Recursive representation and application of transformation matrices of B-spline bases

With a transformation matrix of B-spline bases (abbreviated to BSBT matrix), a B-spline basis can be represented by another B-spline basis. In this paper, we first give the existence conditions and some useful properties of BSBT matrices. Then we propose a recursive formula for BSBT matrices and an efficient method for the computation of BSBT matrices. Based on these results, we further probe into the applications of BSBT matrices in knot insertion and degree elevation of B-spline curves. Using BSBT matrices, we develop a new uniform algorithm for knot insertion and degree elevation of B-spline curves. This algorithm is efficient, general-purpose, and simple to implement. It can be used to insert one knot or multiple knots, raise one degree or multiple degrees, or to insert knot and raise degree simultaneously. Several examples are given to illustrate the stability of the algorithm. The results in the paper show that BSBT matrices can be applied to establish a uniform mathematical model for knot insertion, degree elevation, knot removal and degree reduction of B-spline curves and surfaces, and to provide a general tool for the conversions between different representations of B-spline curves and surfaces.