Distributed Shape Analysis using Alpha and Delaunay-Čech shapes

Alpha shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of a given point set V . For a given point set V in 2 or 3 dimensions, we show that the knowledge of pairwise distances between pairs of points which are closer than a certain distance is sufficient to compute the alpha shape. We also show that the alpha shapes can be computed distributively using only local-neighborhood information. Such distributed algorithms are very useful in domains such as sensor networks, where each point represents a sensing node, the location of which is not necessarily known. It is also known that alpha shapes are boundaries of alpha complexes. We introduce a new geometric object called the Delaunay-Čech complex, which is geometrically more appropriate than an alpha complex for some cases, and show that it is of the same homotopy type as the alpha complex in all dimensions. We further derive an explicit homotopy equivalence for the case of 2 dimensions.