4-8 Subdivision

In this paper we introduce 4-8 subdivision, a new scheme that generalizes the four-directional box spline of class C⁴ to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement automatically generates conforming variable-resolution meshes in contrast to face and vertex split methods which require a postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4-8 scheme are C⁴ continuous almost everywhere, except at extraordinary vertices where they are is C¹-continuous.