange,Newton and Hermite interpolation functions.The Neville and Aitken algorithms are combined to analyze the properties of the affine combination coefficients of the path landmarks,and the exchangeability of the node subscripts is obtained.The smoothness of spline curve is analyzed by numerical example,and the superiority of the algorithm is verified.Secondly,according to the Blossom theory,the algorithm pyramid of the B-spline basis function is deduced by using the symmetry and multi-affinities to get the new value.Based on the symmetrical parallelism,the computational complexity is reduced by inverting the pyramid,and a downward recursive algorithm is obtained.The adjacent paths are overlapped,and a simple proof of the smoothness is achieved.Furthermore,the algorithm is bilinearly interpolated in x and y directions to obtain the rectangular tensor product basis function.The nodes are defined on triangular meshes and extended to the local triangular B-spline surfaces by using the affine invariance of the center of gravity coordinates.Based on the research of the basis function construction theory,this paper provides a new idea and method for the complex curved surface modeling.Especially for the problem of geometric variables needing to be solved in engineering design,the pyramid algorithm based on dynamic programming has important practical value.