Efficient Algorithms for the Implementation of General B-Splines

Nonuniform B-splines are usually computed using the traditional recurrence relation Bi,r(u) = ui − uui+r−1−uiBi,r−1(u) + ui+r − uui+r − ui+1Bi+1,r−1(u).We derive a recurrence relation which relates the rth derivative of Bi,r(ū) to the (r − 1)th derivatives of Bi,r−1(u) and Bi + 1, r − 1u[formula]B(r)i, r(u) is comprised of r + 1 impulses (Dirac functions) at the knots [ūi, ūi + 1, . . . , ūi + r]. The amplitudes of the impulses are found from the recurrence. We show that equally spaced samples of the continuous B-spline function Bi, r(ū) can be computed exactly using recursive summation.