Continuity of non-uniform recursive subdivision surfaces

Since Doo-Sabin and Catmull-Clark surfaces were proposed in 1978, eigenstru cture, convergence and continuity analyses of stationary subdivision have b een performed very well, but it has been very difficult to prove the conver gence and continuity of non-uniform recursive subdivision surfaces (NURSSes , for short) of arbitrary topology. In fact, so far a problem whether or no t there exists the limit surface as well as G(1) continuity of a non-unifor m Catmull-Clark subdivision has not been solved yet. Here the concept of eq uivalent knot spacing is introduced. A new technique for eigenanalysis, con vergence and continuity analyses of non-uniform Catmull-Clark surfaces is p roposed such that the convergence and G(1) continuity of NURSSes at extraor dinary points are proved. In addition, slightly improved rules for NURSSes are developed. This offers us one more alternative for modeling free-form s urfaces of arbitrary topologies with geometric features such as cusps, shar p edges, creases and darts, while elsewhere maintaining the same order of c ontinuity as B-spline surfaces.