Vietoris-Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean n-space En , thought of as a point-cloud sampling of a certain domain D En . The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural "shadow" projection map from the Vietoris-Rips complex to En that has as its image a more accurate n-dimensional approximation to the homotopy type of D. We demonstrate that this projection map is 1-connected for the planar case n = 2. That is, for planar domains, the Vietoris-Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Vietoris-Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-VietorisRips complexes with nearly arbitrary fundamental groups. This topological no...

• Type: Journal
• Year: 2010
• Where: DCG
• Authors: Erin W. Chambers, Vin de Silva, Jeff Erickson, Robert Ghrist
• Keywords: Computer Graphics, DCG 2010, Fundamental Group, Projection Map, Vietoris-Rips Complex