Eigenanalysis and continuity of non-uniform Doo-Sabin surfaces

In computer graphics and computer-aided geometric design, more and more subdivision schemes are being extensively used for free-form surfaces of arbitrary topology. The convergence and continuity analyses of uniform subdivision surfaces have been performed very well, but it is very difficult to prove the convergence and the continuity properties of non-uniform recursive subdivision surfaces (NURSSes, for short) because the subdivision matrix varies at each iteration step. This restricts widespread use of NURSSes, although NURSSes have a lot of advantages over uniform subdivision surfaces. In this paper, the concept of equivalent knot spacing is presented. A new technique for eigenanalysis, convergence and continuity analyses of non-uniform Doo-Sabin surfaces is proposed. Also, an interesting and important fact is found that the subdivision process of nonuniform Doo-Sabin surfaces may diverge sometime