Before revision: A Vietoris--Rips complex is a way to thicken a (possibly discrete) metric space into a larger space containing more topological information. We prove that if C is a convex closed differentiable curve in the plane such that the convex hull of C contains the evolute of C, then the homotopy type of a Vietoris--Rips complex of a subset of n points from C can be computed in time O(n  \log n) .
After revision: We give an O(n \log n) .