On extraordinary rules of quad-based interpolatory subdivision schemes

This article deals with interpolatory subdivision schemes generalizing the tensor-product version of the Dubuc-Deslauriers 4-point scheme to quadrilateral meshes of arbitrary manifold topology. In particular, we focus our attention on an extension of the ^C1 regular stencils that respectively exploits (N+2)-point and (2N+8)-point stencils for the computation of an edge-point and a face-point in the vicinity of an extraordinary vertex of valence N≠4 not lying on a boundary. The aim of our work consists in identifying which constraints are required to be respected by the weights appearing in the above stencils in order to get closed limit surfaces that are ^C1-continuous at extraordinary points, have both principal curvatures bounded and at least one of them nonzero. The obtained constraints are used to easily check these features in the limit surfaces resulting from the application of special extraordinary rules proposed in the literature by different authors. Moreover, the conditions derived on the stencil weights are exploited to design new extraordinary rules that can produce closed limit surfaces of the same quality as the existing proposals, but at a reduced computational cost.