C2 subdivision over triangulations with one extraordinary point

This paper presents a new subdivision scheme that operates over an infinite triangulation, which is regular except for a single extraordinary vertex. The scheme is based on the quartic three-directional Box-spline scheme, and is guaranteed to generate C2 limit functions whenever the valency n of the extraordinary vertex is in the range 4 ≤ n ≤ 20. The new scheme differs from the commonly used subdivision schemes by the fact that it applies special subdivision rules near edges of the original triangulation, which emanate from the extraordinary vertex, and not only in the vicinity of the extraordinary vertex.