An Extension of Chaiken's Algorithm to B-Spline Curves with Knots in Geometric Progression

Chaiken's algorithm is a procedure for inserting new knots into uniform quadratic Bspline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken's algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken's algorithm to B-spline curves of arbitrary degree for knot sequences in geometric and aÆne progression. 1 Subdivision for knots in arithmetic progression Let N k (t) be the B-spline basis function of degree n+ 1 whose support lies over the knot sequence t2k; t2k+2; :::; t2k+2n+4 and let N̂ n+1 k (t) be the B-spline basis function of degree n+1 whose support lies over the re ned knot sequence tk; tk+1; :::; tk+n+2. Since the B-splines form a basis, there exist constants n+1 j such that N k (t) = n+2 X j=0 n+1 j N̂ n+1 j+2k(t): (1) For the degree zero basis functions, the 0s satisfy