Higher-degrees Hybrid Non-uniform Subdivision Surfaces

Non-Uniform Rational B-splines Surfaces can be defined for any degrees and non-uniform knots, but existing subdivision surfaces are either uniform or of a fixed degree. The only existing non-uniform arbitrary degree subdivision is the scheme in Cashman et al. (2009). However, in order to improve the surface quality, the knot insertion strategy in Cashman et al. (2009) has the problem that the limit surface does not change continuously in terms of the perturbation of knot intervals. This paper solves this problem by introducing higher-degree hybrid non-uniform subdivision surfaces (HNUSS), where the first level refinement converts each valence $n$ extraordinary point (EP) into a valence $n$ face (Li et al., 2019). And then, the subdivision scheme can be defined with one step of splitting and several steps of averaging, where most rules are tensor-product of the arbitrary degree B-spline refinement rule with one double knot. We verify that higher-degree HNUSS limit surface is $G^1$ at the EPs if the knot intervals for the spoke edges of an EP are the same and has a higher order of continuity in other regions. In the absence of multiple knots at EPs, we provide a knot insertion strategy to create a uniform region around an EP. Additionally, numerical experiments show that the limit surface has satisfactory shape quality.