Knotenvectors

Knotenvectors are nondecreasing sequences of parameter values used in B-splines and NURBS to define basis functions and parameterization. For a degree p and n+1 control points P0 through Pn, a knot vector U = {u0, u1, ..., u_{n+p+1}} determines the support and spacing of the B-spline basis functions N_i,p(u). Each basis function is defined by the Cox–de Boor recursion, and the curve C(u) = sum_i N_i,p(u) P_i is defined over the parameter domain [u_p, u_{n+1}].

End conditions are often controlled by the knot vector. An open or clamped knot vector has the first and last knots repeated p+1 times, which makes the curve pass through the first and last control points. Knot vectors can be uniform, where interior knots are equally spaced, or nonuniform, where knot spacings vary to alter parameterization and local control.

Continuity at a knot depends on knot multiplicity. If a knot value has multiplicity m, the continuity of the basis functions at that knot is C^{p-m}. Thus increasing multiplicity reduces smoothness at that location; full C^p continuity occurs when all interior knots have multiplicity 1 (no repetition). A knot with multiplicity p+1 can produce a cusp or a breakpoint, and multiplicities greater than p sharply limit possible continuity.

Knotvectors are central to knot insertion, refinement, and degree elevation in CAD and computer graphics. They enable local control of shape without global recomputation and support the representation of complex geometries through NURBS surfaces and volumes. A simple example of an open cubic vector is {0, 0, 0, 0, 1, 1, 1, 1}, which corresponds to a cubic Bezier curve with four control points P0..P3.