In the uniform B-spline case a subdivided component shares many of its individual control points with other components, allowing us to define the totality of control points generated through subdivision as a refinement of the original control polygon. These new control points can be assembled into a new control polygon and refined again by the same methods.
Successive refinements produce a sequence of control polygons that in the limit converge to a curve.
The uniform B-spline curves, surfaces and solids have been extensively studied in the literature and subdivision methods for these objects are well known. We develop here the refinement method for a cubic uniform B-spline curve. The analysis is similar to that presented in the quadratic case however, the refinement algorithm can be specified in a different manner which eventually allows us to use eigenanalysis and directly calculate points on the curve . For a pdf version of these notes look here . Such a curve is shown in the following illustration.