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Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as apoint-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex isa combinatorial simplicial complex based on proximity of neighbors that servesas an easily-computed but high-dimensional approximation to the homotopy typeof $D$. 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 tothe 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 capturesconnectivity and fundamental group data. This implies that the fundamentalgroup 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 proximitydetection leads to `quasi'-Rips complexes with nearly arbitrary fundamentalgroups. This topological noise can be mitigated by examining a pair ofquasi-Rips complexes and using ideas from persistent topology. Finally, we showthat the projection map does not preserve higher-order topological data forplanar sets, nor does it preserve fundamental group data for point sets indimension larger than three.

Rips Complexes of Planar Point Sets