On knot sequences of the B-splines

Abstract

First we study B-splines and their knot sequences and then a stable evaluation of B$_{i,k,t}$ is introduced. The knots of spline may be specified at arbitrary positions, as long as the Schoenberg and Whitney condition relating to the exist satisfied. We study the choice of knot sequence for the B-spline approximations of curves. Our experience with many examples shows that (1) The locations of knots outside interval of approximation do not have a visible affect in the approximations, (2) The locations of knots inside the interval of approximation should be placed evenly in each interval formed by the points of data and (3) To get smoother approximation, if necessary, we should increase the order of spline. This can be done by moving some knots inside of the interval of approximation to the end points of the interval.