Rips Complexes of Planar Point Sets
| dc.creator | Chambers, Erin W. | |
| dc.creator | de Silva, Vin | |
| dc.creator | Erickson, Jeff | |
| dc.creator | Ghrist, Robert | |
| dc.date | 2007-12-03 | |
| dc.date.accessioned | 2026-07-07T08:47:07Z | |
| dc.date.available | 2026-07-07T08:47:07Z | |
| dc.description | Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The 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 Rips complex to $\euc^n$ 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 Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a 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'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-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. | |
| dc.description | 16 pages, 8 figures | |
| dc.identifier | https://arxiv.org/abs/0712.0395 | |
| dc.identifier | http://arxiv.org/abs/0712.0395 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143489 | |
| dc.subject | Geometric Topology | |
| dc.subject | Algebraic Topology | |
| dc.title | Rips Complexes of Planar Point Sets | |
| dc.type | text |