Recursive geometry of the flow complex and topology of the flow complex filtration

The flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in Rk. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have found applications in surface reconstruction, shape matching, and molecular modeling. In this article we give an algorithm for computing the flow complex of weighted points in any dimension. The algorithm reflects the recursive structure of the flow complex. On the basis of the algorithm we establish a topological similarity between flow shapes and the nerve of a corresponding ball set, namely homotopy equivalence. ? Part of this research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program ’Combinatorics, Geometry, and Computation’ (No. GRK 588/2). Also partially supported by the IST Programme of the EU as a Shared-cost RTD(FET Open) Project under Contract No IST-006413 (ACS Algorithms for Complex Shapes). Also supported by NSF CARGO grant DMS0310642. Preprint submitted to Elsevier Science 15 May 2007