The flow complex: A data structure for geometric modeling

Authors:
Joachim Giesen;Matthias John
Affiliations:
Max-Planck-Institut für Informatik, Department 1: Algorithms and Complexity, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany;Institut für Theoretische Informatik, ETH Zurich, Universitätstraße 6, 8092 Zürich, Switzerland
Venue:
Computational Geometry: Theory and Applications
Year:
2008

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Abstract

We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in R^3. The fixed points of this flow are exactly the critical points of the distance function. Our main result is a mathematical characterization and algorithms to compute the stable manifolds, i.e., the inflow regions, of the fixed points. It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set. We call the latter complex the flow complex of the point set. The flow complex is suited for geometric modeling tasks like surface reconstruction.