DIRECTIONS:   In this applet, you create a cubic B-Spline by specifying control points p_i with i=0,1,2, ..., n and a knot-sequence u_j with j=0,1,2,...,m. The number of knots is related to the number of control points by m=n+4. The curve c(t) is defined on the interval u₃ < t < u_m-3, and given by

c(t) = Σ N_i³ p_i

With the basis functions defined iteratively by the relation

N_i^k+1(t) = ((t-u_i)/(u_i+k - u_i)) N_i^k(t) + ((u_i+k+1-t)/(u_i+k+1-u_i+1)) N_i+1^k(t)

with N_i⁰ = 1 when u_i < t < u_i+1 and 0 otherwise.

B-Splines are a generalization of Bezier curves and have many of the same properties but the knot sequence allows one to shape the curve and interpolation properties of the curve at specific points. The applet original chooses after every point to reset t the knots to be equally spaced apart on the interval 0 < t < 1.