Is there any difference between closed and open NU

Question

I am wondering if there is any difference between closed and open nurbs on control and knots.

I checked many documents, but cannot find the exact answer, so asked here.

Answer 1

There is really no difference in terms of control points and knot sequences for open or closed nurbs curves. They both follow the rule "number_of_knots = number_of_control_points + order". However, closed nurbs curves (or periodic nurbs curves) are achieved by having some special arrangement for the control points and knot sequence. For example, the knot sequence [ 0., 0., 0., 0., 0.25, 0.5, 0.75, 1.0, 1.0, 1.0, 1.0] with 7 control points (P1, P2,...to P7) will result in an open cubic B-spline curve. If we change the knot sequence into [ -0.75, -0.5, -0.25, 0., 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75] and make P5=P1, P6=P2 and P7=P3, then the curve will become a periodic cubic B-spline curve with C2 continuity at the joint.