A corner-cutting scheme for hexagonal subdivision surfaces

In their paper about how the duality between subdivision surface schemes leads to higher-degree continuity, Zorin and Schroder (2001) consider only quadrilateral subdivision schemes. The dual of a quadrilateral scheme is again a quadrilateral scheme, while the dual of a triangular scheme is a hexagonal scheme. In this paper we propose such a hexagonal scheme, which can be considered a dual to Kobbelt's (2000) Sqrt(3) scheme for triangular meshes. We introduce recursive subdivision rules for meshes with arbitrary topology, optimizing the surface continuity given a minimal support area. These rules have a simplicity comparable to the Doo-Sabin scheme: only new vertices of one type are introduced and every subdivision step removes the vertices of the previous steps. As hexagonal meshes are not encountered-frequently in practice, we describe two different techniques to convert triangular meshes into hexagonal ones.