Uniform Spline Approach Using Symmetric B-Spline Basis on Closed Periodic Zone

To overcome the weakness of using complicated iterative algorithms in the existing methods of solving for control points of uniform B-spline curve,this paper presents fast parallel algorithms to compute control points of uniform B-spline curves.First we shift naive B-spline basis and establish symmetric B-spline basis(parametric variable is defined on unit interval).Next we use orthogonal properties of complex functions ξ_k(v)=e^ikv to establish orthogonal B-spline basis on closed periodic zone,and derive explicit parallel computing formulas for coefficients of orthogonal B-spline basis.We further use relations among coefficients of orthogonal B-spline basis and coefficients of symmetric B-spline basis(control points of B-spline curve) to achieve explicit parallel computing formulas for control points of B-spline curve.At last,fourth order and third order B-spline approaches are taken as examples to analyze the proposed fast algorithms of the parallel computing formulas.Two examples of constructing B-spline curves from closed and non-closed given points are provided to verify the presented method.Experimental results show that,the method presented in this paper adds symmetric property to simple B-spline basis.It can easily carry out parallel computation for control points of B-spline curves and can improve efficiency of building large-scale B-spline curves.