Knot Insertion and Reparametrization of Interval B-spline Curves

Knot insertion is the operation of obtaining a new representation of a B-spline curve by introducing additional knot values to the defining knot vector. The new curve has control points consisting of the original control points and additional new control points corresponding to the number of new knot values. So knot insertions give additional control points which provide extra shape control without necessarily subdividing the curve. However, if following a knot insertion operation a knot has multiplicity equal to the degree, then the B-spline is split into two B-splines at that knot value. In this paper the concept of knot insertion for analyzing interval Bspline curve has been introduced. The four fixed Kharitonov's polynomials (four fixed B-spline curves) associated with the original interval B-spline curve are obtained. The four fixed Kharitonov's polynomials (four fixed B-spline curves) are subdivided by inserting additional knot values while maintaining an open uniform knot vector. Finally, the required interval control points are obtained from the fixed control points of the four fixed subdivided Kharitonov's polynomials. The problem of parametric interval B-spline curve reparametrization is also discussed. The shape of the curve remains unchanged during the process of reparametrization; only the way the curve is described is altered. If it is important that the degree of the given curve should be kept unchanged, we may choose a linear reparametrization function. Numerical examples are included in order to demonstrate the effectiveness of the proposed method. Index Term— Knot insertion, reparametrization, interval B-spline curve, image processing, CAGD.