Vietoris–Rips Complexes of Planar Point Sets

Abstract

Fix a finite set of points in Euclidean n-space \(\mathbb{E}^{n}\) , thought of as a point-cloud sampling of a certain domain \(D\subset\mathbb{E}^{n}\) . 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 \(\mathbb{E}^{n}\) that has as its image a more accurate n-dimensional approximation to the homotopy type of D.