Nonlinear Weighted Subdivision Schemes

In this paper, we present new variants of both the de Casteljau subdivision algorithm for curves and Doo-Sabin subdivision algorithm for surfaces. Our subdivision schemes are built on nonlinear weighted averaging rules which are induced by monotonic functions. These averaging rules are used instead of midpoint averaging rule in the mentioned well-known subdivision algorithms. The analysis shows that the smoothness of the subdivision schemes for curves is inherited from the smoothness of the function which induces the averaging rule used in the refinement of the schemes. The results show that with our subdivision schemes, both convex surfaces and concave surfaces can be generated by the same scheme. This happens by only interchanging the weights of two adjacent points when computing the edge points in the subdivision refinement. This is an advantage since a designer can adjust the limit shape according to his interests.