General matrix representations for B-splines

Abstract

In this paper, the concept of basis matrix of B-splines is presented. A general matrix representation, which results in an explicitly recursive matrix formula, for nonuniform B-spline curves of an arbitrary degree is also presented by means of Toeplitz matrix. New recursive matrix representations for uniform B-spline curves and Bezier ones of an arbitrary degree are obtained as special cases of that for nonuniform B-spline curves. The recursive formula for basis matrix can be substituted for de Boor-Cox's one for B-splines, and it has better time complexity than de Boor-Cox's formula when used for conversion and computation of B-spline curves and surfaces between different CAD systems. Finally, some applications of the matrix representations are given in the paper.