Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local properties of B-spline curve and the control flexibility of its shape can be further improved, also the segmentation of the curve can be realized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties.This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized Polya polynomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear combination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.