On the Locality of Extracting a 2-Manifold in {IR}3

Algorithms for reconstructing a 2-manifold from a point sample in R^3 based on Voronoi-filtering like CRUST or CoCone still require -- after identifying a set of candidate triangles -- a so-called manifold extraction step which identifies a subset of the candidate triangles to form the final reconstruction surface. Non-locality of the latter step is caused by so-called slivers -- configurations of four almost cocircular points having an empty circumsphere with center close to the manifold surface. We prove that under a certain mild condition -- local uniformity -- which typically holds in practice but can also be enforced theoretically, one can compute a reconstruction using an algorithm whose decisions about the adjacencies of a point only depend on nearby points. While the theoretical proof requires an extremely high sampling density, our prototype implementation, described in a companion paper, performs well on typical sample sets. Due to its local mode of computation, it might be particularly suited for parallel computing or external memory scenarios