Vietoris-rips complexes of planar point sets

Erin W. Chambers, Vin de Silva, Jeff Erickson, Robert Ghrist

Research output: Contribution to journalArticlepeer-review


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 Vietoris-Rips 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"-Vietoris-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Vietoris-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.

Original languageEnglish (US)
Pages (from-to)75-90
Number of pages16
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Jul 2010


  • Quasi-Rips complex
  • Rips complex
  • Topology

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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